Felix' Math Place (Posts about giant steps.)
https://math.fontein.de/tag/giant-steps.atom
2019-11-17T10:38:24Z
Felix Fontein
Nikola
Obtaining Infrastructures from Global Fields.
https://math.fontein.de/2009/07/21/obtaining-infrastructures-from-global-fields/
2009-07-21T09:39:48+02:00
2009-07-21T09:39:48+02:00
Felix Fontein
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<div class="subsection-block">
<h4>
Basics on Global Fields.
</h4>
<div>
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> be a <a href="https://en.wikipedia.org/wiki/Global_field">global field</a>, i.e. an algebraic number field or an algebraic function field with a finite constant field. In the first case, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="18" height="12" src="https://math.fontein.de/formulae/KNpOu8H6kUesAJNA335iDhv878WaoMOAl2.1hw.svgz" alt="k^*" title="k^*"></span> be the roots of unity and <span class="inline-formula"><img class="img-inline-formula img-formula" width="98" height="18" src="https://math.fontein.de/formulae/07Ikr5NtaU8mO3NdHkSF8gmNV1oLtwlxex14ng.svgz" alt="k = k^* \cup \{ 0 \}" title="k = k^* \cup \{ 0 \}"></span>. In the latter case, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="10" height="12" src="https://math.fontein.de/formulae/KIbZSMqheEMj2LBRLlu5emEXwvNaAcODJL5YWQ.svgz" alt="k" title="k"></span> be the exact field of constants.
</p>
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="146" height="18" src="https://math.fontein.de/formulae/d5qdl3Sm8M1V8cakG77iFjLyGIQAEGWhwx17_w.svgz" alt="S = \{ \frakp_1, \dots, \frakp_{n+1} \}" title="S = \{ \frakp_1, \dots, \frakp_{n+1} \}"></span> be the set of infinite places of <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>. If <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> is a number field, the elements of <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/pYWI0gROGxbmepTlPB65gHsm8w4iqZkEdmltJA.svgz" alt="S" title="S"></span> correspond to embeddings of <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> into <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/sznF0_DZfMvBmQiFOcFDqhQZwWHPEvTQYYjJZA.svgz" alt="\C" title="\C"></span> up to complex conjugation. Define <span class="inline-formula"><img class="img-inline-formula img-formula" width="87" height="18" src="https://math.fontein.de/formulae/Nei6CxLwQfowjNsccSIrldsBytPFxLGV7ecJdg.svgz" alt="q := \exp(1)" title="q := \exp(1)"></span>, and for <span class="inline-formula"><img class="img-inline-formula img-formula" width="42" height="15" src="https://math.fontein.de/formulae/YkrPToBqCrBzPlTwf6DsfmrX6y0gpY75eFGZCQ.svgz" alt="\frakp \in S" title="\frakp \in S"></span> let <span class="inline-formula"><img class="img-inline-formula img-formula" width="82" height="12" src="https://math.fontein.de/formulae/9nrXuiBK3LxvKmC8rrfV9KIqllv9saMXmebrYQ.svgz" alt="\sigma : K \to \C" title="\sigma : K \to \C"></span> be a corresponding embedding. Then define <span class="inline-formula"><img class="img-inline-formula img-formula" width="157" height="18" src="https://math.fontein.de/formulae/lF2302034R.Va77PvIioVmnFloaZvzwKfpZc1Q.svgz" alt="\nu_\frakp(f) := -\log \abs{\sigma(f)}" title="\nu_\frakp(f) := -\log \abs{\sigma(f)}"></span> for <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="16" src="https://math.fontein.de/formulae/J4C0dkT_SToe7mWds1p3hHRPKdN2gAL4MA1yiA.svgz" alt="f \in K^*" title="f \in K^*"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="76" height="16" src="https://math.fontein.de/formulae/08tqDXh41TEOULMJuskWqmbZ4KK9vsqzBwu4BQ.svgz" alt="\deg \frakp := 1" title="\deg \frakp := 1"></span> if <span class="inline-formula"><img class="img-inline-formula img-formula" width="77" height="18" src="https://math.fontein.de/formulae/0AFlUhBZu0V_HsWOpqQZ.HcXf4i4S_hBcPP2.Q.svgz" alt="\sigma(K) \subseteq \R" title="\sigma(K) \subseteq \R"></span>, or <span class="inline-formula"><img class="img-inline-formula img-formula" width="76" height="16" src="https://math.fontein.de/formulae/Dcq.9WKok.HkECTlH8mn7VEiouptxU73RKxIlg.svgz" alt="\deg \frakp := 2" title="\deg \frakp := 2"></span> otherwise, and define <span class="inline-formula"><img class="img-inline-formula img-formula" width="63" height="17" src="https://math.fontein.de/formulae/EzQodWQ8.FkO0O.m8GRGK8045ZAR2jvBV8MFGQ.svgz" alt="\G_\frakp := \R" title="\G_\frakp := \R"></span>. If <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> is a function field, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="18" src="https://math.fontein.de/formulae/pRbayn42OvuagwrBaDBhU5fJIR1DslKXeDGKVw.svgz" alt="q := \abs{k}" title="q := \abs{k}"></span>, i.e. <span class="inline-formula"><img class="img-inline-formula img-formula" width="52" height="17" src="https://math.fontein.de/formulae/u91eZyD3hD39_x29yjnmz6tmplnMfPGLN.fhGA.svgz" alt="k = \F_q" title="k = \F_q"></span>; in this case, there exists an element <span class="inline-formula"><img class="img-inline-formula img-formula" width="75" height="18" src="https://math.fontein.de/formulae/.Bt5k1JyqlSa7pyS3sUAqzYGbf3NtUt3fFyhAQ.svgz" alt="x \in K \setminus k" title="x \in K \setminus k"></span> whose poles are exactly the elements of <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/pYWI0gROGxbmepTlPB65gHsm8w4iqZkEdmltJA.svgz" alt="S" title="S"></span>, i.e. are the places of <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> lying above the infinite place of <span class="inline-formula"><img class="img-inline-formula img-formula" width="34" height="18" src="https://math.fontein.de/formulae/HcPS0FWGeKHvrbI26_.0s0MFO_XNC5bemm_8AQ.svgz" alt="k(x)" title="k(x)"></span>. In all cases, <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/pYWI0gROGxbmepTlPB65gHsm8w4iqZkEdmltJA.svgz" alt="S" title="S"></span> is finite and non-empty.
</p>
<p>
For a non-archimedean place <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="13" src="https://math.fontein.de/formulae/fIYgEXdYw02G_7eV0YbxD5xCWpjHSHQUgFLXgw.svgz" alt="\frakp" title="\frakp"></span> of <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="22" height="17" src="https://math.fontein.de/formulae/nJgkqvvky2FRSO_fioYP1cyNWvi9rgZV7TwabQ.svgz" alt="\calO_\frakp" title="\calO_\frakp"></span> be the valuation ring and <span class="inline-formula"><img class="img-inline-formula img-formula" width="22" height="13" src="https://math.fontein.de/formulae/Re3ZJrUdmyjZj47QC84InAOahKlPrD1b3OgfTQ.svgz" alt="\frakm_\frakp" title="\frakm_\frakp"></span> its maximal idea, and denote the discrete valuation by <span class="inline-formula"><img class="img-inline-formula img-formula" width="17" height="13" src="https://math.fontein.de/formulae/BppuLEX0U7plziRQCge7bi3Cp57TujwxK3fZxw.svgz" alt="\nu_\frakp" title="\nu_\frakp"></span>. Then set <span class="inline-formula"><img class="img-inline-formula img-formula" width="164" height="20" src="https://math.fontein.de/formulae/qp1aJc2JbpVhPGkiEBcTFkAnFHdxHFPVVl7U1Q.svgz" alt="\deg \frakp := \log_q \abs{\calO_\frakp / \frakm_\frakp}" title="\deg \frakp := \log_q \abs{\calO_\frakp / \frakm_\frakp}"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="146" height="23" src="https://math.fontein.de/formulae/13DjGje5fhG8A18uluBLMuSi2wZDbGdbnxRdPQ.svgz" alt="\abs{f}_\frakp := q^{-\nu_\frakp(f) \deg \frakp}" title="\abs{f}_\frakp := q^{-\nu_\frakp(f) \deg \frakp}"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="49" height="16" src="https://math.fontein.de/formulae/i_f1WCD_GHQZkGm_PM.SUufJ3kt7TfV2UAJigg.svgz" alt="f \in K" title="f \in K"></span>. Define <span class="inline-formula"><img class="img-inline-formula img-formula" width="62" height="17" src="https://math.fontein.de/formulae/.IO7S668pXeUFuHQJn5PV_IgSFXq9LW_xKrSIw.svgz" alt="\G_\frakp := \Z" title="\G_\frakp := \Z"></span>. In the number field case, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="55" height="12" src="https://math.fontein.de/formulae/CW1fFzQ1XrhlRM71thQfza6e2s9bXcC7p9ez_g.svgz" alt="\G := \R" title="\G := \R"></span>, and otherwise <span class="inline-formula"><img class="img-inline-formula img-formula" width="54" height="12" src="https://math.fontein.de/formulae/Jj5SFC.0ZDpQ44GBCGhFMZT5.avVPCIqJqt_0Q.svgz" alt="\G := \Z" title="\G := \Z"></span>.
</p>
<p>
Denote the set of places of <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> by <span class="inline-formula"><img class="img-inline-formula img-formula" width="26" height="15" src="https://math.fontein.de/formulae/e.p88GcUJmJiMMXsYG4mCg0GROWEY4yzlwD6Sg.svgz" alt="\calP_K" title="\calP_K"></span>. The divisor group of <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> is <span class="inline-formula"><img class="img-inline-formula img-formula" width="157" height="21" src="https://math.fontein.de/formulae/vVelRFQZdD9w4y09QHSRSn_VMKR2vnI_dIr59A.svgz" alt="\Div(K) := \coprod_{\frakp \in \calP} \G_\frakp" title="\Div(K) := \coprod_{\frakp \in \calP} \G_\frakp"></span>, and for <span class="inline-formula"><img class="img-inline-formula img-formula" width="128" height="21" src="https://math.fontein.de/formulae/4_F_n1eR6OohhxJhtakl16H7sJYoerp0Ln_ZvQ.svgz" alt="D = \sum_{\frakp \in \calP_K} n_\frakp \frakp" title="D = \sum_{\frakp \in \calP_K} n_\frakp \frakp"></span> define <span class="inline-formula"><img class="img-inline-formula img-formula" width="195" height="21" src="https://math.fontein.de/formulae/y2cb2e3Kbu.HaEnG0If6HOsWL1Ep0rKGTr8ofA.svgz" alt="\deg D := \sum_{\frakp \in \calP_K} n_\frakp \deg \frakp" title="\deg D := \sum_{\frakp \in \calP_K} n_\frakp \deg \frakp"></span>. This is a homomorphism <span class="inline-formula"><img class="img-inline-formula img-formula" width="141" height="18" src="https://math.fontein.de/formulae/ZNBOoX44UufOfs8BcAzyP8j3SB0tm85TT5fdlQ.svgz" alt="\deg : \Div(K) \to \G" title="\deg : \Div(K) \to \G"></span>; denote its kernel by <span class="inline-formula"><img class="img-inline-formula img-formula" width="66" height="20" src="https://math.fontein.de/formulae/XoXAz6cd8x7maPY1Ss34w8M_r5jXmoqWqLAblg.svgz" alt="\Div^0(K)" title="\Div^0(K)"></span>. For <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="16" src="https://math.fontein.de/formulae/J4C0dkT_SToe7mWds1p3hHRPKdN2gAL4MA1yiA.svgz" alt="f \in K^*" title="f \in K^*"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="252" height="23" src="https://math.fontein.de/formulae/hvkieiZF1BmAn5UhYk6fRWl5FYl3QqlkLriu0g.svgz" alt="(f) := \sum_{\frakp \in \calP_K} \nu_\frakp(f) \frakp \in \Div^0(K)" title="(f) := \sum_{\frakp \in \calP_K} \nu_\frakp(f) \frakp \in \Div^0(K)"></span> is a principal divisor; let the group of all these be denoted by <span class="inline-formula"><img class="img-inline-formula img-formula" width="72" height="18" src="https://math.fontein.de/formulae/ytrEYWa.zKRwtySg6lrqk5KN8vis9Mlz_7V9TQ.svgz" alt="\Princ(K)" title="\Princ(K)"></span>. Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="225" height="18" src="https://math.fontein.de/formulae/XTP3wlQDk3JnZxIMPC7lfEpTt4zRGh4Rx48EzQ.svgz" alt="\Pic(K) := \Div(K) / \Princ(K)" title="\Pic(K) := \Div(K) / \Princ(K)"></span> is the divisor class group of <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="241" height="20" src="https://math.fontein.de/formulae/P9O781MxcSsyF7RNB5E8OIgzwo8O34Ml85m27g.svgz" alt="\Pic^0(K) := \Div^0(K) / \Princ(K)" title="\Pic^0(K) := \Div^0(K) / \Princ(K)"></span> its degree zero part.
</p>
<p>
The support of a divisor <span class="inline-formula"><img class="img-inline-formula img-formula" width="128" height="21" src="https://math.fontein.de/formulae/4_F_n1eR6OohhxJhtakl16H7sJYoerp0Ln_ZvQ.svgz" alt="D = \sum_{\frakp \in \calP_K} n_\frakp \frakp" title="D = \sum_{\frakp \in \calP_K} n_\frakp \frakp"></span> is the set <span class="inline-formula"><img class="img-inline-formula img-formula" width="229" height="18" src="https://math.fontein.de/formulae/XsPqLc_WPkF60vLy.hs7HDNQOWMIZ.lc6ZqFHQ.svgz" alt="\support(D) = \{ \frakp \in \calP_K \mid n_\frakp \neq 0 \}" title="\support(D) = \{ \frakp \in \calP_K \mid n_\frakp \neq 0 \}"></span>. Consider the subgroups
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="425" height="44" src="https://math.fontein.de/formulae/TAGhIh6bKFVw2jVUTu438GRr1_uWvvqEJtZYgg.svgz" alt="\Div_{fin}(K) :={} & \{ D \in \Div(K) \mid \support(D) \cap S = \emptyset \} \\
\text{and} \qquad \Div_\infty(K) :={} & \{ D \in \Div(K) \mid \support(D) \subseteq S \};" title="\Div_{fin}(K) :={} & \{ D \in \Div(K) \mid \support(D) \cap S = \emptyset \} \\
\text{and} \qquad \Div_\infty(K) :={} & \{ D \in \Div(K) \mid \support(D) \subseteq S \};">
</div>
<p>
then <span class="inline-formula"><img class="img-inline-formula img-formula" width="258" height="18" src="https://math.fontein.de/formulae/mIq23nOsWONU1MscdXsLhWBXKNyThCEkRY7G2Q.svgz" alt="\Div(K) = \Div_{fin}(K) \oplus \Div_\infty(K)" title="\Div(K) = \Div_{fin}(K) \oplus \Div_\infty(K)"></span>. Moreover, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="261" height="20" src="https://math.fontein.de/formulae/mJcezOooiecqkcWZYX4IIsl2hYEygrXkUAhsiQ.svgz" alt="\Div_\infty^0(K) := \Div^0(K) \cap \Div_\infty(K)" title="\Div_\infty^0(K) := \Div^0(K) \cap \Div_\infty(K)"></span>.
</p>
<p>
The set <span class="inline-formula"><img class="img-inline-formula img-formula" width="351" height="18" src="https://math.fontein.de/formulae/MwGMqaYusKIQ4BWG0TIXnrCuqyNgkd3M0njd7A.svgz" alt="\calO := \calO_S := \{ f \in K \mid \nu_\frakp(f) \ge 0 \text{ for all } \frakp \in S \}" title="\calO := \calO_S := \{ f \in K \mid \nu_\frakp(f) \ge 0 \text{ for all } \frakp \in S \}"></span> is a Dedekind domain, whose maixmal ideals correspond to the places in <span class="inline-formula"><img class="img-inline-formula img-formula" width="55" height="18" src="https://math.fontein.de/formulae/mw8PJEdSxZDMj6ufiUpYBuvB48RsqddAesS69A.svgz" alt="\calP_K \setminus S" title="\calP_K \setminus S"></span>. Moreover, the fractional ideal group <span class="inline-formula"><img class="img-inline-formula img-formula" width="54" height="18" src="https://math.fontein.de/formulae/SkHGpPqKCrTWYTV.CQ9pQV9oxps8aeWviBJAnQ.svgz" alt="\Id(\calO_S)" title="\Id(\calO_S)"></span> is isomorphic to <span class="inline-formula"><img class="img-inline-formula img-formula" width="81" height="18" src="https://math.fontein.de/formulae/1cDySuFZGcSGS_6hDSRsfa1yotMP1lXyKYtI_w.svgz" alt="\Div_{fin}(K)" title="\Div_{fin}(K)"></span> by <span class="inline-formula"><img class="img-inline-formula img-formula" width="147" height="21" src="https://math.fontein.de/formulae/ZqCcdAVsr29Uz2m2i.AaxNTHPgJ26cmIZf3S7Q.svgz" alt="\divisor(\fraka) = \sum_{\frakp \not\in S} n_\frakp \frakp" title="\divisor(\fraka) = \sum_{\frakp \not\in S} n_\frakp \frakp"></span>, in case <span class="inline-formula"><img class="img-inline-formula img-formula" width="185" height="22" src="https://math.fontein.de/formulae/onOMT41d3oSiJvOCbXw9tcNbxqK9ZRsey1bkRA.svgz" alt="\fraka = \prod_{\frakp \not\in S} (\frakm_\frakp \cap \calO_S)^{-n_\frakp}" title="\fraka = \prod_{\frakp \not\in S} (\frakm_\frakp \cap \calO_S)^{-n_\frakp}"></span>; the inverse is given by the restriction of <span class="inline-formula"><img class="img-inline-formula img-formula" width="191" height="18" src="https://math.fontein.de/formulae/6cu2Mh8LtHY.9X6Up.QZhkt2sErCTpIA6._A_g.svgz" alt="\ideal : \Div(K) \to \Id(\calO_S)" title="\ideal : \Div(K) \to \Id(\calO_S)"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="229" height="22" src="https://math.fontein.de/formulae/UeJUGD2VECP4ukattqYoUNLHSlS4k03gkRhvYg.svgz" alt="\sum n_\frakp \frakp \mapsto \prod_{\frakp \not\in S} (\frakm_\frakp \cap \calO_S)^{-n_\frakp}" title="\sum n_\frakp \frakp \mapsto \prod_{\frakp \not\in S} (\frakm_\frakp \cap \calO_S)^{-n_\frakp}"></span> to <span class="inline-formula"><img class="img-inline-formula img-formula" width="81" height="18" src="https://math.fontein.de/formulae/1cDySuFZGcSGS_6hDSRsfa1yotMP1lXyKYtI_w.svgz" alt="\Div_{fin}(K)" title="\Div_{fin}(K)"></span>. The group of fractional principal ideals <span class="inline-formula"><img class="img-inline-formula img-formula" width="66" height="18" src="https://math.fontein.de/formulae/9yj_2Z2CFmGZIU49BFy.cZGV0cQpXyN4asv4tw.svgz" alt="\PId(\calO_S)" title="\PId(\calO_S)"></span> equals <span class="inline-formula"><img class="img-inline-formula img-formula" width="122" height="18" src="https://math.fontein.de/formulae/CgawV3bzrb_Nt1hWj3HHvW4I_GOfc8kF2oA64Q.svgz" alt="\ideal(\Princ(K))" title="\ideal(\Princ(K))"></span>. The quotient <span class="inline-formula"><img class="img-inline-formula img-formula" width="133" height="18" src="https://math.fontein.de/formulae/wADVab9OCDH6.YLdPzRKrm45iP_I4raST2M9CQ.svgz" alt="\Id(\calO_S) / \PId(\calO_S)" title="\Id(\calO_S) / \PId(\calO_S)"></span> is the ideal class group <span class="inline-formula"><img class="img-inline-formula img-formula" width="63" height="18" src="https://math.fontein.de/formulae/ln_II2vRV4rmdwPdsY3ombMCrsPxSg2D_A68TQ.svgz" alt="\Pic(\calO_S)" title="\Pic(\calO_S)"></span> of <span class="inline-formula"><img class="img-inline-formula img-formula" width="24" height="15" src="https://math.fontein.de/formulae/Gd48AlL0RCLMaMySZsX3OXsmy8NI9JMWM5OAVA.svgz" alt="\calO_S" title="\calO_S"></span>. Putting all these things together, we get the following diagram with exact rows and columns:
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="435" height="370" src="https://math.fontein.de/formulae/qq2MRtJc.Wna.zusQQqB5eijJ.z7xJ.q9U9WQA.svgz" alt="\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] & \\ 0 \ar[r] & \calO_S^* / k^* \ar[r] \ar[d] & \Div^0_\infty(K) \ar[r] \ar[d] & T \ar[r] \ar[d] & 0 \\ 0 \ar[r] & K^* / k^* \ar[r] \ar[d] & \Div^0(K) \ar[r] \ar[d] & \Pic^0(K) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & K^* / \calO_S^* \ar[r] \ar[d] & \Id(\calO_S) \ar[r] \ar[d] & \Pic(\calO_S) \ar[r] \ar[d] & 0 \\ & 0 & H \ar@{=}[r] \ar[d] & H \ar[d] & \\ & & 0 & 0 & }" title="\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] & \\ 0 \ar[r] & \calO_S^* / k^* \ar[r] \ar[d] & \Div^0_\infty(K) \ar[r] \ar[d] & T \ar[r] \ar[d] & 0 \\ 0 \ar[r] & K^* / k^* \ar[r] \ar[d] & \Div^0(K) \ar[r] \ar[d] & \Pic^0(K) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & K^* / \calO_S^* \ar[r] \ar[d] & \Id(\calO_S) \ar[r] \ar[d] & \Pic(\calO_S) \ar[r] \ar[d] & 0 \\ & 0 & H \ar@{=}[r] \ar[d] & H \ar[d] & \\ & & 0 & 0 & }">
</div>
<p>
Here, <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/NlJVsDKqqV5jBHHEceVtse6TyXCHH9OQyWen_Q.svgz" alt="T" title="T"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/592A9cpJqEapTTR0mQzYD2FBigGo_84WF1hl8w.svgz" alt="H" title="H"></span> are essentially defined by the diagram, i.e. are the kernels and cokernels of the respective maps. In the number field case, <span class="inline-formula"><img class="img-inline-formula img-formula" width="49" height="12" src="https://math.fontein.de/formulae/wWOiqh31c10NzfOTYaXDvCARRj_v3mHRt09OWw.svgz" alt="H = 0" title="H = 0"></span>, and in the function field case, <span class="inline-formula"><img class="img-inline-formula img-formula" width="282" height="18" src="https://math.fontein.de/formulae/E.wpEnamztph7v0y.X0_BcL6u9nSqvuQG0PGQg.svgz" alt="H \cong (\deg \frakp \mid \frakp \in \calP_K) / (\deg \frakp \mid \frakp \in S)" title="H \cong (\deg \frakp \mid \frakp \in \calP_K) / (\deg \frakp \mid \frakp \in S)"></span>.
</p>
</div>
</div>
<div class="subsection-block">
<h4>
A Geometry of Numbers in Global Fields.
</h4>
<div>
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="85" height="18" src="https://math.fontein.de/formulae/gL7lmEL4jCjw4m1wGx7njKLP9lwxO1pBcVfs3w.svgz" alt="\fraka \in \Id(\calO_S)" title="\fraka \in \Id(\calO_S)"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="123" height="16" src="https://math.fontein.de/formulae/jX1rTpviKi49ApZmXGl4v9SKrSSEQC_JRtfffg.svgz" alt="t_1, \dots, t_{n+1} \in \G" title="t_1, \dots, t_{n+1} \in \G"></span>. Define
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="402" height="24" src="https://math.fontein.de/formulae/q5Vu5hc8WynssBBosYL97ldh_hp8wVoRHs2V8g.svgz" alt="B(\fraka, (t_1, \dots, t_{n+1})) := \{ f \in \fraka \mid \forall i : \abs{f}_{\frakp_i} \le q^{t_i \deg \frakp_i} \}." title="B(\fraka, (t_1, \dots, t_{n+1})) := \{ f \in \fraka \mid \forall i : \abs{f}_{\frakp_i} \le q^{t_i \deg \frakp_i} \}.">
</div>
<p>
If <span class="inline-formula"><img class="img-inline-formula img-formula" width="268" height="22" src="https://math.fontein.de/formulae/pjcF_JfjlkMIWTApecGfBJ1tlL4ARXKpBFtcSQ.svgz" alt="D := \divisor(\fraka) + \sum_{i=1}^{n+1} t_i \frakp_i \in \Div(K)" title="D := \divisor(\fraka) + \sum_{i=1}^{n+1} t_i \frakp_i \in \Div(K)"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="41" height="18" src="https://math.fontein.de/formulae/.K0eWOd5_T.LzueNG0aT.X.RS356YVDVv7q4pA.svgz" alt="L(D)" title="L(D)"></span> is the <a href="https://en.wikipedia.org/wiki/Riemann%E2%80%93Roch_theorem">Riemann-Roch space</a> of <span class="inline-formula"><img class="img-inline-formula img-formula" width="15" height="12" src="https://math.fontein.de/formulae/.1gXmU_PRktx2Rs2BFoRNw.Q_4Ub6YJ34UcmGw.svgz" alt="D" title="D"></span>, then <span class="inline-formula"><img class="img-inline-formula img-formula" width="211" height="18" src="https://math.fontein.de/formulae/wjHNCGIClGezR6Y9XVYWtMEnERrve01QJF03vQ.svgz" alt="L(D) = B(\fraka, (t_1, \dots, t_{n+1}))" title="L(D) = B(\fraka, (t_1, \dots, t_{n+1}))"></span>. In particular, the set is finite and invariant under multiplication by elements of <span class="inline-formula"><img class="img-inline-formula img-formula" width="10" height="12" src="https://math.fontein.de/formulae/KIbZSMqheEMj2LBRLlu5emEXwvNaAcODJL5YWQ.svgz" alt="k" title="k"></span>; in case <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> is a function field, <span class="inline-formula"><img class="img-inline-formula img-formula" width="41" height="18" src="https://math.fontein.de/formulae/.K0eWOd5_T.LzueNG0aT.X.RS356YVDVv7q4pA.svgz" alt="L(D)" title="L(D)"></span> is a finite-dimensional <span class="inline-formula"><img class="img-inline-formula img-formula" width="10" height="12" src="https://math.fontein.de/formulae/KIbZSMqheEMj2LBRLlu5emEXwvNaAcODJL5YWQ.svgz" alt="k" title="k"></span>-vector space, whose dimension is described by the Riemann-Roch theorem. In the number field case, we can make statements on <span class="inline-formula"><img class="img-inline-formula img-formula" width="41" height="18" src="https://math.fontein.de/formulae/.K0eWOd5_T.LzueNG0aT.X.RS356YVDVv7q4pA.svgz" alt="L(D)" title="L(D)"></span> with Minkowski's Lattice Point Theorem.
</p>
<p>
Consider the map
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="352" height="18" src="https://math.fontein.de/formulae/LHmiCZc8doFovgGiDXbybgzxCoyExSo2jXg02A.svgz" alt="\Psi : K^* \to \G^n, \quad f \mapsto (-\nu_{\frakp_1}(f), \dots, -\nu_{\frakp_n}(f))." title="\Psi : K^* \to \G^n, \quad f \mapsto (-\nu_{\frakp_1}(f), \dots, -\nu_{\frakp_n}(f)).">
</div>
<p>
Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="136" height="18" src="https://math.fontein.de/formulae/XRw8rlkDVj9HTvgL6xd5wr60WpoXEIWreBzlag.svgz" alt="\Lambda := \Psi(\calO^*) \cong \Z^n" title="\Lambda := \Psi(\calO^*) \cong \Z^n"></span> is a lattice by Dirichlet's Unit Theorem, and <span class="inline-formula"><img class="img-inline-formula img-formula" width="107" height="18" src="https://math.fontein.de/formulae/3NF1HoidE4Igcw.k5sUO4m9RsH2rXqgzhhJXfQ.svgz" alt="\ker \Psi|_{\calO^*} = k^*" title="\ker \Psi|_{\calO^*} = k^*"></span>. We get <span class="inline-formula"><img class="img-inline-formula img-formula" width="107" height="14" src="https://math.fontein.de/formulae/WVc9LYPB61srP3wN1TYzKzyYZRZ2aeFPTew1nA.svgz" alt="\calO^* \cong k^* \times \Z^n" title="\calO^* \cong k^* \times \Z^n"></span>, and <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span> is called the <em>unit rank</em> of <span class="inline-formula"><img class="img-inline-formula img-formula" width="24" height="15" src="https://math.fontein.de/formulae/Gd48AlL0RCLMaMySZsX3OXsmy8NI9JMWM5OAVA.svgz" alt="\calO_S" title="\calO_S"></span>. This <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/XY_2LK590TGO0.jb0ZbkmaiBi4kJr1xvp52q2g.svgz" alt="\Lambda" title="\Lambda"></span> will be the lattice for our <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span>-dimensional infrastructure.
</p>
</div>
</div>
<div class="subsection-block">
<h4>
Reduced Ideals.
</h4>
<div>
<p>
The elements of <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span> will be principal reduced fractional ideals, modulo an equivalence relation. We begin by defining minima, which are similar to the ones introduced in the <a href="https://math.fontein.de/2009/07/21/how-to-obtain-reduction-maps-for-n-dimensional-infrastructures/">previous post</a> for lattices.
</p>
<div class="theorem-environment theorem-definition-environment">
<div class="theorem-header theorem-definition-header">
Definition.
</div>
<div class="theorem-content theorem-definition-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="85" height="18" src="https://math.fontein.de/formulae/gL7lmEL4jCjw4m1wGx7njKLP9lwxO1pBcVfs3w.svgz" alt="\fraka \in \Id(\calO_S)" title="\fraka \in \Id(\calO_S)"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="84" height="18" src="https://math.fontein.de/formulae/rnlb5YTpmaOJdPNhMB.kd4byo7UjGADJpklbmA.svgz" alt="\mu \in \fraka \setminus \{ 0 \}" title="\mu \in \fraka \setminus \{ 0 \}"></span>. We say that <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="11" src="https://math.fontein.de/formulae/QGPIFfjz60I1ozjZngbI3x6aeMfrAlc1UlQkzg.svgz" alt="\mu" title="\mu"></span> is a <em>minimum</em> of <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="8" src="https://math.fontein.de/formulae/7tKsFfQJIH5P5Se7T.JEYI_Aa3rHmfwW5_gmtA.svgz" alt="\fraka" title="\fraka"></span> if every <span class="inline-formula"><img class="img-inline-formula img-formula" width="84" height="18" src="https://math.fontein.de/formulae/tyhim4xdr1Y6QVJLMwui5wfcXbc.PR7EihHItg.svgz" alt="f \in \fraka \setminus \{ 0 \}" title="f \in \fraka \setminus \{ 0 \}"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="92" height="21" src="https://math.fontein.de/formulae/waBicgQWbivtcQGpvVlHbd_JfPAXOHIPISuikw.svgz" alt="\abs{f}_{\frakp_i} \le \abs{\mu}_{\frakp_i}" title="\abs{f}_{\frakp_i} \le \abs{\mu}_{\frakp_i}"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="6" height="12" src="https://math.fontein.de/formulae/S43oPTrFqmoVC.yqOcgzvrroaMU3pS7pa40ROQ.svgz" alt="i" title="i"></span> satisfies <span class="inline-formula"><img class="img-inline-formula img-formula" width="92" height="21" src="https://math.fontein.de/formulae/TLTIcQxQTgftxY5M7Q5maQpIzQFODnJJ1z1ynw.svgz" alt="\abs{f}_{\frakp_i} = \abs{\mu}_{\frakp_i}" title="\abs{f}_{\frakp_i} = \abs{\mu}_{\frakp_i}"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="6" height="12" src="https://math.fontein.de/formulae/S43oPTrFqmoVC.yqOcgzvrroaMU3pS7pa40ROQ.svgz" alt="i" title="i"></span>. Denote the set of all minima of <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="8" src="https://math.fontein.de/formulae/7tKsFfQJIH5P5Se7T.JEYI_Aa3rHmfwW5_gmtA.svgz" alt="\fraka" title="\fraka"></span> by <span class="inline-formula"><img class="img-inline-formula img-formula" width="33" height="18" src="https://math.fontein.de/formulae/a5L2Nri_zgXxzkoL68qknZ.bYJjXTDAgp3qpsA.svgz" alt="\calC(\fraka)" title="\calC(\fraka)"></span>.
</p>
</div>
</div>
<p>
Using them, we can define reduced ideals:
</p>
<div class="theorem-environment theorem-definition-environment">
<div class="theorem-header theorem-definition-header">
Definition.
</div>
<div class="theorem-content theorem-definition-content">
<p>
An ideal <span class="inline-formula"><img class="img-inline-formula img-formula" width="85" height="18" src="https://math.fontein.de/formulae/gL7lmEL4jCjw4m1wGx7njKLP9lwxO1pBcVfs3w.svgz" alt="\fraka \in \Id(\calO_S)" title="\fraka \in \Id(\calO_S)"></span> is said to be <em>reduced</em> if <span class="inline-formula"><img class="img-inline-formula img-formula" width="39" height="12" src="https://math.fontein.de/formulae/1fL8eO8_1O9Cq5CnzEcBZKqiALr_Koa_HS3pfw.svgz" alt="1 \in \fraka" title="1 \in \fraka"></span> is a minimum. Write <span class="inline-formula"><img class="img-inline-formula img-formula" width="71" height="18" src="https://math.fontein.de/formulae/BFayH5LpRi4Skv7aBBNkQ1TNd2qg5tsQ54jCiQ.svgz" alt="\Red_S(K)" title="\Red_S(K)"></span> for the set of all reduced ideals of <span class="inline-formula"><img class="img-inline-formula img-formula" width="24" height="15" src="https://math.fontein.de/formulae/Gd48AlL0RCLMaMySZsX3OXsmy8NI9JMWM5OAVA.svgz" alt="\calO_S" title="\calO_S"></span>. For <span class="inline-formula"><img class="img-inline-formula img-formula" width="85" height="18" src="https://math.fontein.de/formulae/FLiIyKzOvqu7SGMzT_MVmK9cAGgHUkKRIsH_tw.svgz" alt="\frakb \in \Id(\calO_S)" title="\frakb \in \Id(\calO_S)"></span> let <span class="inline-formula"><img class="img-inline-formula img-formula" width="360" height="18" src="https://math.fontein.de/formulae/Hdrj9KLUJZ.G2ig.3YjdqGMaEVWV.VldnLW6kQ.svgz" alt="\Red_S(\frakb) := \{ \fraka \in \Red_S(K) \mid \exists f \in K^* : f \fraka = \frakb \}" title="\Red_S(\frakb) := \{ \fraka \in \Red_S(K) \mid \exists f \in K^* : f \fraka = \frakb \}"></span>.
</p>
</div>
</div>
<p>
The equivalence relation we need is defined by
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="375" height="22" src="https://math.fontein.de/formulae/fwevKBVPG0KeC_48Bq6WDVBAxyUwNViAIElW0Q.svgz" alt="\fraka \sim_S \fraka' :\Leftrightarrow \exists f \in K^* : \fraka = f \fraka' \wedge \forall \frakp \in S : \abs{f}_\frakp = 1" title="\fraka \sim_S \fraka' :\Leftrightarrow \exists f \in K^* : \fraka = f \fraka' \wedge \forall \frakp \in S : \abs{f}_\frakp = 1">
</div>
<p>
for <span class="inline-formula"><img class="img-inline-formula img-formula" width="107" height="18" src="https://math.fontein.de/formulae/78AKR2rGEjATumRU_PeSByliNbIbgKCwUOu8Ng.svgz" alt="\fraka, \fraka' \in \Id(\calO_S)" title="\fraka, \fraka' \in \Id(\calO_S)"></span>. We then get the following results:
</p>
<div class="theorem-environment theorem-theorem-environment qed">
<div class="theorem-header theorem-theorem-header">
Theorem.
</div>
<div class="theorem-content theorem-theorem-content">
<ol class="enum-level-1">
<li>We have that <span class="inline-formula"><img class="img-inline-formula img-formula" width="71" height="18" src="https://math.fontein.de/formulae/BFayH5LpRi4Skv7aBBNkQ1TNd2qg5tsQ54jCiQ.svgz" alt="\Red_S(K)" title="\Red_S(K)"></span> is a finite set.</li>
<li>In case <span class="inline-formula"><img class="img-inline-formula img-formula" width="71" height="16" src="https://math.fontein.de/formulae/boAVNScKJIebxM4cdQYAUMj_mTPuVqYFZrx5lA.svgz" alt="\deg \frakp = 1" title="\deg \frakp = 1"></span> for some <span class="inline-formula"><img class="img-inline-formula img-formula" width="42" height="15" src="https://math.fontein.de/formulae/YkrPToBqCrBzPlTwf6DsfmrX6y0gpY75eFGZCQ.svgz" alt="\frakp \in S" title="\frakp \in S"></span>, we get <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="16" src="https://math.fontein.de/formulae/S8dVyJOC4.cqjOEsj12FjkHflummwyejSDgKrA.svgz" alt="\fraka \sim_S \fraka'" title="\fraka \sim_S \fraka'"></span> for <span class="inline-formula"><img class="img-inline-formula img-formula" width="113" height="18" src="https://math.fontein.de/formulae/ccvGZvU_xBs0Wrcf1vJx_FRGnHJDKIgQODNJRA.svgz" alt="\fraka, \fraka' \in \Red(K)" title="\fraka, \fraka' \in \Red(K)"></span> if, and only if, <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="13" src="https://math.fontein.de/formulae/93bWlaBNvwplBtY0E8Walgzam.TwOXiNnSbjng.svgz" alt="\fraka = \fraka'" title="\fraka = \fraka'"></span>.</li>
<li>We have that <span class="inline-formula"><img class="img-inline-formula img-formula" width="23" height="12" src="https://math.fontein.de/formulae/v5Ij2805wr7U8f4RL1Z26Pdl9MspPzKHTBcNSw.svgz" alt="\calO^*" title="\calO^*"></span> acts on <span class="inline-formula"><img class="img-inline-formula img-formula" width="33" height="18" src="https://math.fontein.de/formulae/a5L2Nri_zgXxzkoL68qknZ.bYJjXTDAgp3qpsA.svgz" alt="\calC(\fraka)" title="\calC(\fraka)"></span> by multiplication.</li>
<li>
<p>
The map
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="256" height="39" src="https://math.fontein.de/formulae/hR7tS0irNffslQjYI_wXGw3l2AiR7Jy.PdH.LQ.svgz" alt="\calC(\fraka) / \calO^* \to \Red(\fraka), \quad \mu \calO^* \mapsto \frac{1}{\mu} \fraka" title="\calC(\fraka) / \calO^* \to \Red(\fraka), \quad \mu \calO^* \mapsto \frac{1}{\mu} \fraka">
</div>
<p>
is a bijection.
</p>
</li>
<li>If <span class="inline-formula"><img class="img-inline-formula img-formula" width="91" height="18" src="https://math.fontein.de/formulae/x4RJMpg8MsqczxLx4jy_zh1QXX37.ca8mzGxiw.svgz" alt="\fraka \in \Red(K)" title="\fraka \in \Red(K)"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="75" height="18" src="https://math.fontein.de/formulae/oUjOsyU.jAw20T.ybYDUnrn..qoAwsoHFfhvGg.svgz" alt="\frakb \in \Id(\calO)" title="\frakb \in \Id(\calO)"></span> satisfies <span class="inline-formula"><img class="img-inline-formula img-formula" width="52" height="15" src="https://math.fontein.de/formulae/AsrXh9T6kshJKTUETtUh4w0UaDnagb.bTIzrlA.svgz" alt="\fraka \sim_S \frakb" title="\fraka \sim_S \frakb"></span>, then <span class="inline-formula"><img class="img-inline-formula img-formula" width="84" height="18" src="https://math.fontein.de/formulae/Xyy75QRlnBDsv0IHFVYyggRJKFyLOznHo2uxAg.svgz" alt="\frakb \in \Red(\fraka)" title="\frakb \in \Red(\fraka)"></span>.</li>
</ol>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
The proofs of these and the following results or hints to the proofs can be found <a href="http://arxiv.org/abs/0809.1685">here</a>. We next construct the map <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="12" src="https://math.fontein.de/formulae/DxlmDzniFM_09XzQnKCoRu8h2JBxVd5JriHzmg.svgz" alt="d" title="d"></span>:
</p>
<div class="theorem-environment theorem-theorem-environment qed">
<div class="theorem-header theorem-theorem-header">
Theorem (Infrastructure, Part I).
</div>
<div class="theorem-content theorem-theorem-content">
<p>
Fix an ideal <span class="inline-formula"><img class="img-inline-formula img-formula" width="75" height="18" src="https://math.fontein.de/formulae/TFrVZydKSxtrVP7E8cjWib_.ru6gd0P.d6LJxg.svgz" alt="\fraka \in \Id(\calO)" title="\fraka \in \Id(\calO)"></span>. Define <span class="inline-formula"><img class="img-inline-formula img-formula" width="135" height="18" src="https://math.fontein.de/formulae/gbcmvEUmB2_pJQk379RyXxX4cREOc.8rMae0rQ.svgz" alt="X_\fraka := \Red(\fraka)/_{\sim_S}" title="X_\fraka := \Red(\fraka)/_{\sim_S}"></span> and define
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="303" height="23" src="https://math.fontein.de/formulae/D5YduM0xzrurBM1Xdh9AccASuQGhTk5O21y4wA.svgz" alt="d_\fraka : X \to \G^n / \Lambda, \quad [\tfrac{1}{\mu} \fraka]_{\sim_S} \mapsto \Psi(\mu) + \Lambda." title="d_\fraka : X \to \G^n / \Lambda, \quad [\tfrac{1}{\mu} \fraka]_{\sim_S} \mapsto \Psi(\mu) + \Lambda.">
</div>
<p>
Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="18" height="15" src="https://math.fontein.de/formulae/BoErCEo_Kshiv7bnZBb176lWHc.ywQs51SFtJQ.svgz" alt="d_\fraka" title="d_\fraka"></span> is well-defined and injective.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
For <span class="inline-formula"><img class="img-inline-formula img-formula" width="78" height="17" src="https://math.fontein.de/formulae/PU8BIb5ceNVC1QIH69dZM_8tM4hrntIhIb3XoA.svgz" alt="a, a' \in K^*" title="a, a' \in K^*"></span>, write
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="260" height="22" src="https://math.fontein.de/formulae/ZMZW6G510VhjeoA2QMfKC7mxVMtDUr9QcSQNqQ.svgz" alt="a \sim_S a' :\Longleftrightarrow \forall \frakp \in S : \abs{a}_\frakp = \abs{a'}_\frakp." title="a \sim_S a' :\Longleftrightarrow \forall \frakp \in S : \abs{a}_\frakp = \abs{a'}_\frakp.">
</div>
<p>
Define <span class="inline-formula"><img class="img-inline-formula img-formula" width="107" height="21" src="https://math.fontein.de/formulae/NPsVZjI8hdgyWOQdnQZDzdgVVARTDLTsm6OC5g.svgz" alt="\hat{X} := \calC(\fraka)/_{\sim_S}" title="\hat{X} := \calC(\fraka)/_{\sim_S}"></span> and
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="220" height="21" src="https://math.fontein.de/formulae/vj5RQkywBSA_.dHj4nB59CLPe5BcntRhaziPww.svgz" alt="\hat{d} : \hat{X} \to \G^n, \quad [\mu]_\sim \mapsto \Psi(\mu)." title="\hat{d} : \hat{X} \to \G^n, \quad [\mu]_\sim \mapsto \Psi(\mu).">
</div>
<p>
Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="47" height="21" src="https://math.fontein.de/formulae/yml2qjrayo6LrmUSPXGQKCPOl6cveZ75yQZwuw.svgz" alt="(\hat{X}, \hat{d})" title="(\hat{X}, \hat{d})"></span> is the unrolled version of <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="18" src="https://math.fontein.de/formulae/jIn.uZYNabR16.oruR4TBCDUy7VpBBZ9LQZ0OQ.svgz" alt="(X, d)" title="(X, d)"></span>: if <span class="inline-formula"><img class="img-inline-formula img-formula" width="121" height="18" src="https://math.fontein.de/formulae/9579ccqe2Syxupv12lz8CB_iRlC6xAsuWQ63OA.svgz" alt="\pi : \G^n \to \G^n / \Lambda" title="\pi : \G^n \to \G^n / \Lambda"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="82" height="14" src="https://math.fontein.de/formulae/4f5VmTE4UWS69sDpvCVPYATBuG5Sf30s4CXuPw.svgz" alt="x \mapsto x + \Lambda" title="x \mapsto x + \Lambda"></span> is the projection, and <span class="inline-formula"><img class="img-inline-formula img-formula" width="87" height="20" src="https://math.fontein.de/formulae/as.ohNwIxl6T0mmlhY2Q4UUkK_kC6Db9S12i0w.svgz" alt="\psi : \hat{X} \to X" title="\psi : \hat{X} \to X"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="104" height="23" src="https://math.fontein.de/formulae/2BOuR6eBoxXFHEPt5sDe7E5LGYEEYWaEJArvcw.svgz" alt="[\mu]_\sim \mapsto [\frac{1}{\mu} \fraka]_\sim" title="[\mu]_\sim \mapsto [\frac{1}{\mu} \fraka]_\sim"></span>, then the following diagram commutes:
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="132" height="112" src="https://math.fontein.de/formulae/W6l_i7ArzVuzt_3qedjpJeYcQU_W6gLQa.zsYg.svgz" alt="\xymatrix{ \hat{X} \ar[d]_{\psi} \ar[r]^{\hat{d}} & \G^n \ar[d]^{\pi} \\ X \ar[r]_{d} & \G^n/\Lambda }" title="\xymatrix{ \hat{X} \ar[d]_{\psi} \ar[r]^{\hat{d}} & \G^n \ar[d]^{\pi} \\ X \ar[r]_{d} & \G^n/\Lambda }">
</div>
<p>
In particular, <span class="inline-formula"><img class="img-inline-formula img-formula" width="39" height="21" src="https://math.fontein.de/formulae/RnhbfieFDqbJimZmhqU9GTozCJg_bPc7F78VNA.svgz" alt="\hat{d}(\hat{X})" title="\hat{d}(\hat{X})"></span> is the set <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="17" src="https://math.fontein.de/formulae/trC8ROiBHCvA1Mxjz0DMydgaaneA8jHc1Qwtww.svgz" alt="\hat{X}" title="\hat{X}"></span> from the <a href="https://math.fontein.de/2009/07/21/how-to-obtain-reduction-maps-for-n-dimensional-infrastructures/">previous post</a>.
</p>
</div>
</div>
<div class="subsection-block">
<h4>
The Reduction Map, <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/M.ji_f0zsVnTLyaKegBkRJLOeqrrt6brNnapOQ.svgz" alt="f" title="f"></span>-Representations, and the Infrastructure.
</h4>
<div>
<p>
We proceed by defining <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/M.ji_f0zsVnTLyaKegBkRJLOeqrrt6brNnapOQ.svgz" alt="f" title="f"></span>-representations, as giving these is equivalent to give a reduction map. Fix an ideal <span class="inline-formula"><img class="img-inline-formula img-formula" width="85" height="18" src="https://math.fontein.de/formulae/gL7lmEL4jCjw4m1wGx7njKLP9lwxO1pBcVfs3w.svgz" alt="\fraka \in \Id(\calO_S)" title="\fraka \in \Id(\calO_S)"></span>.
</p>
<p>
First, define for <span class="inline-formula"><img class="img-inline-formula img-formula" width="79" height="17" src="https://math.fontein.de/formulae/bMsKIabN7qdfW02XS0TukmbtlrPoIqIuzl1Zqw.svgz" alt="f, f' \in K^*" title="f, f' \in K^*"></span>
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="545" height="23" src="https://math.fontein.de/formulae/zQW.FFClkXe6sEXKB7kw1gESOV6fPy0MQtRXvQ.svgz" alt="f \le_S f' :\Longleftrightarrow (\abs{f}_{\frakp_{n+1}}, \abs{f}_{\frakp_1}, \dots, \abs{f}_{\frakp_n}) \le_{\ell ex} (\abs{f'}_{\frakp_{n+1}}, \abs{f'}_{\frakp_1}, \dots, \abs{f'}_{\frakp_n})," title="f \le_S f' :\Longleftrightarrow (\abs{f}_{\frakp_{n+1}}, \abs{f}_{\frakp_1}, \dots, \abs{f}_{\frakp_n}) \le_{\ell ex} (\abs{f'}_{\frakp_{n+1}}, \abs{f'}_{\frakp_1}, \dots, \abs{f'}_{\frakp_n}),">
</div>
<p>
where <span class="inline-formula"><img class="img-inline-formula img-formula" width="35" height="14" src="https://math.fontein.de/formulae/272urPy4p00s7_NbVXJQCBdlCm0bfpuzIwY0JQ.svgz" alt="\le_{\ell ex}" title="\le_{\ell ex}"></span> is the lexicographic order on <span class="inline-formula"><img class="img-inline-formula img-formula" width="40" height="14" src="https://math.fontein.de/formulae/yo3D3yFnt1B05mNyv8ERQPdEDUGS70ry2cm.kQ.svgz" alt="\R^{n+1}" title="\R^{n+1}"></span>.
</p>
<div class="theorem-environment theorem-definition-environment">
<div class="theorem-header theorem-definition-header">
Definition.
</div>
<div class="theorem-content theorem-definition-content">
<p>
A tuple <span class="inline-formula"><img class="img-inline-formula img-formula" width="305" height="18" src="https://math.fontein.de/formulae/95JuQxs0jLkCPlqCN6Myt016Lga0dAu8EYRxAA.svgz" alt="([\frakb]_{\sim_S}, (t_1, \dots, t_n)) \in \Red_S(\fraka)/_{\sim_S} \times \G^n" title="([\frakb]_{\sim_S}, (t_1, \dots, t_n)) \in \Red_S(\fraka)/_{\sim_S} \times \G^n"></span> is said to be an <em><span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/M.ji_f0zsVnTLyaKegBkRJLOeqrrt6brNnapOQ.svgz" alt="f" title="f"></span>-representation</em> if <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="11" src="https://math.fontein.de/formulae/o.f5nkkZC1PEov1Bjz5CUC3EQ.MnIDCsxksltQ.svgz" alt="1" title="1"></span> is a smallest element of
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="189" height="18" src="https://math.fontein.de/formulae/Vgr6wbwf7CODuppXVStJc.fFULwVVDqNEW7V2Q.svgz" alt="B(\frakb, (t_1, \dots, t_n, 0)) \setminus \{ 0 \}" title="B(\frakb, (t_1, \dots, t_n, 0)) \setminus \{ 0 \}">
</div>
<p>
with respect to <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="14" src="https://math.fontein.de/formulae/zlfN63Ix.fiq0WuW6KUdG.zhODns6kOCg.RygA.svgz" alt="\le" title="\le"></span>. Denote the set of all <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/M.ji_f0zsVnTLyaKegBkRJLOeqrrt6brNnapOQ.svgz" alt="f" title="f"></span>-representations by <span class="inline-formula"><img class="img-inline-formula img-formula" width="62" height="19" src="https://math.fontein.de/formulae/c3hFmxM0nuDcAuSb50LixMHUOjBtQizTQH_SgQ.svgz" alt="\fRep(\fraka)" title="\fRep(\fraka)"></span>.
</p>
</div>
</div>
<p>
One quickly sees that this is well-defined. We have two auxilliary results:
</p>
<div class="theorem-environment theorem-lemma-environment qed">
<div class="theorem-header theorem-lemma-header">
Lemma (Uniqueness).
</div>
<div class="theorem-content theorem-lemma-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="265" height="19" src="https://math.fontein.de/formulae/380PfzgmZzjClcyD8zqQtkMde2hmJ9rOlb5v8g.svgz" alt="A = ([\frakb]_{\sim_S}, (t_1, \dots, t_n)) \in \fRep(\fraka)" title="A = ([\frakb]_{\sim_S}, (t_1, \dots, t_n)) \in \fRep(\fraka)"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="16" src="https://math.fontein.de/formulae/J4C0dkT_SToe7mWds1p3hHRPKdN2gAL4MA1yiA.svgz" alt="f \in K^*" title="f \in K^*"></span> such that
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="425" height="24" src="https://math.fontein.de/formulae/Pp3zLWCelO1qLTPp6tklCT33UgqFn9kIjvzE0w.svgz" alt="B = ([\tfrac{1}{f} \frakb]_{\sim_S}, (t_1 + \nu_{\frakp_1}(f), \dots, t_n + \nu_{\frakp_n}(f))) \in \fRep(\fraka)." title="B = ([\tfrac{1}{f} \frakb]_{\sim_S}, (t_1 + \nu_{\frakp_1}(f), \dots, t_n + \nu_{\frakp_n}(f))) \in \fRep(\fraka).">
</div>
<p>
Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="61" height="21" src="https://math.fontein.de/formulae/9kNBDTj0GVjzsBvYSNRMfUfWfT802Vo4eUE1ng.svgz" alt="\abs{f}_\frakp = 1" title="\abs{f}_\frakp = 1"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="42" height="15" src="https://math.fontein.de/formulae/YkrPToBqCrBzPlTwf6DsfmrX6y0gpY75eFGZCQ.svgz" alt="\frakp \in S" title="\frakp \in S"></span>, i.e. <span class="inline-formula"><img class="img-inline-formula img-formula" width="51" height="12" src="https://math.fontein.de/formulae/cMqXrgz2BI0S11nKbCPB3c_uEQeJQHWtGKkJ8g.svgz" alt="A = B" title="A = B"></span>.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<div class="theorem-environment theorem-lemma-environment qed">
<div class="theorem-header theorem-lemma-header">
Lemma (Reduction).
</div>
<div class="theorem-content theorem-lemma-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="166" height="18" src="https://math.fontein.de/formulae/8nwO5C86aIxlICGolzzVDRy.amNNN7_ddqiBMA.svgz" alt="v = (v_1, \dots, v_n) \in \G^n" title="v = (v_1, \dots, v_n) \in \G^n"></span>. Then there exists a smallest <span class="inline-formula"><img class="img-inline-formula img-formula" width="43" height="13" src="https://math.fontein.de/formulae/VAXSOlQYnpjdhdvx.KqlICSKW4iZQe6mlBVvPg.svgz" alt="\ell \in \G" title="\ell \in \G"></span> such that <span class="inline-formula"><img class="img-inline-formula img-formula" width="273" height="18" src="https://math.fontein.de/formulae/itZDbrKHkEOL9KuaTdZWJhb0Z2M9TNZaWlDqTA.svgz" alt="B_\ell := B(\fraka, (v_1, \dots, v_n, \ell)) \setminus \{ 0 \} \neq \emptyset" title="B_\ell := B(\fraka, (v_1, \dots, v_n, \ell)) \setminus \{ 0 \} \neq \emptyset"></span>. If <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="11" src="https://math.fontein.de/formulae/QGPIFfjz60I1ozjZngbI3x6aeMfrAlc1UlQkzg.svgz" alt="\mu" title="\mu"></span> is minimal with respect to <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="14" src="https://math.fontein.de/formulae/zlfN63Ix.fiq0WuW6KUdG.zhODns6kOCg.RygA.svgz" alt="\le" title="\le"></span> in that <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="15" src="https://math.fontein.de/formulae/j0a55wkBPqZKflQEb.noEqsh0kkwQvybOzqNfg.svgz" alt="B_\ell" title="B_\ell"></span>, then
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="387" height="24" src="https://math.fontein.de/formulae/yMhlKmcK2Q3JCL1bohVOiyivqy5AoVE.ODCG6Q.svgz" alt="([\tfrac{1}{\mu} \fraka]_{\sim_S}, (v_1 + \nu_{\frakp_1}(\mu), \dots, v_n + \nu_{\frakp_n}(\mu))) \in \fRep(\fraka)" title="([\tfrac{1}{\mu} \fraka]_{\sim_S}, (v_1 + \nu_{\frakp_1}(\mu), \dots, v_n + \nu_{\frakp_n}(\mu))) \in \fRep(\fraka)">
</div>
<p>
and <span class="inline-formula"><img class="img-inline-formula img-formula" width="389" height="18" src="https://math.fontein.de/formulae/5IPvR.3d06qhbDqAUPecQgKHZ_ZX2lWHgMrmWA.svgz" alt="\Phi(\mu) + (v_1 + \nu_{\frakp_1}(\mu), \dots, v_n + \nu_{\frakp_n}(\mu)) + \Lambda = v + \Lambda" title="\Phi(\mu) + (v_1 + \nu_{\frakp_1}(\mu), \dots, v_n + \nu_{\frakp_n}(\mu)) + \Lambda = v + \Lambda"></span>.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
From that, we get the following result:
</p>
<div class="theorem-environment theorem-theorem-environment qed">
<div class="theorem-header theorem-theorem-header">
Theorem (Infrastructure, Part II).
</div>
<div class="theorem-content theorem-theorem-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="75" height="18" src="https://math.fontein.de/formulae/TFrVZydKSxtrVP7E8cjWib_.ru6gd0P.d6LJxg.svgz" alt="\fraka \in \Id(\calO)" title="\fraka \in \Id(\calO)"></span>. Then the map
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="389" height="53" src="https://math.fontein.de/formulae/wjpOIljChP7zOyYY_HVkhAJssFlmVnWSIdOuBA.svgz" alt="\Phi :{} & \fRep(\fraka) \to \G^n / \Lambda \\
& ([\tfrac{1}{\mu} \fraka]_{\sim_S}, (t_1, \dots, t_n)) \mapsto \Psi(\mu) + (t_1, \dots, t_n) + \Lambda" title="\Phi :{} & \fRep(\fraka) \to \G^n / \Lambda \\
& ([\tfrac{1}{\mu} \fraka]_{\sim_S}, (t_1, \dots, t_n)) \mapsto \Psi(\mu) + (t_1, \dots, t_n) + \Lambda">
</div>
<p>
is a bijection.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
This allows to equip <span class="inline-formula"><img class="img-inline-formula img-formula" width="62" height="19" src="https://math.fontein.de/formulae/c3hFmxM0nuDcAuSb50LixMHUOjBtQizTQH_SgQ.svgz" alt="\fRep(\fraka)" title="\fRep(\fraka)"></span> with a group operation. We will see that the group operation of <span class="inline-formula"><img class="img-inline-formula img-formula" width="77" height="19" src="https://math.fontein.de/formulae/FsI.7Kq0dhyWl_LI6FeTXtBfU1V_mVewFrEPoA.svgz" alt="\fRep(\calO_S)" title="\fRep(\calO_S)"></span> can be described in a very explicit form. This extends to a broader interpretation of the infrastructure, whence we will do this in the next section.
</p>
<p>
Before ending this section, we want to state a result which shows that <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/M.ji_f0zsVnTLyaKegBkRJLOeqrrt6brNnapOQ.svgz" alt="f" title="f"></span>-representations are small.
</p>
<div class="theorem-environment theorem-theorem-environment qed">
<div class="theorem-header theorem-theorem-header">
Theorem.
</div>
<div class="theorem-content theorem-theorem-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="228" height="19" src="https://math.fontein.de/formulae/Tkgzw5gXffLM2uEFjkXqu6socPUWzB1ZpDN78w.svgz" alt="([\frakb]_{\sim_S}, (t_1, \dots, t_n)) \in \fRep(\fraka)" title="([\frakb]_{\sim_S}, (t_1, \dots, t_n)) \in \fRep(\fraka)"></span>, then <span class="inline-formula"><img class="img-inline-formula img-formula" width="80" height="18" src="https://math.fontein.de/formulae/WJwq50Gwrq7XLYIzqHDglrvtk1MY70VyoOAhWg.svgz" alt="\divisor(\frakb) \ge 0" title="\divisor(\frakb) \ge 0"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="14" src="https://math.fontein.de/formulae/cRmhOSsIGH9x_kDrRPR8gFqaBPim5SnsfdzQ5A.svgz" alt="t_i \ge 0" title="t_i \ge 0"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="6" height="12" src="https://math.fontein.de/formulae/S43oPTrFqmoVC.yqOcgzvrroaMU3pS7pa40ROQ.svgz" alt="i" title="i"></span> and
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="228" height="52" src="https://math.fontein.de/formulae/jmuLIvtK8cBLy3uBATz..ESZo55DFLcVyKgfKg.svgz" alt="\deg \divisor(\fraka) + \sum_{i=1}^n t_i \deg \frakp_i \le \kappa," title="\deg \divisor(\fraka) + \sum_{i=1}^n t_i \deg \frakp_i \le \kappa,">
</div>
<p>
where
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="382" height="53" src="https://math.fontein.de/formulae/RyTFQnm7sYZJYxd9mT7axllLIFKnI6Uousd4YQ.svgz" alt="\kappa := \begin{cases} g + \deg \frakp_{n+1} - 1 & \text{if } K \text{ is a function field} \\ s \log \tfrac{2}{\pi} + \tfrac{1}{2} \log \abs{\Delta} & \text{if } K \text{ is a number field;} \end{cases}" title="\kappa := \begin{cases} g + \deg \frakp_{n+1} - 1 & \text{if } K \text{ is a function field} \\ s \log \tfrac{2}{\pi} + \tfrac{1}{2} \log \abs{\Delta} & \text{if } K \text{ is a number field;} \end{cases}">
</div>
<p>
here, <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="11" src="https://math.fontein.de/formulae/UxCAbnAvo.LhIVRUfLjfwIlPAfhwrVfYLg_Adw.svgz" alt="g" title="g"></span> is the genus of <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> in case <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> is a function field, and in case <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> is a number field, <span class="inline-formula"><img class="img-inline-formula img-formula" width="8" height="8" src="https://math.fontein.de/formulae/ovmFNULo3EfWiVyyUOID62AW15.M_94ZehNNBw.svgz" alt="s" title="s"></span> denotes the number of places of degree two and <span class="inline-formula"><img class="img-inline-formula img-formula" width="15" height="12" src="https://math.fontein.de/formulae/yACQU7OyIs1vDOS4Gffc.71vOz9QXxbT5QQVxA.svgz" alt="\Delta" title="\Delta"></span> is the discriminant of <span class="inline-formula"><img class="img-inline-formula img-formula" width="24" height="15" src="https://math.fontein.de/formulae/Gd48AlL0RCLMaMySZsX3OXsmy8NI9JMWM5OAVA.svgz" alt="\calO_S" title="\calO_S"></span>.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
Therefore, <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/M.ji_f0zsVnTLyaKegBkRJLOeqrrt6brNnapOQ.svgz" alt="f" title="f"></span>-representations are small.
</p>
</div>
</div>
<div class="subsection-block">
<h4>
The Infrastructure and the Divisor Class Group.
</h4>
<div>
<p>
Assume for a moment that <span class="inline-formula"><img class="img-inline-formula img-formula" width="99" height="16" src="https://math.fontein.de/formulae/8W2KF4D6KCWXPxEojsaPISvWagaRhl6x1G2QOQ.svgz" alt="\deg \frakp_{n+1} = 1" title="\deg \frakp_{n+1} = 1"></span>, or that <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> is a number field. Then we have a short exact sequence
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="384" height="31" src="https://math.fontein.de/formulae/FpRjw5Jql5_P46_ETLS2E2YSNFb75_l9NPBd.g.svgz" alt="\xymatrix{ 0 \ar[r] & T \ar[r] & \Pic^0(K) \ar[r] & \Pic(\calO_S) \ar[r] & 0, }" title="\xymatrix{ 0 \ar[r] & T \ar[r] & \Pic^0(K) \ar[r] & \Pic(\calO_S) \ar[r] & 0, }">
</div>
<p>
and <span class="inline-formula"><img class="img-inline-formula img-formula" width="166" height="19" src="https://math.fontein.de/formulae/L9mAXhl7CaAdfbGjEPwfhtRvcSG1m2IYKqJFfw.svgz" alt="T \cong \G^n / \Lambda \cong \fRep(\fraka)" title="T \cong \G^n / \Lambda \cong \fRep(\fraka)"></span>. This means that the divisor class group <span class="inline-formula"><img class="img-inline-formula img-formula" width="63" height="20" src="https://math.fontein.de/formulae/u0_navDCuz614cXLn3RMPT.As5xD3N0zgJmWNA.svgz" alt="\Pic^0(K)" title="\Pic^0(K)"></span> is covered by copies of <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="18" src="https://math.fontein.de/formulae/y7inFt29Lt_rp.PbK0jo9HaZeKFqRkt83i7w7w.svgz" alt="\G^n/\Lambda" title="\G^n/\Lambda"></span>, where the copies are indexed by the elements of the divisor class group. If <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="8" src="https://math.fontein.de/formulae/7tKsFfQJIH5P5Se7T.JEYI_Aa3rHmfwW5_gmtA.svgz" alt="\fraka" title="\fraka"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="13" src="https://math.fontein.de/formulae/.b4y0U4e.Uzpi4mI3pt7aDYT.GuVZhqWPdByjA.svgz" alt="\fraka'" title="\fraka'"></span> are in the same ideal class, <span class="inline-formula"><img class="img-inline-formula img-formula" width="23" height="15" src="https://math.fontein.de/formulae/xC1LdNDYgU6jjWCdoDarRwdEOnidJ1TzXhW5IQ.svgz" alt="X_\fraka" title="X_\fraka"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="28" height="15" src="https://math.fontein.de/formulae/VkWb9fT8b1O.kk_lATb3XcmJWQPNWaNm1OL5ig.svgz" alt="X_{\fraka'}" title="X_{\fraka'}"></span> differ by a translation, i.e. they give essentially the same infrastructure; in fact, <span class="inline-formula"><img class="img-inline-formula img-formula" width="152" height="19" src="https://math.fontein.de/formulae/sqm6M9uPfTB3KxC_3EYi.aJ0MCRheKsTrxCrBA.svgz" alt="\fRep(\fraka) = \fRep(\fraka')" title="\fRep(\fraka) = \fRep(\fraka')"></span>. Hence, one could get the idea to cover <span class="inline-formula"><img class="img-inline-formula img-formula" width="63" height="20" src="https://math.fontein.de/formulae/u0_navDCuz614cXLn3RMPT.As5xD3N0zgJmWNA.svgz" alt="\Pic^0(K)" title="\Pic^0(K)"></span> by <span class="inline-formula"><img class="img-inline-formula img-formula" width="62" height="19" src="https://math.fontein.de/formulae/c3hFmxM0nuDcAuSb50LixMHUOjBtQizTQH_SgQ.svgz" alt="\fRep(\fraka)" title="\fRep(\fraka)"></span>, where <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="8" src="https://math.fontein.de/formulae/7tKsFfQJIH5P5Se7T.JEYI_Aa3rHmfwW5_gmtA.svgz" alt="\fraka" title="\fraka"></span> ranges over the distinct ideal classes, i.e. by <span class="inline-formula"><img class="img-inline-formula img-formula" width="239" height="23" src="https://math.fontein.de/formulae/N8sdJcvY6Cu2LiE1NR54w0I4fIAiEBOCxpZvag.svgz" alt="\fRep(K) := \bigcup_{\fraka \in \Id(\calO_S)} \fRep(\fraka)" title="\fRep(K) := \bigcup_{\fraka \in \Id(\calO_S)} \fRep(\fraka)"></span>. It turns out that this is indeed the case, and the arithmetic on <span class="inline-formula"><img class="img-inline-formula img-formula" width="77" height="19" src="https://math.fontein.de/formulae/FsI.7Kq0dhyWl_LI6FeTXtBfU1V_mVewFrEPoA.svgz" alt="\fRep(\calO_S)" title="\fRep(\calO_S)"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="63" height="20" src="https://math.fontein.de/formulae/u0_navDCuz614cXLn3RMPT.As5xD3N0zgJmWNA.svgz" alt="\Pic^0(K)" title="\Pic^0(K)"></span> turn out to be the same under the bijection we get.
</p>
<p>
In case <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> is a function field and <span class="inline-formula"><img class="img-inline-formula img-formula" width="99" height="16" src="https://math.fontein.de/formulae/IAbhKw6xoQBmD6Cyn5n7tNVEnjrEX9j4lYCuUA.svgz" alt="\deg \frakp_{n+1} > 1" title="\deg \frakp_{n+1} > 1"></span>, we have <span class="inline-formula"><img class="img-inline-formula img-formula" width="81" height="18" src="https://math.fontein.de/formulae/Ug.neSkKhCTS3ZCaRZsFPDowjUviydhDBGpWNw.svgz" alt="T \not\cong \G^n / \Lambda" title="T \not\cong \G^n / \Lambda"></span> in general (this is the case if, and only if, <span class="inline-formula"><img class="img-inline-formula img-formula" width="279" height="18" src="https://math.fontein.de/formulae/I2K3pJuhn1aXsVjwGshBOKAl26V3zmvHqO28uw.svgz" alt="\deg \frakp_{n+1} = \gcd(\deg \frakp_1, \dots, \frakp_n, \frakp_{n+1})" title="\deg \frakp_{n+1} = \gcd(\deg \frakp_1, \dots, \frakp_n, \frakp_{n+1})"></span>), and <span class="inline-formula"><img class="img-inline-formula img-formula" width="153" height="20" src="https://math.fontein.de/formulae/v.utT9O5gI8UbgdbCRD.KpiWVBp0aA.C0dIh0Q.svgz" alt="\Pic^0(K) \to \Pic(\calO_S)" title="\Pic^0(K) \to \Pic(\calO_S)"></span> does not needs to be surjective. It would be nice to change the above sequence to
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="411" height="31" src="https://math.fontein.de/formulae/HlsOKl7LwqS3IsuEuQ.epfaLF6W2zXh_16MN4Q.svgz" alt="\xymatrix{ 0 \ar[r] & \G^n/\Lambda \ar[r] & \Pic^0(K) \ar[r] & \Pic(\calO_S) \ar[r] & 0 }" title="\xymatrix{ 0 \ar[r] & \G^n/\Lambda \ar[r] & \Pic^0(K) \ar[r] & \Pic(\calO_S) \ar[r] & 0 }">
</div>
<p>
in any case, but this is not possible with <span class="inline-formula"><img class="img-inline-formula img-formula" width="63" height="20" src="https://math.fontein.de/formulae/u0_navDCuz614cXLn3RMPT.As5xD3N0zgJmWNA.svgz" alt="\Pic^0(K)" title="\Pic^0(K)"></span> as it is; we have to replace it by something bigger. It turns out that the right replacement is <span class="inline-formula"><img class="img-inline-formula img-formula" width="124" height="18" src="https://math.fontein.de/formulae/NPOiamtoiH7z_5TR0sTfqSJHVy6v86q.eIwsxg.svgz" alt="\Pic(K) / \ggen{[\frakp_{n+1}]}" title="\Pic(K) / \ggen{[\frakp_{n+1}]}"></span>, which is canonically isomorphic to <span class="inline-formula"><img class="img-inline-formula img-formula" width="63" height="20" src="https://math.fontein.de/formulae/u0_navDCuz614cXLn3RMPT.As5xD3N0zgJmWNA.svgz" alt="\Pic^0(K)" title="\Pic^0(K)"></span> in case <span class="inline-formula"><img class="img-inline-formula img-formula" width="240" height="18" src="https://math.fontein.de/formulae/ab2BOtm5J7ZXE.vdxE8uAeG0DKHFSIDriQZfAg.svgz" alt="\deg \frakp_{n+1} = \gcd(\deg \frakp \mid \frakp \in \calP_K)" title="\deg \frakp_{n+1} = \gcd(\deg \frakp \mid \frakp \in \calP_K)"></span>. We then get the diagram
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="476" height="103" src="https://math.fontein.de/formulae/EUqZO_EBf7lGOwP_pD_if9fRCAUe..HWD0OEwQ.svgz" alt="\xymatrix{ 0 \ar[r] & T \ar[r] \ar@{^(->}[d] & \Pic^0(K) \ar@{^(->}[d] \ar[r] & \Pic(\calO_S) \ar@{=}[d] & \\ 0 \ar[r] & \G^n/\Lambda \ar[r] & \Pic(K) / \ggen{[\frakp_{n+1}]} \ar[r] & \Pic(\calO_K) \ar[r] & 0 }" title="\xymatrix{ 0 \ar[r] & T \ar[r] \ar@{^(->}[d] & \Pic^0(K) \ar@{^(->}[d] \ar[r] & \Pic(\calO_S) \ar@{=}[d] & \\ 0 \ar[r] & \G^n/\Lambda \ar[r] & \Pic(K) / \ggen{[\frakp_{n+1}]} \ar[r] & \Pic(\calO_K) \ar[r] & 0 }">
</div>
<p>
with exact rows.
</p>
<p>
The complete result is stated in the following theorem:
</p>
<div class="theorem-environment theorem-theorem-environment qed">
<div class="theorem-header theorem-theorem-header">
Theorem (Infrastructure, Part III).
</div>
<div class="theorem-content theorem-theorem-content">
<ol class="enum-level-1">
<li>
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> be a number field. Then the map
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="476" height="79" src="https://math.fontein.de/formulae/Dvj9.29XntuETzkQP1OIHr4q.5SqhmaX.Bv86g.svgz" alt="\Phi :{} & \fRep(K) \to \Pic^0(K), \\
& ([\frakb]_{\sim_S}, (t_1, \dots, t_n)) \mapsto \biggl[ \divisor(\frakb) + \sum_{i=1}^n t_i \frakp_i - \frac{\dots}{\deg \frakp_{n+1}} \frakp_{n+1} \biggr]," title="\Phi :{} & \fRep(K) \to \Pic^0(K), \\
& ([\frakb]_{\sim_S}, (t_1, \dots, t_n)) \mapsto \biggl[ \divisor(\frakb) + \sum_{i=1}^n t_i \frakp_i - \frac{\dots}{\deg \frakp_{n+1}} \frakp_{n+1} \biggr],">
</div>
<p>
where <span class="inline-formula"><img class="img-inline-formula img-formula" width="21" height="2" src="https://math.fontein.de/formulae/Lua5qdEcW2z676uaXCKQzctmtOizd0TydttZfw.svgz" alt="\dots" title="\dots"></span> equals <span class="inline-formula"><img class="img-inline-formula img-formula" width="204" height="20" src="https://math.fontein.de/formulae/Qbxhsq6VbpoXDnRxyCj86pQgaeRlDOLDPQQv2Q.svgz" alt="\deg \divisor(\frakb) + \sum_{i=1}^n t_i \deg \frakp_i" title="\deg \divisor(\frakb) + \sum_{i=1}^n t_i \deg \frakp_i"></span>, is a bijection.
</p>
</li>
<li>
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> be a function field. Then the map
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="424" height="79" src="https://math.fontein.de/formulae/qO_EkI0q6IWwCb506ghUsxtomqIs1YSLdiO9dA.svgz" alt="\Phi :{} & \fRep(K) \to \Pic(K) / \ggen{[\frakp_{n+1}]}, \\
& ([\frakb]_{\sim_S}, (t_1, \dots, t_n)) \mapsto \biggl[ \divisor(\frakb) + \sum_{i=1}^n t_i \frakp_i \biggr] + \ggen{[\frakp_{n+1}]}" title="\Phi :{} & \fRep(K) \to \Pic(K) / \ggen{[\frakp_{n+1}]}, \\
& ([\frakb]_{\sim_S}, (t_1, \dots, t_n)) \mapsto \biggl[ \divisor(\frakb) + \sum_{i=1}^n t_i \frakp_i \biggr] + \ggen{[\frakp_{n+1}]}">
</div>
<p>
is a bijection.
</p>
</li>
</ol>
<p>
Moreover, <span class="inline-formula"><img class="img-inline-formula img-formula" width="80" height="20" src="https://math.fontein.de/formulae/.larI3KDqPpOO0583Mac9Ii4HsLT6OV2vlvZ.g.svgz" alt="\Phi|_{\fRep(\calO_S)}" title="\Phi|_{\fRep(\calO_S)}"></span> is a group homomorphism, where the group structure on <span class="inline-formula"><img class="img-inline-formula img-formula" width="77" height="19" src="https://math.fontein.de/formulae/FsI.7Kq0dhyWl_LI6FeTXtBfU1V_mVewFrEPoA.svgz" alt="\fRep(\calO_S)" title="\fRep(\calO_S)"></span> is the one induced by the bijection <span class="inline-formula"><img class="img-inline-formula img-formula" width="149" height="19" src="https://math.fontein.de/formulae/GvpSe_a7dFl3HBLGFPzc7eWyYy_GWb0NG3VZyA.svgz" alt="\fRep(\calO_S) \to \G^n/\Lambda" title="\fRep(\calO_S) \to \G^n/\Lambda"></span>.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
Finally, we explicitly describe the group operation induced by this bijection on <span class="inline-formula"><img class="img-inline-formula img-formula" width="69" height="19" src="https://math.fontein.de/formulae/t5AOObF8PuHvUGJUAQGn5JB7gZlW3ufudb7kMQ.svgz" alt="\fRep(K)" title="\fRep(K)"></span> without using the bijection itself.
</p>
<div class="theorem-environment theorem-theorem-environment qed">
<div class="theorem-header theorem-theorem-header">
Theorem.
</div>
<div class="theorem-content theorem-theorem-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/h1YPUpOWbsd4f48ktllbLrRQbkfVpcJAOVaMmA.svgz" alt="\Phi" title="\Phi"></span> be the bijection from the previous theorem, and let <span class="inline-formula"><img class="img-inline-formula img-formula" width="182" height="18" src="https://math.fontein.de/formulae/CcLOWfOL6fkVgtlVfZ.4iZa1_zAj0jUBNNvqXg.svgz" alt="A = ([\fraka]_{\sim_S}, (t_1, \dots, t_n))" title="A = ([\fraka]_{\sim_S}, (t_1, \dots, t_n))"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="282" height="19" src="https://math.fontein.de/formulae/DqwJiP07t0fjoSqywbczPanUVoQ3EohAkPR3JQ.svgz" alt="A' = ([\fraka']_{\sim_S}, (t'_1, \dots, t'_n)) \in \fRep(K)" title="A' = ([\fraka']_{\sim_S}, (t'_1, \dots, t'_n)) \in \fRep(K)"></span>.
</p>
<ol class="enum-level-1">
<li>
<p>
Set <span class="inline-formula"><img class="img-inline-formula img-formula" width="323" height="18" src="https://math.fontein.de/formulae/tM42zbyW1ulfdNBztCAJGpF0E_YXO5EA8GkpPA.svgz" alt="B_\ell := B(\fraka \fraka', (t_1 + t'_1, \dots, t_n + t'_n, \ell)) \setminus \{ 0 \}" title="B_\ell := B(\fraka \fraka', (t_1 + t'_1, \dots, t_n + t'_n, \ell)) \setminus \{ 0 \}"></span> for <span class="inline-formula"><img class="img-inline-formula img-formula" width="43" height="13" src="https://math.fontein.de/formulae/VAXSOlQYnpjdhdvx.KqlICSKW4iZQe6mlBVvPg.svgz" alt="\ell \in \G" title="\ell \in \G"></span>. There exists a minimal <span class="inline-formula"><img class="img-inline-formula img-formula" width="7" height="12" src="https://math.fontein.de/formulae/TxqokZWmJwdp4cC9v4vThIX8YQePrhEiYeCmGw.svgz" alt="\ell" title="\ell"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="53" height="17" src="https://math.fontein.de/formulae/EkOqccQu9awmchrbLe6gwvbBrL1TN5x.KW1v3w.svgz" alt="B_\ell \neq \emptyset" title="B_\ell \neq \emptyset"></span>, and if <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="11" src="https://math.fontein.de/formulae/QGPIFfjz60I1ozjZngbI3x6aeMfrAlc1UlQkzg.svgz" alt="\mu" title="\mu"></span> is a smallest element of <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="15" src="https://math.fontein.de/formulae/j0a55wkBPqZKflQEb.noEqsh0kkwQvybOzqNfg.svgz" alt="B_\ell" title="B_\ell"></span> with respect to <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="14" src="https://math.fontein.de/formulae/zlfN63Ix.fiq0WuW6KUdG.zhODns6kOCg.RygA.svgz" alt="\le" title="\le"></span>, we get
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="404" height="24" src="https://math.fontein.de/formulae/CJViOZzY5FZ.iGx6ab10GlcZDPx0ldUTC57gRw.svgz" alt="B := ([\tfrac{1}{\mu} \fraka \fraka']_{\sim_S}, (t_i + t'_i + \nu_{\frakp_i}(\mu))_{i=1,\dots,n}) \in \fRep(K)" title="B := ([\tfrac{1}{\mu} \fraka \fraka']_{\sim_S}, (t_i + t'_i + \nu_{\frakp_i}(\mu))_{i=1,\dots,n}) \in \fRep(K)">
</div>
<p>
with <span class="inline-formula"><img class="img-inline-formula img-formula" width="171" height="18" src="https://math.fontein.de/formulae/NgnULQj8Dm4mzqkvOJW6vIFKtzRyPIm6UARAUg.svgz" alt="\Phi(A) + \Phi(A') = \Phi(B)" title="\Phi(A) + \Phi(A') = \Phi(B)"></span>.
</p>
</li>
<li>
<p>
Set <span class="inline-formula"><img class="img-inline-formula img-formula" width="282" height="19" src="https://math.fontein.de/formulae/zdJ7eNXYyB7liyz9kyAtGOmIZ0ZK3tOwK3xUKg.svgz" alt="B_\ell := B(\fraka^{-1}, (-t_1, \dots, -t_n, \ell)) \setminus \{ 0 \}" title="B_\ell := B(\fraka^{-1}, (-t_1, \dots, -t_n, \ell)) \setminus \{ 0 \}"></span> for <span class="inline-formula"><img class="img-inline-formula img-formula" width="43" height="13" src="https://math.fontein.de/formulae/VAXSOlQYnpjdhdvx.KqlICSKW4iZQe6mlBVvPg.svgz" alt="\ell \in \G" title="\ell \in \G"></span>. There exists a minimal <span class="inline-formula"><img class="img-inline-formula img-formula" width="7" height="12" src="https://math.fontein.de/formulae/TxqokZWmJwdp4cC9v4vThIX8YQePrhEiYeCmGw.svgz" alt="\ell" title="\ell"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="53" height="17" src="https://math.fontein.de/formulae/EkOqccQu9awmchrbLe6gwvbBrL1TN5x.KW1v3w.svgz" alt="B_\ell \neq \emptyset" title="B_\ell \neq \emptyset"></span>, and if <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="11" src="https://math.fontein.de/formulae/QGPIFfjz60I1ozjZngbI3x6aeMfrAlc1UlQkzg.svgz" alt="\mu" title="\mu"></span> is a smallest element of <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="15" src="https://math.fontein.de/formulae/j0a55wkBPqZKflQEb.noEqsh0kkwQvybOzqNfg.svgz" alt="B_\ell" title="B_\ell"></span> with respect to <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="14" src="https://math.fontein.de/formulae/zlfN63Ix.fiq0WuW6KUdG.zhODns6kOCg.RygA.svgz" alt="\le" title="\le"></span>, we get
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="388" height="24" src="https://math.fontein.de/formulae/eyJgHWO32JmXm8RJvXMcQyUxUw4j0UekgbmWSw.svgz" alt="C := ([\tfrac{1}{\mu} \fraka^{-1}]_{\sim_S}, (-t_i + \nu_{\frakp_i}(\mu))_{i=1,\dots,n}) \in \fRep(K)" title="C := ([\tfrac{1}{\mu} \fraka^{-1}]_{\sim_S}, (-t_i + \nu_{\frakp_i}(\mu))_{i=1,\dots,n}) \in \fRep(K)">
</div>
<p>
with <span class="inline-formula"><img class="img-inline-formula img-formula" width="118" height="18" src="https://math.fontein.de/formulae/TlMkVSY9yEi0FPxqZser9ssSQJxXP.VNmKfsOw.svgz" alt="-\Phi(A) = \Phi(C)" title="-\Phi(A) = \Phi(C)"></span>.
</p>
</li>
</ol>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
This shows that the <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span>-dimensional infrastructure we defined has a very close connection to the arithmetic of the divisor class group. This connection was first shown for real hyperelliptic curves by H.-G. Rück and S. Paulus, <a href="http://citeseer.ist.psu.edu/old/717445.html">“Real and Imaginary Quadratic Representations of Hyperelliptic Function Fields”</a>. The first relation between the infrastructure of number fields and the Arakelov divisor class group was described by R. Schoof in his paper <a href="http://www.mat.uniroma2.it/%20schoof/papers.html">Computing Arakelov class groups</a>.
</p>
</div>
</div>
<div class="subsection-block">
<h4>
What about... Baby Steps?
</h4>
<div>
<p>
As I <a href="https://math.fontein.de/2009/07/20/n-dimensional-infrastructures/">mentioned</a>, there is no known construction for baby steps in general <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span>-dimensional infrastructures, but there exists a construction for infrastructures obtained from global fields. I want to describe this construction here.
</p>
<p>
For <span class="inline-formula"><img class="img-inline-formula img-formula" width="135" height="18" src="https://math.fontein.de/formulae/mb_keKTeRRSaOW25TmOvrADH12EBVcCsdRZZaA.svgz" alt="i \in \{ 1, \dots, n + 1 \}" title="i \in \{ 1, \dots, n + 1 \}"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="91" height="18" src="https://math.fontein.de/formulae/x4RJMpg8MsqczxLx4jy_zh1QXX37.ca8mzGxiw.svgz" alt="\fraka \in \Red(K)" title="\fraka \in \Red(K)"></span>, consider
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="369" height="49" src="https://math.fontein.de/formulae/GEFO0OuBORGbGZN9I0TfzUgAaUOfwMZkYlKq0w.svgz" alt="B_\ell := \biggl\{ f \in \fraka \;\biggm|\begin{matrix} \abs{f}_{\frakp_j} \le 1 \text{ for all } j \neq i, \\ \exists j' : \abs{f}_{\frakp_{j'}} < 1, \; \abs{f}_{\frakp_i} \le \ell \end{matrix} \biggr\} \setminus \{ 0 \}" title="B_\ell := \biggl\{ f \in \fraka \;\biggm|\begin{matrix} \abs{f}_{\frakp_j} \le 1 \text{ for all } j \neq i, \\ \exists j' : \abs{f}_{\frakp_{j'}} < 1, \; \abs{f}_{\frakp_i} \le \ell \end{matrix} \biggr\} \setminus \{ 0 \}">
</div>
<p>
for <span class="inline-formula"><img class="img-inline-formula img-formula" width="40" height="13" src="https://math.fontein.de/formulae/sajLq_SKey_MIohLJPM5_HqE8PXy.kXIJ5uy2A.svgz" alt="\ell > 0" title="\ell > 0"></span>. There exists a minimal <span class="inline-formula"><img class="img-inline-formula img-formula" width="7" height="12" src="https://math.fontein.de/formulae/TxqokZWmJwdp4cC9v4vThIX8YQePrhEiYeCmGw.svgz" alt="\ell" title="\ell"></span> such that <span class="inline-formula"><img class="img-inline-formula img-formula" width="53" height="17" src="https://math.fontein.de/formulae/EkOqccQu9awmchrbLe6gwvbBrL1TN5x.KW1v3w.svgz" alt="B_\ell \neq \emptyset" title="B_\ell \neq \emptyset"></span>. In case <span class="inline-formula"><img class="img-inline-formula img-formula" width="77" height="16" src="https://math.fontein.de/formulae/7nLkGdqjymBbMaeZYrhlVTPkB_VbhMLtgnr3OQ.svgz" alt="\deg \frakp_i = 1" title="\deg \frakp_i = 1"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="15" src="https://math.fontein.de/formulae/j0a55wkBPqZKflQEb.noEqsh0kkwQvybOzqNfg.svgz" alt="B_\ell" title="B_\ell"></span> contains exactly one <span class="inline-formula"><img class="img-inline-formula img-formula" width="18" height="12" src="https://math.fontein.de/formulae/KNpOu8H6kUesAJNA335iDhv878WaoMOAl2.1hw.svgz" alt="k^*" title="k^*"></span>-orbit, which gives a unique element <span class="inline-formula"><img class="img-inline-formula img-formula" width="53" height="16" src="https://math.fontein.de/formulae/0CSrTer_ub5jt1.SZGn6MG.Xo0HZnKGCIPotVA.svgz" alt="\mu \in B_\ell" title="\mu \in B_\ell"></span>. Otherwise, one has to add an order (lexicographic order as <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="14" src="https://math.fontein.de/formulae/zlfN63Ix.fiq0WuW6KUdG.zhODns6kOCg.RygA.svgz" alt="\le" title="\le"></span> above) to chose an element. In any case, define <span class="inline-formula"><img class="img-inline-formula img-formula" width="157" height="23" src="https://math.fontein.de/formulae/EHy07EKsAeh.t.oOy.6n143KAuWkeXru1B5U7w.svgz" alt="\bs_i([\fraka]_{\sim_S}) := [\frac{1}{\mu} \fraka]_{\sim_S}" title="\bs_i([\fraka]_{\sim_S}) := [\frac{1}{\mu} \fraka]_{\sim_S}"></span>; then this gives a function <span class="inline-formula"><img class="img-inline-formula img-formula" width="149" height="18" src="https://math.fontein.de/formulae/uglS6HP9glDYoTAyzdZdOQBvb2H5ELUbM7ecaw.svgz" alt="\Red(K) \to \Red(K)" title="\Red(K) \to \Red(K)"></span> resp. <span class="inline-formula"><img class="img-inline-formula img-formula" width="135" height="18" src="https://math.fontein.de/formulae/IAOfyju7ub4kbK4_frE_sk0.qPQG_l21ngDXbw.svgz" alt="\Red(\frakb) \to \Red(\frakb)" title="\Red(\frakb) \to \Red(\frakb)"></span> for any <span class="inline-formula"><img class="img-inline-formula img-formula" width="75" height="18" src="https://math.fontein.de/formulae/oUjOsyU.jAw20T.ybYDUnrn..qoAwsoHFfhvGg.svgz" alt="\frakb \in \Id(\calO)" title="\frakb \in \Id(\calO)"></span>. Opposed to the one-dimensional case, this function neither has to be injective nor surjective, as examples below will show.
</p>
<p>
We begin with a “small” example: the infrastructure <span class="inline-formula"><img class="img-inline-formula img-formula" width="88" height="18" src="https://math.fontein.de/formulae/0Z3wf8UEABI_w.saxEgF.WLXOjfaU74UB9q74A.svgz" alt="(X_{\calO_S}, d_{\calO_S})" title="(X_{\calO_S}, d_{\calO_S})"></span> of the function field defined by <span class="inline-formula"><img class="img-inline-formula img-formula" width="187" height="18" src="https://math.fontein.de/formulae/qqYKumhfcxSH28LGZvWFTlpz5zD0_R6gdgnfRg.svgz" alt="y^3 = x^6 + x^5 + x^4 + 4 x^2" title="y^3 = x^6 + x^5 + x^4 + 4 x^2"></span> over <span class="inline-formula"><img class="img-inline-formula img-formula" width="19" height="15" src="https://math.fontein.de/formulae/JRmYwLNj1fIeE5gF_0gmpYMEJCopJn4E7T8xIw.svgz" alt="\F_7" title="\F_7"></span>. The red arrows show <span class="inline-formula"><img class="img-inline-formula img-formula" width="25" height="15" src="https://math.fontein.de/formulae/K9oXYeywYUBv1jiz4sHBzlQUviMaTYhTSTfqzw.svgz" alt="\bs_1" title="\bs_1"></span>, the blue arrows <span class="inline-formula"><img class="img-inline-formula img-formula" width="25" height="15" src="https://math.fontein.de/formulae/NIA9tFFIrMaHX_GmwKgKJPN4RIht9jN6YudJVw.svgz" alt="\bs_2" title="\bs_2"></span> and the green arrows <span class="inline-formula"><img class="img-inline-formula img-formula" width="25" height="15" src="https://math.fontein.de/formulae/UfXA2REOqUGjEC_hP0XvyrgOn6Vn2lOK6JES_Q.svgz" alt="\bs_3" title="\bs_3"></span>. The small black circles denote usual minima, the larger black circles denote elements of <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/XY_2LK590TGO0.jb0ZbkmaiBi4kJr1xvp52q2g.svgz" alt="\Lambda" title="\Lambda"></span>, and the shaded areas denote translates of an fundamental parallelepiped of <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/XY_2LK590TGO0.jb0ZbkmaiBi4kJr1xvp52q2g.svgz" alt="\Lambda" title="\Lambda"></span>:
</p>
<div class="center-block">
<p>
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</p>
</div>
<p>
The second example depicts the infrastructure <span class="inline-formula"><img class="img-inline-formula img-formula" width="88" height="18" src="https://math.fontein.de/formulae/0Z3wf8UEABI_w.saxEgF.WLXOjfaU74UB9q74A.svgz" alt="(X_{\calO_S}, d_{\calO_S})" title="(X_{\calO_S}, d_{\calO_S})"></span> of the function field defined by <span class="inline-formula"><img class="img-inline-formula img-formula" width="205" height="18" src="https://math.fontein.de/formulae/xsntax.MUCJgPB9BZb6AP4QiS0wHOdHvfs19qw.svgz" alt="y^3 = x^6 + 5 x^5 + 6 x^4 + 5 x^2" title="y^3 = x^6 + 5 x^5 + 6 x^4 + 5 x^2"></span> over <span class="inline-formula"><img class="img-inline-formula img-formula" width="19" height="15" src="https://math.fontein.de/formulae/JRmYwLNj1fIeE5gF_0gmpYMEJCopJn4E7T8xIw.svgz" alt="\F_7" title="\F_7"></span>, with the same notation:
</p>
<div class="center-block">
<p>
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</p>
</div>
</div>
</div>
</div>
n-dimensional Infrastructures.
https://math.fontein.de/2009/07/20/n-dimensional-infrastructures/
2009-07-20T08:40:46+02:00
2009-07-20T08:40:46+02:00
Felix Fontein
<div>
<div>
<p>
For <a href="https://math.fontein.de/2009/07/20/one-dimensional-infrastructures/">one-dimensional infrastructures</a>, we have a circle <span class="inline-formula"><img class="img-inline-formula img-formula" width="47" height="18" src="https://math.fontein.de/formulae/Blo1FniNoktf2EKQy0CWZxv_r_wmqWWcM_Pc9A.svgz" alt="\R/R\Z" title="\R/R\Z"></span> together with a finite, non-empty set <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span> and an injective map <span class="inline-formula"><img class="img-inline-formula img-formula" width="115" height="18" src="https://math.fontein.de/formulae/nG5tOQuC_AiH8MK1kd7kHlmnr7XkfyU3Es3Tew.svgz" alt="d : X \to \R/R\Z" title="d : X \to \R/R\Z"></span>. Now <span class="inline-formula"><img class="img-inline-formula img-formula" width="114" height="18" src="https://math.fontein.de/formulae/je5sWDSzyU2LwOWbO2BPl7MrRpZMNNPO4l_UJg.svgz" alt="\R/R\Z = \R^n / \Lambda" title="\R/R\Z = \R^n / \Lambda"></span>, where <span class="inline-formula"><img class="img-inline-formula img-formula" width="43" height="11" src="https://math.fontein.de/formulae/AI3jLOaiR4OxpS_JgU78KWnQ5jO4jP6BS3hJew.svgz" alt="n = 1" title="n = 1"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/XY_2LK590TGO0.jb0ZbkmaiBi4kJr1xvp52q2g.svgz" alt="\Lambda" title="\Lambda"></span> is the one-dimensional <a href="https://en.wikipedia.org/wiki/Lattice_%28group%29">lattice</a> <span class="inline-formula"><img class="img-inline-formula img-formula" width="61" height="12" src="https://math.fontein.de/formulae/ehoSwEBlGxIA1X6WzvZdPs3rJo291gV1954Ubw.svgz" alt="\Lambda = R \Z" title="\Lambda = R \Z"></span>. Hence, one could say that an <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span>-dimensional infrastructure is a torus <span class="inline-formula"><img class="img-inline-formula img-formula" width="44" height="18" src="https://math.fontein.de/formulae/W.nVj7Zi0tCSZE0Q2eogNxSUumdXLxoOTJEzCw.svgz" alt="\R^n/\Lambda" title="\R^n/\Lambda"></span> together with <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="111" height="18" src="https://math.fontein.de/formulae/2pzYFU99Q.eG3U9ebfYQlDgAMSYK7GaQKzO8cg.svgz" alt="d : X \to \R^n/\Lambda" title="d : X \to \R^n/\Lambda"></span> as above. From the discussion in the remarks of <a href="https://math.fontein.de/2009/07/20/one-dimensional-infrastructures/">this post</a> we see that we need some kind of reduction map to define giant steps (and also <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/M.ji_f0zsVnTLyaKegBkRJLOeqrrt6brNnapOQ.svgz" alt="f" title="f"></span>-representations) in the one-dimensional case, even though there a pretty canonical reduction map is given. In the case of <span class="inline-formula"><img class="img-inline-formula img-formula" width="44" height="18" src="https://math.fontein.de/formulae/W.nVj7Zi0tCSZE0Q2eogNxSUumdXLxoOTJEzCw.svgz" alt="\R^n/\Lambda" title="\R^n/\Lambda"></span>, we do not have something similar to a given positive direction. Moreover, definiting the “nearest” element of a finite subset of <span class="inline-formula"><img class="img-inline-formula img-formula" width="44" height="18" src="https://math.fontein.de/formulae/W.nVj7Zi0tCSZE0Q2eogNxSUumdXLxoOTJEzCw.svgz" alt="\R^n/\Lambda" title="\R^n/\Lambda"></span> to some <span class="inline-formula"><img class="img-inline-formula img-formula" width="72" height="18" src="https://math.fontein.de/formulae/dSbvCoREOPLKRZwVZS3oVPS8tuJgehNh0C1G7w.svgz" alt="t \in \R^n/\Lambda" title="t \in \R^n/\Lambda"></span> is even more complicated and offers more choices which appear more or less obvious. Only the selection of different norms on <span class="inline-formula"><img class="img-inline-formula img-formula" width="22" height="12" src="https://math.fontein.de/formulae/bZVXuWhECmRV9pmTW_AgHZeLTOYax7svHPDCxQ.svgz" alt="\R^n" title="\R^n"></span> lead to several possible definitions of such a map. Hence, we should require such a map in the definition:
</p>
<div class="theorem-environment theorem-definition-environment">
<div class="theorem-header theorem-definition-header">
Definition.
</div>
<div class="theorem-content theorem-definition-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="58" height="15" src="https://math.fontein.de/formulae/EBTY1cww2.XbyZp0zeo5cPIgVso4OVcWdqrNBg.svgz" alt="\Lambda \subseteq \R^n" title="\Lambda \subseteq \R^n"></span> be a lattice. Then, an <em><span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span>-dimensional infrastructure</em> is a non-empty finite set <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span> together with an injective map <span class="inline-formula"><img class="img-inline-formula img-formula" width="111" height="18" src="https://math.fontein.de/formulae/g3EkzKjEux19bH4SPfjRnoZOQOaLVxwmoBd9LA.svgz" alt="d : X \to \R^n / \Lambda" title="d : X \to \R^n / \Lambda"></span> and another map <span class="inline-formula"><img class="img-inline-formula img-formula" width="128" height="18" src="https://math.fontein.de/formulae/HrdulxFFRS4ZXBKuq8UY4ZHuLaDvvXLDps7nhA.svgz" alt="red : \R^n/\Lambda \to X" title="red : \R^n/\Lambda \to X"></span> satisfying <span class="inline-formula"><img class="img-inline-formula img-formula" width="104" height="15" src="https://math.fontein.de/formulae/3TNT2dOT2EXA3eU7nQ52xNGV4zlFqL_Gje1h0Q.svgz" alt="red \circ d = \id_X" title="red \circ d = \id_X"></span>.
</p>
</div>
</div>
<p>
Again, as in the one-dimensional case, one can define giant steps:
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="327" height="19" src="https://math.fontein.de/formulae/xcADjETDmD0MW7QB7UvF10FS_sfkTdN3JEhDcA.svgz" alt="\gs(x, x') := red(d(x) + d(x')), \quad x, x' \in X." title="\gs(x, x') := red(d(x) + d(x')), \quad x, x' \in X.">
</div>
<p>
Moreover, one gets the same relation between reduction maps and <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/M.ji_f0zsVnTLyaKegBkRJLOeqrrt6brNnapOQ.svgz" alt="f" title="f"></span>-representations, whence we define
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="518" height="20" src="https://math.fontein.de/formulae/QJN0BpLAOPpkagUbNlY5LvYA.o626.T2VVV4Gg.svgz" alt="\fRep := \fRep(X, d, red) := \{ (x, f) \in X \times \R^n \mid red(d(x) + f) = x \}." title="\fRep := \fRep(X, d, red) := \{ (x, f) \in X \times \R^n \mid red(d(x) + f) = x \}.">
</div>
<p>
Then the map
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="379" height="20" src="https://math.fontein.de/formulae/D49Sf0cS7t_ihscO0AypGfxiMGwMb7s_UPr.HQ.svgz" alt="\Psi : \fRep(X, d, red) \to \R^n/\Lambda, \quad (x, f) \mapsto d(x) + f" title="\Psi : \fRep(X, d, red) \to \R^n/\Lambda, \quad (x, f) \mapsto d(x) + f">
</div>
<p>
is a bijection, and we can use this bijection to equip <span class="inline-formula"><img class="img-inline-formula img-formula" width="119" height="19" src="https://math.fontein.de/formulae/R1_fB035Gv00zHSrkUbMpoa1r9AqRh2GRWmwZA.svgz" alt="\fRep(X, d, red)" title="\fRep(X, d, red)"></span> with a group law by
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="527" height="20" src="https://math.fontein.de/formulae/Qb8TPwpT4rhUfYu0KDJMlkhUtHka.2Xqjfko4g.svgz" alt="(x, f) + (x', f') = \Psi^{-1}(\Psi(x, f) + \Psi(x', f')), \quad (x, f), (x', f') \in \fRep." title="(x, f) + (x', f') = \Psi^{-1}(\Psi(x, f) + \Psi(x', f')), \quad (x, f), (x', f') \in \fRep.">
</div>
</div>
<div class="subsection-block">
<h4>
Discrete Infrastructure.
</h4>
<div>
<p>
We say that an <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span>-dimensional infrastructure <span class="inline-formula"><img class="img-inline-formula img-formula" width="80" height="18" src="https://math.fontein.de/formulae/Q3Wi7WBtdKSnv1gaLvYnlftQgM.JrmfkFL4zUA.svgz" alt="(X, d, red)" title="(X, d, red)"></span> with lattice <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/XY_2LK590TGO0.jb0ZbkmaiBi4kJr1xvp52q2g.svgz" alt="\Lambda" title="\Lambda"></span> is <em>discrete</em> if <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="15" src="https://math.fontein.de/formulae/J8WZLQNTj4hsLiLptic9ZbHN6du478Qep2spzQ.svgz" alt="\Lambda \subseteq \Z^n" title="\Lambda \subseteq \Z^n"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="105" height="18" src="https://math.fontein.de/formulae/dulT4cQCEDJN0_jeueOyj_j1g565IBQsowZkQA.svgz" alt="d(X) \subseteq \Z^n/\Lambda" title="d(X) \subseteq \Z^n/\Lambda"></span> and if <span class="inline-formula"><img class="img-inline-formula img-formula" width="26" height="12" src="https://math.fontein.de/formulae/x9uDaMcRWmFnhjgyf0CLBlJpQRUKxXbuUM0.1Q.svgz" alt="red" title="red"></span> does not depends on fractions. To make the last part more precise, define
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="400" height="18" src="https://math.fontein.de/formulae/JKoXxyyo.1bxGK5IEGxnKqkfKl3j4oaKZqWa_g.svgz" alt="floor : \R^n \to \Z^n, \quad (x_1, \dots, x_n) \mapsto (\floor{x_1}, \dots, \floor{x_n});" title="floor : \R^n \to \Z^n, \quad (x_1, \dots, x_n) \mapsto (\floor{x_1}, \dots, \floor{x_n});">
</div>
<p>
if <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="15" src="https://math.fontein.de/formulae/J8WZLQNTj4hsLiLptic9ZbHN6du478Qep2spzQ.svgz" alt="\Lambda \subseteq \Z^n" title="\Lambda \subseteq \Z^n"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="42" height="16" src="https://math.fontein.de/formulae/Q8X_7IMZ1ugMeaxcLpoOFcNVOGvObyos0zkHdw.svgz" alt="floor" title="floor"></span> induces a map <span class="inline-formula"><img class="img-inline-formula img-formula" width="114" height="18" src="https://math.fontein.de/formulae/ezDX1zs_B0Oxw1nxBFaxeD4yWQHXmYEtZpA1BA.svgz" alt="\R^n/\Lambda \to \Z^n/\Lambda" title="\R^n/\Lambda \to \Z^n/\Lambda"></span>. Now, that <span class="inline-formula"><img class="img-inline-formula img-formula" width="26" height="12" src="https://math.fontein.de/formulae/x9uDaMcRWmFnhjgyf0CLBlJpQRUKxXbuUM0.1Q.svgz" alt="red" title="red"></span> does not depends on fractions simply means that <span class="inline-formula"><img class="img-inline-formula img-formula" width="26" height="12" src="https://math.fontein.de/formulae/x9uDaMcRWmFnhjgyf0CLBlJpQRUKxXbuUM0.1Q.svgz" alt="red" title="red"></span> factors through <span class="inline-formula"><img class="img-inline-formula img-formula" width="42" height="16" src="https://math.fontein.de/formulae/Q8X_7IMZ1ugMeaxcLpoOFcNVOGvObyos0zkHdw.svgz" alt="floor" title="floor"></span>, i.e. that we can write <span class="inline-formula"><img class="img-inline-formula img-formula" width="139" height="17" src="https://math.fontein.de/formulae/CeSRWK1TFxKHHlvAweaVdHlbiJTGcoz_Djhzew.svgz" alt="red = red' \circ floor" title="red = red' \circ floor"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="132" height="18" src="https://math.fontein.de/formulae/PBjkn_5kwQiJO8emTZMeiVo0MwIjRFS52ww5jg.svgz" alt="red' : \Z^n/\Lambda \to X" title="red' : \Z^n/\Lambda \to X"></span>.
</p>
<p>
Moreover, if in the following we specify discrete infrastructures, we often just define <span class="inline-formula"><img class="img-inline-formula img-formula" width="26" height="12" src="https://math.fontein.de/formulae/x9uDaMcRWmFnhjgyf0CLBlJpQRUKxXbuUM0.1Q.svgz" alt="red" title="red"></span> for values in <span class="inline-formula"><img class="img-inline-formula img-formula" width="43" height="18" src="https://math.fontein.de/formulae/IoA0BuM80DWvbyrrE5zoooQiZuyPkzKWW0aUjw.svgz" alt="\Z^n/\Lambda" title="\Z^n/\Lambda"></span>. In that case, for elements <span class="inline-formula"><img class="img-inline-formula img-formula" width="134" height="18" src="https://math.fontein.de/formulae/SqfdPeerp6a_.odVZ628BM6KQn5o2H143I7cjg.svgz" alt="v \in \R^n/\Lambda \setminus \Z^n/\Lambda" title="v \in \R^n/\Lambda \setminus \Z^n/\Lambda"></span>, define <span class="inline-formula"><img class="img-inline-formula img-formula" width="182" height="18" src="https://math.fontein.de/formulae/xNh3ezpEzYlYodQEtb_z0.wTSgDEPHvueO.HKw.svgz" alt="red(v) := red(floor(v))" title="red(v) := red(floor(v))"></span>.
</p>
<p>
In case <span class="inline-formula"><img class="img-inline-formula img-formula" width="80" height="18" src="https://math.fontein.de/formulae/Q3Wi7WBtdKSnv1gaLvYnlftQgM.JrmfkFL4zUA.svgz" alt="(X, d, red)" title="(X, d, red)"></span> is discrete, consider the subset
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="464" height="20" src="https://math.fontein.de/formulae/YM76NrqV.ThBKY_h_GfwB6buGvQ5.8umAg9MKQ.svgz" alt="\fRep_{disc} := \fRep_{disc}(X, d, red) := \{ (x, f) \in \fRep \mid f \in \Z^n \}." title="\fRep_{disc} := \fRep_{disc}(X, d, red) := \{ (x, f) \in \fRep \mid f \in \Z^n \}.">
</div>
<p>
Then
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="224" height="23" src="https://math.fontein.de/formulae/tcrxpNF0qHVEFMEuPhG3SqFRVcB_Delchx2yGw.svgz" alt="\Psi|_{\fRep_{disc}} : \fRep_{disc} \to \Z^n/\Lambda" title="\Psi|_{\fRep_{disc}} : \fRep_{disc} \to \Z^n/\Lambda">
</div>
<p>
is an isomorphism.
</p>
</div>
</div>
<div class="subsection-block">
<h4>
Finite Abelian Groups as Infrastructures.
</h4>
<div>
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/1SEBa2mo2npdO9VAigPQfAPRoqEDUsQLVvcYYg.svgz" alt="G" title="G"></span> be a finite abelian group, generated by <span class="inline-formula"><img class="img-inline-formula img-formula" width="74" height="11" src="https://math.fontein.de/formulae/Fu0_nlKIc0UUBt4FnNjm2anOMMT5GEx2bVAbSA.svgz" alt="g_1, \dots, g_n" title="g_1, \dots, g_n"></span>. Consider the <em>relation lattice</em> <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="15" src="https://math.fontein.de/formulae/J8WZLQNTj4hsLiLptic9ZbHN6du478Qep2spzQ.svgz" alt="\Lambda \subseteq \Z^n" title="\Lambda \subseteq \Z^n"></span> for <span class="inline-formula"><img class="img-inline-formula img-formula" width="74" height="11" src="https://math.fontein.de/formulae/Fu0_nlKIc0UUBt4FnNjm2anOMMT5GEx2bVAbSA.svgz" alt="g_1, \dots, g_n" title="g_1, \dots, g_n"></span>, defined by
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="235" height="52" src="https://math.fontein.de/formulae/UYMavUU63zLuPsqTjsoh87WUDSEexQRlKibJRQ.svgz" alt="(v_1, \dots, v_n) \in \Lambda \Leftrightarrow \prod_{i=1}^n g_i^{v_i} = 1." title="(v_1, \dots, v_n) \in \Lambda \Leftrightarrow \prod_{i=1}^n g_i^{v_i} = 1.">
</div>
<p>
Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/XY_2LK590TGO0.jb0ZbkmaiBi4kJr1xvp52q2g.svgz" alt="\Lambda" title="\Lambda"></span> is the kernel of <span class="inline-formula"><img class="img-inline-formula img-formula" width="63" height="12" src="https://math.fontein.de/formulae/OvmVVs0xwElKEGt057JVsho6qkpQM5sKP5N14g.svgz" alt="\Z^n \to G" title="\Z^n \to G"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="182" height="20" src="https://math.fontein.de/formulae/BICwYv67dMLULMbbQUwKQKl28GMRLtlEdGZQkw.svgz" alt="(v_1, \dots, v_n) \mapsto \prod_{i=1}^n g_i^{v_i}" title="(v_1, \dots, v_n) \mapsto \prod_{i=1}^n g_i^{v_i}"></span>, and
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="334" height="52" src="https://math.fontein.de/formulae/vNYGsJMYcKofWj_bu5_0DxQPGPX_d7x9oZI2Qw.svgz" alt="\varphi : \Z^n/\Lambda \to G, \quad (v_1, \dots, v_n) + \Lambda \mapsto \prod_{i=1}^n g_i^{v_i}" title="\varphi : \Z^n/\Lambda \to G, \quad (v_1, \dots, v_n) + \Lambda \mapsto \prod_{i=1}^n g_i^{v_i}">
</div>
<p>
is a group isomorphism. Define <span class="inline-formula"><img class="img-inline-formula img-formula" width="59" height="12" src="https://math.fontein.de/formulae/ub9MldwOl6ws9FT1tGUr_i5MqFAjBYd_m7_07g.svgz" alt="X := G" title="X := G"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="68" height="18" src="https://math.fontein.de/formulae/JhMNWD62vva7slnEGXf4.OAgqL8Iy6LwHgfcZQ.svgz" alt="d := \varphi^{-1}" title="d := \varphi^{-1}"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="66" height="16" src="https://math.fontein.de/formulae/jxsJUomgXcUIpnv7cESAhTunDKoE48PiyW5m1w.svgz" alt="red := \varphi" title="red := \varphi"></span> (or, more precisely, <span class="inline-formula"><img class="img-inline-formula img-formula" width="125" height="16" src="https://math.fontein.de/formulae/vt9NvN1f2qmwPBK1Bl3MXr91ivZV8FT6ypsx8w.svgz" alt="red := \varphi \circ floor" title="red := \varphi \circ floor"></span>); then <span class="inline-formula"><img class="img-inline-formula img-formula" width="80" height="18" src="https://math.fontein.de/formulae/Q3Wi7WBtdKSnv1gaLvYnlftQgM.JrmfkFL4zUA.svgz" alt="(X, d, red)" title="(X, d, red)"></span> is an <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span>-dimensional infrastructure. Moreover, for <span class="inline-formula"><img class="img-inline-formula img-formula" width="67" height="17" src="https://math.fontein.de/formulae/viaMtbvDfwHuEUdUVjYivnoijvqaRVIdjhQBPA.svgz" alt="g, g' \in G" title="g, g' \in G"></span>, we have <span class="inline-formula"><img class="img-inline-formula img-formula" width="107" height="18" src="https://math.fontein.de/formulae/zDjZNIktpEwn5gRaOzktD8bNttmPlir0QI4JGQ.svgz" alt="\gs(g, g') = g g'" title="\gs(g, g') = g g'"></span>, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="11" src="https://math.fontein.de/formulae/oALhjAUt95L9B1GjlImGOqnVYMSkaj0JkVGFrg.svgz" alt="\gs" title="\gs"></span> equals the group operation of <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/1SEBa2mo2npdO9VAigPQfAPRoqEDUsQLVvcYYg.svgz" alt="G" title="G"></span>. Hence, every finite abelian group can be seen in a natural way as an infrastructure.
</p>
<p>
Moreover, this shows that <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="12" src="https://math.fontein.de/formulae/DxlmDzniFM_09XzQnKCoRu8h2JBxVd5JriHzmg.svgz" alt="d" title="d"></span> can be thought of as an analogue to the discrete logarithm map, and <span class="inline-formula"><img class="img-inline-formula img-formula" width="26" height="12" src="https://math.fontein.de/formulae/x9uDaMcRWmFnhjgyf0CLBlJpQRUKxXbuUM0.1Q.svgz" alt="red" title="red"></span> is an analogue of the power map <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="15" src="https://math.fontein.de/formulae/qej5hFjgd7bFGsmrDphlLl5M_iRT2UuwuRhZmA.svgz" alt="n \mapsto g^n" title="n \mapsto g^n"></span>. In particular, we obtained the goal described in <a href="https://math.fontein.de/2009/07/20/the-discrete-logarithm-problem-and-generalizations/">the first post of this series</a>: we found a generalization of the discrete logarithm problem to a non-associative algebraic structure. In the next post, I will how such infrastructures can be obtained from global fields; this gives a rich source of examples for <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span>-dimensional infrastructures.
</p>
</div>
</div>
<div class="subsection-block">
<h4>
What about baby steps?
</h4>
<div>
<p>
Note that in the above discussion, I simply ignored baby steps. In the one-dimensional case, <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/eCr2xk03b8XPN3FP16iK5HS4b8Ql_90EmVknKw.svgz" alt="\R" title="\R"></span> has a canonical direction (namely the positive one) and so has <span class="inline-formula"><img class="img-inline-formula img-formula" width="47" height="18" src="https://math.fontein.de/formulae/Blo1FniNoktf2EKQy0CWZxv_r_wmqWWcM_Pc9A.svgz" alt="\R/R\Z" title="\R/R\Z"></span>, whence saying “go to the next element” makes sense. Opposed to that, in <span class="inline-formula"><img class="img-inline-formula img-formula" width="22" height="12" src="https://math.fontein.de/formulae/bZVXuWhECmRV9pmTW_AgHZeLTOYax7svHPDCxQ.svgz" alt="\R^n" title="\R^n"></span>, there are infinitely many directions, no one better than another. Even if we fix a direction, “go to the next element in that direction” seems to not really make sense. So far, I have not seen any definition of baby steps in this case which works for <em>all</em> <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span>-dimensional infrastructures.
</p>
<p>
Note that in the case of infrastructures obtained from global fields, one has some kind of canonical baby steps (even though there are still some choices left). In fact, there are <span class="inline-formula"><img class="img-inline-formula img-formula" width="41" height="13" src="https://math.fontein.de/formulae/fgRY_0KZ1v9Dol4ARGUMspWZdQu629FDAEQ9MA.svgz" alt="n + 1" title="n + 1"></span> of them. To define them, though, one needs more information than just <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="12" src="https://math.fontein.de/formulae/DxlmDzniFM_09XzQnKCoRu8h2JBxVd5JriHzmg.svgz" alt="d" title="d"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="26" height="12" src="https://math.fontein.de/formulae/x9uDaMcRWmFnhjgyf0CLBlJpQRUKxXbuUM0.1Q.svgz" alt="red" title="red"></span>: one needs information about a <span class="inline-formula"><img class="img-inline-formula img-formula" width="55" height="18" src="https://math.fontein.de/formulae/AjCyaw2HagCBfGhDL8_wwlnYMLlETAC11TNcKw.svgz" alt="(n + 1)" title="(n + 1)"></span>-st dimension, both for constructing the reduction map and for defining baby steps.
</p>
</div>
</div>
</div>
Interpreting One-dimensional Infrastructures as Groups: f-Representations.
https://math.fontein.de/2009/07/20/interpreting-one-dimensional-infrastructures-as-groups-f-representations/
2009-07-20T03:46:16+02:00
2009-07-20T03:46:16+02:00
Felix Fontein
<div>
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="18" src="https://math.fontein.de/formulae/jIn.uZYNabR16.oruR4TBCDUy7VpBBZ9LQZ0OQ.svgz" alt="(X, d)" title="(X, d)"></span> be a <a href="https://math.fontein.de/2009/07/20/one-dimensional-infrastructures/">one-dimensional infrastructure</a> of circumference <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="13" src="https://math.fontein.de/formulae/3rzByTx4xORnQaqA2WwgTLGv7jy1L9T1w3rfrA.svgz" alt="R > 0" title="R > 0"></span>. We have seen that we obtain two operations <span class="inline-formula"><img class="img-inline-formula img-formula" width="91" height="12" src="https://math.fontein.de/formulae/9TS8oPy0xMOhXLq6knO8ZhLeq2sRv6fgaG07og.svgz" alt="\bs : X \to X" title="\bs : X \to X"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="128" height="16" src="https://math.fontein.de/formulae/G4CeG3K_JpkwrP6tNiBT9fA26inh2Lnw7NcM2A.svgz" alt="\gs : X \times X \to X" title="\gs : X \times X \to X"></span> together with a reduction map <span class="inline-formula"><img class="img-inline-formula img-formula" width="131" height="18" src="https://math.fontein.de/formulae/7uVOvafbYDZ.azBEIJ99WwgR2fMwW92fZH9vcw.svgz" alt="red : \R/R\Z \to X" title="red : \R/R\Z \to X"></span>, and we have <span class="inline-formula"><img class="img-inline-formula img-formula" width="219" height="18" src="https://math.fontein.de/formulae/qyWzkkBZtH72MecTwkH0ybGdQQiUhO8ieiV_uQ.svgz" alt="\gs(x, x') = red(d(x) + d(x'))" title="\gs(x, x') = red(d(x) + d(x'))"></span> for <span class="inline-formula"><img class="img-inline-formula img-formula" width="71" height="17" src="https://math.fontein.de/formulae/6xyNyeYpuoPTNTVtwZOf4vkY0hV1tOW9JbrSMA.svgz" alt="x, x' \in X" title="x, x' \in X"></span>. This gives us a binary operation which is in general not associative. For several reasons, it would be interesting to embed the infrastructure into a group. An obvious choice for such a group would be <span class="inline-formula"><img class="img-inline-formula img-formula" width="47" height="18" src="https://math.fontein.de/formulae/Blo1FniNoktf2EKQy0CWZxv_r_wmqWWcM_Pc9A.svgz" alt="\R/R\Z" title="\R/R\Z"></span>, as we can identify <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span> as a subset of <span class="inline-formula"><img class="img-inline-formula img-formula" width="47" height="18" src="https://math.fontein.de/formulae/Blo1FniNoktf2EKQy0CWZxv_r_wmqWWcM_Pc9A.svgz" alt="\R/R\Z" title="\R/R\Z"></span> by identifying it with <span class="inline-formula"><img class="img-inline-formula img-formula" width="39" height="18" src="https://math.fontein.de/formulae/Sj0POmFJuFR.MovanqxlnUHsUhGBDR2XEk0PPA.svgz" alt="d(X)" title="d(X)"></span>. Such an interpretation of the infrastructure as part of a “circular group” has first been considered by Hendrik Lenstra in his 1980 paper <em>On the computation of regulators and class numbers of quadratic fields</em>.
</p>
<p>
The idea is to consider pairs <span class="inline-formula"><img class="img-inline-formula img-formula" width="114" height="18" src="https://math.fontein.de/formulae/Io.wTDjp8c.2yG8sTS_yrfV3nU7FjCeopga6zg.svgz" alt="(x, f) \in X \times \R" title="(x, f) \in X \times \R"></span>; the map <span class="inline-formula"><img class="img-inline-formula img-formula" width="149" height="18" src="https://math.fontein.de/formulae/gi2nz3UDDBifIglu2RIZcsmqROIH.7F5YIT6mg.svgz" alt="d : X \times \R \to \R/R\Z" title="d : X \times \R \to \R/R\Z"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="135" height="18" src="https://math.fontein.de/formulae/v_ZAstfoak7ltp_vcEQotQXzPdgmrYLYSI.9NA.svgz" alt="(x, f) \mapsto d(x) + f" title="(x, f) \mapsto d(x) + f"></span> is obviously surjective, but far from being injective. Hence, it would be a good idea to restrict <span class="inline-formula"><img class="img-inline-formula img-formula" width="51" height="14" src="https://math.fontein.de/formulae/Xmclc81TOBSnKhIH_.k2uHdhgIjxVx6DyPIjCg.svgz" alt="X \times \R" title="X \times \R"></span> to a subset which makes this map both injective and surjective. We want this subset to contain <span class="inline-formula"><img class="img-inline-formula img-formula" width="64" height="18" src="https://math.fontein.de/formulae/OlFcMlh_Fsh.8H3cjlrvoanyu0Jv2ZGXqMBE1g.svgz" alt="X \times \{ 0 \}" title="X \times \{ 0 \}"></span>; note that <span class="inline-formula"><img class="img-inline-formula img-formula" width="214" height="20" src="https://math.fontein.de/formulae/IxOw61yH1tUcscQF3R0T3JmnZeCAT15YVlUJJw.svgz" alt="d|_{X \times \{ 0 \}} : X \times \{ 0 \} \to \R/R\Z" title="d|_{X \times \{ 0 \}} : X \times \{ 0 \} \to \R/R\Z"></span> is injective and essentially equals <span class="inline-formula"><img class="img-inline-formula img-formula" width="115" height="18" src="https://math.fontein.de/formulae/nG5tOQuC_AiH8MK1kd7kHlmnr7XkfyU3Es3Tew.svgz" alt="d : X \to \R/R\Z" title="d : X \to \R/R\Z"></span> by the identification <span class="inline-formula"><img class="img-inline-formula img-formula" width="108" height="18" src="https://math.fontein.de/formulae/e6r4Zg3rbnFCGfvjoGdUvYiLlKrjqRqGwnWDaA.svgz" alt="X \leftrightarrow X \times \{ 0 \}" title="X \leftrightarrow X \times \{ 0 \}"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="78" height="18" src="https://math.fontein.de/formulae/gbsQjMngL8V5PxsPHBbtRJWWgkOUFZ0dui6l8g.svgz" alt="x \mapsto (x, 0)" title="x \mapsto (x, 0)"></span>. We obtain the following definition:
</p>
<div class="theorem-environment theorem-definition-environment">
<div class="theorem-header theorem-definition-header">
Definition.
</div>
<div class="theorem-content theorem-definition-content">
<p>
A set of <em><span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/M.ji_f0zsVnTLyaKegBkRJLOeqrrt6brNnapOQ.svgz" alt="f" title="f"></span>-representations</em> of <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="18" src="https://math.fontein.de/formulae/jIn.uZYNabR16.oruR4TBCDUy7VpBBZ9LQZ0OQ.svgz" alt="(X, d)" title="(X, d)"></span> is a set <span class="inline-formula"><img class="img-inline-formula img-formula" width="113" height="18" src="https://math.fontein.de/formulae/JLk9azd7c8rxFn73o3l86x7y1ha6sHyb04svdg.svgz" alt="\fRep \subseteq X \times \R" title="\fRep \subseteq X \times \R"></span> such that <span class="inline-formula"><img class="img-inline-formula img-formula" width="174" height="21" src="https://math.fontein.de/formulae/_Iv5CzWlp.PnSbBJ91M_sCuCRz9pAhhDNUNPZA.svgz" alt="d|_{\fRep} : \fRep \to \R/R\Z" title="d|_{\fRep} : \fRep \to \R/R\Z"></span> is bijective and such that <span class="inline-formula"><img class="img-inline-formula img-formula" width="127" height="19" src="https://math.fontein.de/formulae/qpR3O1Idfi976xJWT4kVN9mB.kHLend4qlAK9Q.svgz" alt="X \times \{ 0 \} \subseteq \fRep" title="X \times \{ 0 \} \subseteq \fRep"></span>.
</p>
</div>
</div>
<p>
But how to obtain such a subset? In fact, we already have all ingredients ready, as the following proposition shows, by relating <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/M.ji_f0zsVnTLyaKegBkRJLOeqrrt6brNnapOQ.svgz" alt="f" title="f"></span>-representations to reduction maps:
</p>
<div class="theorem-environment theorem-theorem-environment qed">
<div class="theorem-header theorem-theorem-header">
Theorem.
</div>
<div class="theorem-content theorem-theorem-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="18" src="https://math.fontein.de/formulae/jIn.uZYNabR16.oruR4TBCDUy7VpBBZ9LQZ0OQ.svgz" alt="(X, d)" title="(X, d)"></span> be a one-dimensional infrastructure. Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/AdTxuIawp9Z_8j4FFnLDOtejF7gGMlzudPQwOA.svgz" alt="A" title="A"></span> be the set of reduction maps and <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/Ih7V6Mp4twAjwq7Tr86Dt9U5c5VQJd0XWRPlqw.svgz" alt="B" title="B"></span> be the set of <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/M.ji_f0zsVnTLyaKegBkRJLOeqrrt6brNnapOQ.svgz" alt="f" title="f"></span>-representations for <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="18" src="https://math.fontein.de/formulae/jIn.uZYNabR16.oruR4TBCDUy7VpBBZ9LQZ0OQ.svgz" alt="(X, d)" title="(X, d)"></span>. The map
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="448" height="18" src="https://math.fontein.de/formulae/LZHF1H5EtmrYp1IH_po.0l6_Yz6BW__LsChA6g.svgz" alt="\Phi : A \to B, \quad red \mapsto \{ (x, f) \in X \times \R \mid red(d(x) + f) = x \}" title="\Phi : A \to B, \quad red \mapsto \{ (x, f) \in X \times \R \mid red(d(x) + f) = x \}">
</div>
<p>
is a bijection, with its inverse given by
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="294" height="22" src="https://math.fontein.de/formulae/T7ds5nXuYRNeEUuWosW2yGskMZAqAtGX1FTw1A.svgz" alt="\Psi : B \to A, \quad \fRep \mapsto \pi_1 \circ (d|_{\fRep})^{-1}," title="\Psi : B \to A, \quad \fRep \mapsto \pi_1 \circ (d|_{\fRep})^{-1},">
</div>
<p>
where <span class="inline-formula"><img class="img-inline-formula img-formula" width="127" height="15" src="https://math.fontein.de/formulae/JDGCu8MByBzzSdc8jILrpfv_21cSwOgqgd7dEg.svgz" alt="\pi_1 : X \times \R \to X" title="\pi_1 : X \times \R \to X"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="80" height="18" src="https://math.fontein.de/formulae/AslAIEgabJjx8Uj8TEQI88jQtqRdCzexdfPFQg.svgz" alt="(x, f) \mapsto x" title="(x, f) \mapsto x"></span> is the projection onto the first component.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
Now fix one corresponding choice of <span class="inline-formula"><img class="img-inline-formula img-formula" width="26" height="12" src="https://math.fontein.de/formulae/x9uDaMcRWmFnhjgyf0CLBlJpQRUKxXbuUM0.1Q.svgz" alt="red" title="red"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="39" height="18" src="https://math.fontein.de/formulae/ZdLndZ0gc4MIQeEo5cZ_OlfT.ZXD_Iwao.uGOw.svgz" alt="\fRep" title="\fRep"></span>, and let <span class="inline-formula"><img class="img-inline-formula img-formula" width="140" height="19" src="https://math.fontein.de/formulae/pLudFkfwD8DvRqQGf1KzuGrsd3ogAn5ZLuD84Q.svgz" alt="\psi : \fRep \to \R/R\Z" title="\psi : \fRep \to \R/R\Z"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="210" height="18" src="https://math.fontein.de/formulae/ThMeDyzNpo9NIqCxR8emuFrmygOLnfZsu1fO1g.svgz" alt="(x, f) \mapsto d(x, f) = d(x) + f" title="(x, f) \mapsto d(x, f) = d(x) + f"></span> be the associated bijection. Then
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="454" height="20" src="https://math.fontein.de/formulae/YGVYC_fBSrvaNkGH.FIUyMdLlSi3dWqtA1MFPg.svgz" alt="\gs(x, x') = red(d(x) + d(x')) = \pi_1(\psi^{-1}(\psi(x, 0) + \psi(x', 0)))" title="\gs(x, x') = red(d(x) + d(x')) = \pi_1(\psi^{-1}(\psi(x, 0) + \psi(x', 0)))">
</div>
<p>
for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="71" height="17" src="https://math.fontein.de/formulae/6xyNyeYpuoPTNTVtwZOf4vkY0hV1tOW9JbrSMA.svgz" alt="x, x' \in X" title="x, x' \in X"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="113" height="18" src="https://math.fontein.de/formulae/zWIeYkakh26hWK0tkuYLvsALguVRq5F2AjpQHg.svgz" alt="\pi_1 : (x, f) \mapsto x" title="\pi_1 : (x, f) \mapsto x"></span> being the projection. Moreover, using the bijection <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="16" src="https://math.fontein.de/formulae/5YQhZpth_qEZfoZavf10MlpXOOEYGw_G8IF6hw.svgz" alt="\psi" title="\psi"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="39" height="18" src="https://math.fontein.de/formulae/ZdLndZ0gc4MIQeEo5cZ_OlfT.ZXD_Iwao.uGOw.svgz" alt="\fRep" title="\fRep"></span> becomes a group which we will write additively by <span class="inline-formula"><img class="img-inline-formula img-formula" width="330" height="19" src="https://math.fontein.de/formulae/AhIGcrEO81dHVoEezQN8G8NKT5Ut4_YI1sy_RA.svgz" alt="(x, f) + (x', f') := \psi^{-1}(\psi(x, f) + \psi(x', f'))" title="(x, f) + (x', f') := \psi^{-1}(\psi(x, f) + \psi(x', f'))"></span>, which extends the giant steps.
</p>
<p>
Finally, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="26" height="12" src="https://math.fontein.de/formulae/x9uDaMcRWmFnhjgyf0CLBlJpQRUKxXbuUM0.1Q.svgz" alt="red" title="red"></span> be the map we originally defined for infrastructures, i.e. <span class="inline-formula"><img class="img-inline-formula img-formula" width="408" height="19" src="https://math.fontein.de/formulae/i7CtrahQdZCnw7kZSPwudK_mJFruNYrpSQnr8g.svgz" alt="red(r) = d^{-1}(r - \min\{ f \in \R \mid f \ge 0, \; r - f \in d(X) \})" title="red(r) = d^{-1}(r - \min\{ f \in \R \mid f \ge 0, \; r - f \in d(X) \})"></span> for <span class="inline-formula"><img class="img-inline-formula img-formula" width="77" height="18" src="https://math.fontein.de/formulae/mj2kbkiwypGaXg.XiN3goFRxVzHCZq.bCSDT_w.svgz" alt="r \in \R/R\Z" title="r \in \R/R\Z"></span>. In that case, one can compute the group operation on <span class="inline-formula"><img class="img-inline-formula img-formula" width="39" height="18" src="https://math.fontein.de/formulae/ZdLndZ0gc4MIQeEo5cZ_OlfT.ZXD_Iwao.uGOw.svgz" alt="\fRep" title="\fRep"></span> without having to evalute <span class="inline-formula"><img class="img-inline-formula img-formula" width="31" height="18" src="https://math.fontein.de/formulae/UKfY5rSkxqQgOBUaEcS_3PIx.zir93Cm77GQeQ.svgz" alt="\psi^{-1}" title="\psi^{-1}"></span> or <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="12" src="https://math.fontein.de/formulae/DxlmDzniFM_09XzQnKCoRu8h2JBxVd5JriHzmg.svgz" alt="d" title="d"></span>, with only computing baby and giant steps and relative distances, i.e. <span class="inline-formula"><img class="img-inline-formula img-formula" width="151" height="18" src="https://math.fontein.de/formulae/z6J4jyXCFjpFuWIcONZWa1mezkhMaHMzdq0obQ.svgz" alt="d(\bs(x)) - d(x) \ge 0" title="d(\bs(x)) - d(x) \ge 0"></span> resp. <span class="inline-formula"><img class="img-inline-formula img-formula" width="233" height="18" src="https://math.fontein.de/formulae/HDPAjvMe3wjtscMSN9QHDCgCEXn1QJrqc6SHSA.svgz" alt="d(x) + d(x') - d(\gs(x, x')) \ge 0" title="d(x) + d(x') - d(\gs(x, x')) \ge 0"></span> for <span class="inline-formula"><img class="img-inline-formula img-formula" width="71" height="17" src="https://math.fontein.de/formulae/6xyNyeYpuoPTNTVtwZOf4vkY0hV1tOW9JbrSMA.svgz" alt="x, x' \in X" title="x, x' \in X"></span>:
</p>
<div class="theorem-environment theorem-theorem-environment qed">
<div class="theorem-header theorem-theorem-header">
Theorem.
</div>
<div class="theorem-content theorem-theorem-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="296" height="18" src="https://math.fontein.de/formulae/444DA5wDuZpW_HPDa_EJdjJAt33mr02ljvH_JA.svgz" alt="D_{\min} := \min\{ d(\bs(x)) - d(x) \mid x \in X \}" title="D_{\min} := \min\{ d(\bs(x)) - d(x) \mid x \in X \}"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="302" height="18" src="https://math.fontein.de/formulae/UxZ2D85B7U5sIdb4DobJz.KR4eTD5vY1L4JhuQ.svgz" alt="D_{\max} := \max\{ d(\bs(x)) - d(x) \mid x \in X \}" title="D_{\max} := \max\{ d(\bs(x)) - d(x) \mid x \in X \}"></span>, and let <span class="inline-formula"><img class="img-inline-formula img-formula" width="163" height="19" src="https://math.fontein.de/formulae/tXIG4Pom18JL2fTWdPR6cgeMQzdmz_GluaIOdg.svgz" alt="(x, f), (x', f') \in \fRep" title="(x, f), (x', f') \in \fRep"></span>. Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="116" height="18" src="https://math.fontein.de/formulae/siNBg5oZBRNK6P8ougkHtHpQnrdzM4FcZn8jAg.svgz" alt="(x, f) + (x', f')" title="(x, f) + (x', f')"></span> can be computed with one giant step computation and at most <span class="inline-formula"><img class="img-inline-formula img-formula" width="65" height="32" src="https://math.fontein.de/formulae/jJjE.v5hAfLPOfIBDsHO9CeJYI4k0yiUWzQpJA.svgz" alt="\ceil{\frac{3 D_{\max}}{D_{\min}}}" title="\ceil{\frac{3 D_{\max}}{D_{\min}}}"></span> baby step computations as follows:
</p>
<ol class="enum-level-1">
<li>Compute <span class="inline-formula"><img class="img-inline-formula img-formula" width="110" height="18" src="https://math.fontein.de/formulae/D6eoDJviw6kyRYqGF8vZ_sG3fKVDBCALU11ZfQ.svgz" alt="x'' := \gs(x, x')" title="x'' := \gs(x, x')"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="332" height="18" src="https://math.fontein.de/formulae/7EOcmptRXW.HK.vZSB6HWl9V8B0SBzoPXHzyBA.svgz" alt="f'' := f + f' + ( d(x) + d(x') - d(\gs(x, x')) )" title="f'' := f + f' + ( d(x) + d(x') - d(\gs(x, x')) )"></span>.</li>
<li>Compute <span class="inline-formula"><img class="img-inline-formula img-formula" width="102" height="18" src="https://math.fontein.de/formulae/YvHpXutWo9a2aBJB2t4Tgc.UQT_QgQ3JoC_ptA.svgz" alt="x''' := \bs(x'')" title="x''' := \bs(x'')"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="217" height="18" src="https://math.fontein.de/formulae/C.xcNI7DXBfKUOPlzH007byHHcxs7VTgmRH4xQ.svgz" alt="f''' := f'' - ( d(x''') - d(x'') )" title="f''' := f'' - ( d(x''') - d(x'') )"></span>.</li>
<li>If <span class="inline-formula"><img class="img-inline-formula img-formula" width="56" height="17" src="https://math.fontein.de/formulae/maUzh7in2CMSkh51cw31o9l8qWhB8UG3PPOJ_g.svgz" alt="f''' \ge 0" title="f''' \ge 0"></span>, set <span class="inline-formula"><img class="img-inline-formula img-formula" width="158" height="18" src="https://math.fontein.de/formulae/OlSIs1CxYNexwAGAKerF5sNST5.FOgNhGixKsw.svgz" alt="(x'', f'') := (x''', f''')" title="(x'', f'') := (x''', f''')"></span> and go to step 2.</li>
<li>Return the pair <span class="inline-formula"><img class="img-inline-formula img-formula" width="60" height="18" src="https://math.fontein.de/formulae/cs4DHmcL9DEY7FAXwpnRT3zeQNWZDoZP3x_4.A.svgz" alt="(x'', f'')" title="(x'', f'')"></span>.</li>
</ol>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
In the case of infrastructures obtained from global fields, we are also able to describe the group operation in <span class="inline-formula"><img class="img-inline-formula img-formula" width="39" height="18" src="https://math.fontein.de/formulae/ZdLndZ0gc4MIQeEo5cZ_OlfT.ZXD_Iwao.uGOw.svgz" alt="\fRep" title="\fRep"></span> without having to evaluate <span class="inline-formula"><img class="img-inline-formula img-formula" width="31" height="18" src="https://math.fontein.de/formulae/UKfY5rSkxqQgOBUaEcS_3PIx.zir93Cm77GQeQ.svgz" alt="\psi^{-1}" title="\psi^{-1}"></span> or <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="12" src="https://math.fontein.de/formulae/DxlmDzniFM_09XzQnKCoRu8h2JBxVd5JriHzmg.svgz" alt="d" title="d"></span>. In fact, evaluating <span class="inline-formula"><img class="img-inline-formula img-formula" width="31" height="18" src="https://math.fontein.de/formulae/UKfY5rSkxqQgOBUaEcS_3PIx.zir93Cm77GQeQ.svgz" alt="\psi^{-1}" title="\psi^{-1}"></span> or <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="12" src="https://math.fontein.de/formulae/DxlmDzniFM_09XzQnKCoRu8h2JBxVd5JriHzmg.svgz" alt="d" title="d"></span> is essentially the discrete logarithm problem, for which so far no general polynomial methods for solving it exist in the case of global fields.
</p>
</div>
One-dimensional Infrastructures.
https://math.fontein.de/2009/07/20/one-dimensional-infrastructures/
2009-07-20T03:45:16+02:00
2009-07-20T03:45:16+02:00
Felix Fontein
<div>
<div>
<p>
One-dimensional <a href="https://math.fontein.de/2009/07/20/the-discrete-logarithm-problem-and-generalizations/">infrastructures</a> first appeared in the 1970's in <a href="https://en.wikipedia.org/wiki/Daniel_Shanks">Daniel Shanks</a>' work on real <a href="https://en.wikipedia.org/wiki/Quadratic_field">quadratic number fields</a> <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="21" src="https://math.fontein.de/formulae/EltK2sKQFKn4YqNRyjBDD_dgDAMcKHePNMbcsw.svgz" alt="\Q(\sqrt{D})" title="\Q(\sqrt{D})"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="13" src="https://math.fontein.de/formulae/Z79FFiacv4NkY8kdF4WxDId6gY1aL5OvP20P1g.svgz" alt="D > 1" title="D > 1"></span> a squarefree integer, when he tried to fasten <a href="https://en.wikipedia.org/wiki/Dirichlet%27s_unit_theorem">regulator computations</a>. The previous algorithms used <a href="https://en.wikipedia.org/wiki/Continued_fraction">continued fraction expansion</a> to obtain the regulator in <span class="inline-formula"><img class="img-inline-formula img-formula" width="83" height="20" src="https://math.fontein.de/formulae/ARt0q92lqMsx5YyEhFCJXR83kGQRTgPVTfetuw.svgz" alt="\calO(D^{1/2 + \varepsilon})" title="\calO(D^{1/2 + \varepsilon})"></span> binary operation, <span class="inline-formula"><img class="img-inline-formula img-formula" width="41" height="12" src="https://math.fontein.de/formulae/ly6Ap7KW4SRUC02VaqDollTKyMXjq8.lKwPx8g.svgz" alt="\varepsilon > 0" title="\varepsilon > 0"></span> arbitrary. Shanks found out that one can obtain a multiplication like operation, which he dubbed <em>giant steps</em>, as opposed to the <em>baby steps</em> taken by one step in the continued fraction expansion. He described a <a href="https://en.wikipedia.org/wiki/Baby-step_giant-step">baby step-giant step method</a> to compute the regulator in <span class="inline-formula"><img class="img-inline-formula img-formula" width="83" height="20" src="https://math.fontein.de/formulae/y0kAotp.Y.33xp333Es1Z3yIRCRaaVQHdXqEfg.svgz" alt="\calO(D^{1/4 + \varepsilon})" title="\calO(D^{1/4 + \varepsilon})"></span> binary operations, requiring <span class="inline-formula"><img class="img-inline-formula img-formula" width="83" height="20" src="https://math.fontein.de/formulae/y0kAotp.Y.33xp333Es1Z3yIRCRaaVQHdXqEfg.svgz" alt="\calO(D^{1/4 + \varepsilon})" title="\calO(D^{1/4 + \varepsilon})"></span> bytes of storage. His methods were analysed, written up more clearly and extended by various people, including <a href="https://en.wikipedia.org/wiki/Hendrik_Lenstra">Hendrik Lenstra</a>, Hugh Williams, <a href="https://de.wikipedia.org/wiki/Johannes_Buchmann">Johannes Buchmann</a>, Rene Schoof, and many others. Extensions of the method to function fields exist as well, most notably due to the work of Andreas Stein and Renate Scheidler.
</p>
<p>
I begin with giving an abstract definition of a one-dimensional infrastructure.
</p>
<div class="theorem-environment theorem-definition-environment">
<div class="theorem-header theorem-definition-header">
Definition (One-dimensional infrastructure).
</div>
<div class="theorem-content theorem-definition-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="13" src="https://math.fontein.de/formulae/3rzByTx4xORnQaqA2WwgTLGv7jy1L9T1w3rfrA.svgz" alt="R > 0" title="R > 0"></span> be a real number. A <em>one-dimensional infrastructure</em> of circumference <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/vYu2wcShMlDKJ.IrmXbb2no6ZOHI_2bIn_7ZWQ.svgz" alt="R" title="R"></span> is a pair <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="18" src="https://math.fontein.de/formulae/jIn.uZYNabR16.oruR4TBCDUy7VpBBZ9LQZ0OQ.svgz" alt="(X, d)" title="(X, d)"></span>, where <span class="inline-formula"><img class="img-inline-formula img-formula" width="49" height="17" src="https://math.fontein.de/formulae/vroX4zy.EF6Jd8J4slYfcooFDuZqaHdjbpIajg.svgz" alt="X \neq \emptyset" title="X \neq \emptyset"></span> is a finite set and <span class="inline-formula"><img class="img-inline-formula img-formula" width="115" height="18" src="https://math.fontein.de/formulae/nG5tOQuC_AiH8MK1kd7kHlmnr7XkfyU3Es3Tew.svgz" alt="d : X \to \R/R\Z" title="d : X \to \R/R\Z"></span> is an injective map.
</p>
</div>
</div>
<p>
If you interpret <span class="inline-formula"><img class="img-inline-formula img-formula" width="47" height="18" src="https://math.fontein.de/formulae/Blo1FniNoktf2EKQy0CWZxv_r_wmqWWcM_Pc9A.svgz" alt="\R/R\Z" title="\R/R\Z"></span> as a circle of circumference <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/vYu2wcShMlDKJ.IrmXbb2no6ZOHI_2bIn_7ZWQ.svgz" alt="R" title="R"></span> (think of it as folding up the real line, such that two numbers whose difference is an integer multiple of <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/vYu2wcShMlDKJ.IrmXbb2no6ZOHI_2bIn_7ZWQ.svgz" alt="R" title="R"></span> are identified), a one-dimensional infrastructure can be seen as a circle with a finite number of dots on it. The map <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="12" src="https://math.fontein.de/formulae/DxlmDzniFM_09XzQnKCoRu8h2JBxVd5JriHzmg.svgz" alt="d" title="d"></span> gives the <em>distance</em> between <span class="inline-formula"><img class="img-inline-formula img-formula" width="78" height="18" src="https://math.fontein.de/formulae/cf9RwGv48TV3aumzEka23ZzlJQCLpCAsU1WTag.svgz" alt="0 \in \R/R\Z" title="0 \in \R/R\Z"></span> and some element <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="13" src="https://math.fontein.de/formulae/NI69X9WcENA0aHM1mBDy4f04R45xllROP0rk7A.svgz" alt="x \in X" title="x \in X"></span> on the circle, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="12" src="https://math.fontein.de/formulae/DxlmDzniFM_09XzQnKCoRu8h2JBxVd5JriHzmg.svgz" alt="d" title="d"></span> is called the <em>distance map</em>.
</p>
<p>
Now one can define two operations on a one-dimensional infrastructure. Due to Shanks' nomenclature, these are called <em>baby steps</em> and <em>giant steps</em>. To define a baby step, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="13" src="https://math.fontein.de/formulae/NI69X9WcENA0aHM1mBDy4f04R45xllROP0rk7A.svgz" alt="x \in X" title="x \in X"></span>. Then consider the set <span class="inline-formula"><img class="img-inline-formula img-formula" width="308" height="18" src="https://math.fontein.de/formulae/a0Rs24n9Ft3byBtKakQxdpktgzRH6i8OKcvyXg.svgz" alt="F_x := \{ f \in \R \mid f > 0, \; d(x) + f \in d(X) \}" title="F_x := \{ f \in \R \mid f > 0, \; d(x) + f \in d(X) \}"></span>. It is non-empty as <span class="inline-formula"><img class="img-inline-formula img-formula" width="56" height="15" src="https://math.fontein.de/formulae/2pvzcuCpnuVmjF.XbCJMdhjxBbACaDR5KBnBkw.svgz" alt="R \in F_x" title="R \in F_x"></span> and bounded from below. Moreover, it is discrete as <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span> is finite; therefore, <span class="inline-formula"><img class="img-inline-formula img-formula" width="92" height="16" src="https://math.fontein.de/formulae/hCm3McVn3oo.MqoGIB.CERyqshZnI6TMTgva4w.svgz" alt="f := \min F_x" title="f := \min F_x"></span> exists and <span class="inline-formula"><img class="img-inline-formula img-formula" width="126" height="18" src="https://math.fontein.de/formulae/CXMsyG77aX8zaKByU84i1YFLSuz7MOuqgySNlw.svgz" alt="d(x) + f \in d(X)" title="d(x) + f \in d(X)"></span>, say <span class="inline-formula"><img class="img-inline-formula img-formula" width="127" height="18" src="https://math.fontein.de/formulae/j9yJOxVtP8FiM_OTGFjp05Ra1VwKf6wufY1HTw.svgz" alt="d(x) + f = d(x')" title="d(x) + f = d(x')"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="53" height="14" src="https://math.fontein.de/formulae/CH2_T5Qtn0BevZi3FMa39PM_XpL7YIVSuGtzGg.svgz" alt="x' \in X" title="x' \in X"></span>. In that case, we define <span class="inline-formula"><img class="img-inline-formula img-formula" width="84" height="18" src="https://math.fontein.de/formulae/4DBCTjlf.qxyFA1GJrM4hD2r_lZ2OZNtJdL4vw.svgz" alt="\bs(x) := x'" title="\bs(x) := x'"></span>. This gives a bijective map <span class="inline-formula"><img class="img-inline-formula img-formula" width="91" height="12" src="https://math.fontein.de/formulae/9TS8oPy0xMOhXLq6knO8ZhLeq2sRv6fgaG07og.svgz" alt="\bs : X \to X" title="\bs : X \to X"></span> which, in case <span class="inline-formula"><img class="img-inline-formula img-formula" width="58" height="18" src="https://math.fontein.de/formulae/OmoWSbeos56GEDtDgYRSrEIfYCXXMRFGRTIUpg.svgz" alt="\abs{X} > 1" title="\abs{X} > 1"></span>, has no fixed points. If <span class="inline-formula"><img class="img-inline-formula img-formula" width="47" height="18" src="https://math.fontein.de/formulae/Blo1FniNoktf2EKQy0CWZxv_r_wmqWWcM_Pc9A.svgz" alt="\R/R\Z" title="\R/R\Z"></span> is interpreted as a circle and <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span> identified with <span class="inline-formula"><img class="img-inline-formula img-formula" width="39" height="18" src="https://math.fontein.de/formulae/Sj0POmFJuFR.MovanqxlnUHsUhGBDR2XEk0PPA.svgz" alt="d(X)" title="d(X)"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="17" height="12" src="https://math.fontein.de/formulae/h4LronIpmA8uO2JxOYyV73usVbaSr5mMKE4hQg.svgz" alt="\bs" title="\bs"></span> will send each point to the “next one” in positive direction on the circle.
</p>
<p>
To define giant steps, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="71" height="17" src="https://math.fontein.de/formulae/6xyNyeYpuoPTNTVtwZOf4vkY0hV1tOW9JbrSMA.svgz" alt="x, x' \in X" title="x, x' \in X"></span>. For that, note that <span class="inline-formula"><img class="img-inline-formula img-formula" width="47" height="18" src="https://math.fontein.de/formulae/Blo1FniNoktf2EKQy0CWZxv_r_wmqWWcM_Pc9A.svgz" alt="\R/R\Z" title="\R/R\Z"></span> is naturally a group, whence we can add <span class="inline-formula"><img class="img-inline-formula img-formula" width="33" height="18" src="https://math.fontein.de/formulae/mC.FPQdv97asVdrZ_Qn3M8fsKSUn95MR5D55cQ.svgz" alt="d(x)" title="d(x)"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="38" height="18" src="https://math.fontein.de/formulae/Fs9EtIOEzeX.OxdJGTawEec4.FXbB6Br.rH1dA.svgz" alt="d(x')" title="d(x')"></span>. Now <span class="inline-formula"><img class="img-inline-formula img-formula" width="162" height="18" src="https://math.fontein.de/formulae/yy0aJhuksGAEk.orNgX9S6RQWpjWsUHqkUcTqw.svgz" alt="d(x) + d(x') \in \R/R\Z" title="d(x) + d(x') \in \R/R\Z"></span>, but in general <span class="inline-formula"><img class="img-inline-formula img-formula" width="154" height="18" src="https://math.fontein.de/formulae/DBltW4O5rmxbN3GFF616vZyNYZcaIRGhctTgtw.svgz" alt="d(x) + d(x') \not\in d(X)" title="d(x) + d(x') \not\in d(X)"></span>. But we can use a similar trick as in the baby step case: we jump back to the previous point of <span class="inline-formula"><img class="img-inline-formula img-formula" width="39" height="18" src="https://math.fontein.de/formulae/Sj0POmFJuFR.MovanqxlnUHsUhGBDR2XEk0PPA.svgz" alt="d(X)" title="d(X)"></span>. For that, define <span class="inline-formula"><img class="img-inline-formula img-formula" width="385" height="18" src="https://math.fontein.de/formulae/J8K__sakqK4VtIleueAC7f.NK8gUIOzYJwWs7w.svgz" alt="F_{x,x'} := \{ f \in \R \mid f \ge 0, \; d(x) + d(x') - f \in d(X) \}" title="F_{x,x'} := \{ f \in \R \mid f \ge 0, \; d(x) + d(x') - f \in d(X) \}"></span>. It is bounded from above, non-empty and discrete, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="112" height="17" src="https://math.fontein.de/formulae/q96IARnJ_et7bnV3ecwMMxEmdRB_qH0PQQMlzQ.svgz" alt="f := \max F_{x,x'}" title="f := \max F_{x,x'}"></span> exists with <span class="inline-formula"><img class="img-inline-formula img-formula" width="191" height="18" src="https://math.fontein.de/formulae/ASdZK1yM_t13.BexW7NJRbDqZBxEfjdGUjxpSA.svgz" alt="d(x) + d(x') - f' \in d(X)" title="d(x) + d(x') - f' \in d(X)"></span>, say <span class="inline-formula"><img class="img-inline-formula img-formula" width="181" height="18" src="https://math.fontein.de/formulae/1rPauIsDHOlYB270vBDzFPO0wBeyenL4d2eWWw.svgz" alt="d(x) + d(x') - f = d(y)" title="d(x) + d(x') - f = d(y)"></span> for <span class="inline-formula"><img class="img-inline-formula img-formula" width="47" height="16" src="https://math.fontein.de/formulae/Ye9G7eto4Mwdqzh.TSSLf2uhBkiDBrxVHWboww.svgz" alt="y \in X" title="y \in X"></span>; then we define <span class="inline-formula"><img class="img-inline-formula img-formula" width="101" height="18" src="https://math.fontein.de/formulae/j_fWRvuxRH35nzVNmGQ8xX95.QLVnkz8w6sYsw.svgz" alt="\gs(x, x') := y" title="\gs(x, x') := y"></span>. This gives a binary operation <span class="inline-formula"><img class="img-inline-formula img-formula" width="128" height="16" src="https://math.fontein.de/formulae/G4CeG3K_JpkwrP6tNiBT9fA26inh2Lnw7NcM2A.svgz" alt="\gs : X \times X \to X" title="\gs : X \times X \to X"></span> which is in general not associative.
</p>
<p>
But even though, we have
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="202" height="19" src="https://math.fontein.de/formulae/3mKzV0ZqyL6K2.eDnPH7QWt0h0_1wKKbjOpzKQ.svgz" alt="d(\gs(x, x')) \approx d(x) + d(x')" title="d(\gs(x, x')) \approx d(x) + d(x')">
</div>
<p>
in general, assuming that <span class="inline-formula"><img class="img-inline-formula img-formula" width="275" height="18" src="https://math.fontein.de/formulae/FOsEhNIkWUV38JWlaB8omqEAnIVrKbtPLlnyMQ.svgz" alt="D := \max\{ d(\bs(x)) - d(x) \mid x \in X \}" title="D := \max\{ d(\bs(x)) - d(x) \mid x \in X \}"></span> is small (here, we identify <span class="inline-formula"><img class="img-inline-formula img-formula" width="187" height="18" src="https://math.fontein.de/formulae/KzxEtRsqITx5roeHMawXNqFyaf9whzdxNfsv7g.svgz" alt="d(\bs(x)) - d(x) \in \R/R\Z" title="d(\bs(x)) - d(x) \in \R/R\Z"></span> with its smallest non-negative representant). More precisely, we have <span class="inline-formula"><img class="img-inline-formula img-formula" width="234" height="18" src="https://math.fontein.de/formulae/SRMNWPpjM1RCOPCuuJMysm_yRyIGORbxBvXWXA.svgz" alt="d(\gs(x, x')) + f = d(x) + d(x')" title="d(\gs(x, x')) + f = d(x) + d(x')"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="82" height="16" src="https://math.fontein.de/formulae/tZysG1T.Nb27xZGQtQzlzqJraviuOos9o4ADig.svgz" alt="0 \le f < D" title="0 \le f < D"></span>, whence the giant step operation is “almost” associative.
</p>
</div>
<div class="subsection-block">
<h4>
Finite Cyclic Groups as One-dimensional Infrastructures.
</h4>
<div>
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="60" height="18" src="https://math.fontein.de/formulae/AUxU.uYx74LzIzYpwLzeHxHWYbg0FPld39Uc.A.svgz" alt="G = \ggen{g}" title="G = \ggen{g}"></span> be a finite cyclic group of order <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/vYu2wcShMlDKJ.IrmXbb2no6ZOHI_2bIn_7ZWQ.svgz" alt="R" title="R"></span>. For <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="13" src="https://math.fontein.de/formulae/.OtVteTvb00JDHGGHFdaeG4RN9McHmkT6VlVRg.svgz" alt="h \in G" title="h \in G"></span>, one can write <span class="inline-formula"><img class="img-inline-formula img-formula" width="53" height="16" src="https://math.fontein.de/formulae/SS5F080MrOXcz0uxgpD7ost5c_hvGxit6tLCUg.svgz" alt="h = g^n" title="h = g^n"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="44" height="13" src="https://math.fontein.de/formulae/XpBPwowNNTbUIL5g3SqKnKd1BnfcrcAa2nmWmg.svgz" alt="n \in \Z" title="n \in \Z"></span>; note that <span class="inline-formula"><img class="img-inline-formula img-formula" width="146" height="20" src="https://math.fontein.de/formulae/0osFtUmaQVfmH2DH3dOTjU8g2S1O3nk.2Q3DEA.svgz" alt="n = \log_g h \in \Z/R\Z" title="n = \log_g h \in \Z/R\Z"></span> is the discrete logarithm of <span class="inline-formula"><img class="img-inline-formula img-formula" width="10" height="12" src="https://math.fontein.de/formulae/BhIgQy.yStNYm82Tu7qgDRTe5Zg2.7Iux8hXFw.svgz" alt="h" title="h"></span> with respect to <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="11" src="https://math.fontein.de/formulae/UxCAbnAvo.LhIVRUfLjfwIlPAfhwrVfYLg_Adw.svgz" alt="g" title="g"></span>. Hence, we get the isomorphism <span class="inline-formula"><img class="img-inline-formula img-formula" width="84" height="18" src="https://math.fontein.de/formulae/25c.3JHlWO64JwmB6YjlCE6aNxkr1BUQEfQHqA.svgz" alt="G \cong \Z/R\Z" title="G \cong \Z/R\Z"></span> induced by <span class="inline-formula"><img class="img-inline-formula img-formula" width="133" height="20" src="https://math.fontein.de/formulae/XuU8d9MAK5Q8tGU5m83lFBogAp2k0SyABwXH8w.svgz" alt="\log_g : G \to \Z/R\Z" title="\log_g : G \to \Z/R\Z"></span>. As <span class="inline-formula"><img class="img-inline-formula img-formula" width="117" height="18" src="https://math.fontein.de/formulae/LokI.7V7Hb2SYxaNF68zMhqh2LMWKewxA0S.kg.svgz" alt="\Z/R\Z \subseteq \R/R\Z" title="\Z/R\Z \subseteq \R/R\Z"></span>, we get the injective map <span class="inline-formula"><img class="img-inline-formula img-formula" width="112" height="18" src="https://math.fontein.de/formulae/rkdYzeSn1P3NRP9KI60A1AuhVgeX_PRr_YfR8w.svgz" alt="d : G \to \R/R\Z" title="d : G \to \R/R\Z"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="82" height="19" src="https://math.fontein.de/formulae/8uqxidKt3gqqsRZ9tg7_mE7RkaGhiFGcUVUMRg.svgz" alt="h \mapsto \log_g h" title="h \mapsto \log_g h"></span>, turning <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="18" src="https://math.fontein.de/formulae/TMTR.Ps6eLAw469tQ3ocOk5zxUp.Wna_oBuWMg.svgz" alt="(G, d)" title="(G, d)"></span> into a one-dimensional infrastructure.
</p>
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="69" height="17" src="https://math.fontein.de/formulae/.CCmaN.vPHTHCcdCtxvHd0RgyAIh_ePYqZTnYg.svgz" alt="h, h' \in G" title="h, h' \in G"></span>; then we get <span class="inline-formula"><img class="img-inline-formula img-formula" width="84" height="18" src="https://math.fontein.de/formulae/.ShW4b_ISKZV_nHXC4HUKeyJMYl8wg3kEPuWPQ.svgz" alt="\bs(h) = g h" title="\bs(h) = g h"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="112" height="18" src="https://math.fontein.de/formulae/Bg9gfxWtFQlcZHTQ867JkOaB18vKrdsafqlBAg.svgz" alt="\gs(h, h') = h h'" title="\gs(h, h') = h h'"></span>, i.e. baby steps are multiplications by the generator <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="11" src="https://math.fontein.de/formulae/UxCAbnAvo.LhIVRUfLjfwIlPAfhwrVfYLg_Adw.svgz" alt="g" title="g"></span> and the giant steps equals the group operation. In particular, this provides an example for giant steps being associative.
</p>
<p>
Therefore, one-dimensional infrastructures can be seen as generalizations of finite cyclic groups.
</p>
</div>
</div>
<div class="subsection-block">
<h4>
Remarks.
</h4>
<div>
<p>
Finally, we want to sketch some ideas, which will allow generalizing infrastructures to higher dimensions. For that, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="18" src="https://math.fontein.de/formulae/jIn.uZYNabR16.oruR4TBCDUy7VpBBZ9LQZ0OQ.svgz" alt="(X, d)" title="(X, d)"></span> be a one-dimensional infrastructure.
</p>
<p>
First, define the map <span class="inline-formula"><img class="img-inline-formula img-formula" width="131" height="18" src="https://math.fontein.de/formulae/7uVOvafbYDZ.azBEIJ99WwgR2fMwW92fZH9vcw.svgz" alt="red : \R/R\Z \to X" title="red : \R/R\Z \to X"></span> as follows. For <span class="inline-formula"><img class="img-inline-formula img-formula" width="77" height="18" src="https://math.fontein.de/formulae/mj2kbkiwypGaXg.XiN3goFRxVzHCZq.bCSDT_w.svgz" alt="r \in \R/R\Z" title="r \in \R/R\Z"></span>, define <span class="inline-formula"><img class="img-inline-formula img-formula" width="283" height="18" src="https://math.fontein.de/formulae/D0w3SxTnGpTLKFi6KEvbJboaNnJo3TGqncdVyQ.svgz" alt="F_r := \{ f \in \R \mid f \ge 0, \; r - f \in d(X) \}" title="F_r := \{ f \in \R \mid f \ge 0, \; r - f \in d(X) \}"></span>. Again, <span class="inline-formula"><img class="img-inline-formula img-formula" width="19" height="15" src="https://math.fontein.de/formulae/0vyYT9s2N5af8Lwn40Zerv2n7ifN.WTeSN9YBg.svgz" alt="F_r" title="F_r"></span> is non-empty, bounded from below and discrete, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="91" height="16" src="https://math.fontein.de/formulae/0IYcnr_xd8tPNyZE99gyjW0vWTZx2EHgc1a1og.svgz" alt="f := \min F_r" title="f := \min F_r"></span> exists and <span class="inline-formula"><img class="img-inline-formula img-formula" width="101" height="18" src="https://math.fontein.de/formulae/ZsluJrC07rtQHcgIQNNrXcGlBTgdqfNDle6MhQ.svgz" alt="r - f \in d(X)" title="r - f \in d(X)"></span>, say <span class="inline-formula"><img class="img-inline-formula img-formula" width="97" height="18" src="https://math.fontein.de/formulae/n.tN8iLOC0kxfQH_DQdSwfkBUd.yGY0psXBM9g.svgz" alt="r - f = d(x)" title="r - f = d(x)"></span> for some <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="13" src="https://math.fontein.de/formulae/NI69X9WcENA0aHM1mBDy4f04R45xllROP0rk7A.svgz" alt="x \in X" title="x \in X"></span>. Define <span class="inline-formula"><img class="img-inline-formula img-formula" width="87" height="18" src="https://math.fontein.de/formulae/uqmy8fxHxPrD4qnBXpimBLscWCJhfTvHTl03xw.svgz" alt="red(r) := x" title="red(r) := x"></span>. Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="26" height="12" src="https://math.fontein.de/formulae/x9uDaMcRWmFnhjgyf0CLBlJpQRUKxXbuUM0.1Q.svgz" alt="red" title="red"></span> satisfies <span class="inline-formula"><img class="img-inline-formula img-formula" width="104" height="15" src="https://math.fontein.de/formulae/3TNT2dOT2EXA3eU7nQ52xNGV4zlFqL_Gje1h0Q.svgz" alt="red \circ d = \id_X" title="red \circ d = \id_X"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="219" height="18" src="https://math.fontein.de/formulae/qyWzkkBZtH72MecTwkH0ybGdQQiUhO8ieiV_uQ.svgz" alt="\gs(x, x') = red(d(x) + d(x'))" title="\gs(x, x') = red(d(x) + d(x'))"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="71" height="17" src="https://math.fontein.de/formulae/6xyNyeYpuoPTNTVtwZOf4vkY0hV1tOW9JbrSMA.svgz" alt="x, x' \in X" title="x, x' \in X"></span>.
</p>
<p>
If <span class="inline-formula"><img class="img-inline-formula img-formula" width="136" height="18" src="https://math.fontein.de/formulae/ggSvS5nmK24uYEyjTS0Drq81ozcg.U4o8e4_oQ.svgz" alt="red' : \R/R\Z \to X" title="red' : \R/R\Z \to X"></span> would be any other map satisfying <span class="inline-formula"><img class="img-inline-formula img-formula" width="109" height="16" src="https://math.fontein.de/formulae/3wrbtfNBrBsw7jGLOPNgUbOsTpO3GkbRiu22sA.svgz" alt="red' \circ d = \id_X" title="red' \circ d = \id_X"></span>, one would obtain another giant step function <span class="inline-formula"><img class="img-inline-formula img-formula" width="133" height="17" src="https://math.fontein.de/formulae/Iv0k49XMXta.rB4c4uiLViMOte3_UVnjENeJ.Q.svgz" alt="\gs' : X \times X \to X" title="\gs' : X \times X \to X"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="212" height="18" src="https://math.fontein.de/formulae/mCqNz9L1AIyyaPSKO5klor2Hu8kVYhNJq5z.2Q.svgz" alt="(x, x') \mapsto red'(d(x) + d(x'))" title="(x, x') \mapsto red'(d(x) + d(x'))"></span>. In case <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span> comes from a finite cyclic group, as above, <span class="inline-formula"><img class="img-inline-formula img-formula" width="21" height="17" src="https://math.fontein.de/formulae/4aw.4k5f2mCsKgGkWhGPfRpiHkn4W4BIalfbRQ.svgz" alt="\gs'" title="\gs'"></span> would again be the group operation. If this is not the case, <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="11" src="https://math.fontein.de/formulae/oALhjAUt95L9B1GjlImGOqnVYMSkaj0JkVGFrg.svgz" alt="\gs" title="\gs"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="21" height="17" src="https://math.fontein.de/formulae/4aw.4k5f2mCsKgGkWhGPfRpiHkn4W4BIalfbRQ.svgz" alt="\gs'" title="\gs'"></span> could be two distinct binary operations on <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span>. If <span class="inline-formula"><img class="img-inline-formula img-formula" width="31" height="13" src="https://math.fontein.de/formulae/DtOMzCdRPvNQu5ZYKFZqPUOFZACx2OxO6ZCokA.svgz" alt="red'" title="red'"></span> satisfies <span class="inline-formula"><img class="img-inline-formula img-formula" width="108" height="18" src="https://math.fontein.de/formulae/LkSV57Bzt3iA7WB52Zyd9R3pzPJHgdTTbNh04g.svgz" alt="d(red'(r)) \approx r" title="d(red'(r)) \approx r"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="77" height="18" src="https://math.fontein.de/formulae/mj2kbkiwypGaXg.XiN3goFRxVzHCZq.bCSDT_w.svgz" alt="r \in \R/R\Z" title="r \in \R/R\Z"></span>, we would also have
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="339" height="19" src="https://math.fontein.de/formulae/pUDs5.9vcSsEhTqKlOykOPC95vE5FO6j.2HHUA.svgz" alt="d(\gs'(x, x')) \approx d(x) + d(x') \text{ for all } x, x' \in X." title="d(\gs'(x, x')) \approx d(x) + d(x') \text{ for all } x, x' \in X.">
</div>
<p>
This shows that our choice of <span class="inline-formula"><img class="img-inline-formula img-formula" width="26" height="12" src="https://math.fontein.de/formulae/x9uDaMcRWmFnhjgyf0CLBlJpQRUKxXbuUM0.1Q.svgz" alt="red" title="red"></span> is rather random; we could also define <span class="inline-formula"><img class="img-inline-formula img-formula" width="179" height="19" src="https://math.fontein.de/formulae/xmkRztnJNbzrteP4vrZnukywxuKkNUfVVxuuhg.svgz" alt="red(r) = d^{-1}(d(x) + f)" title="red(r) = d^{-1}(d(x) + f)"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="298" height="18" src="https://math.fontein.de/formulae/9ohe_dGvijIVIUEMRIieKRTJxS67_cIQN178LQ.svgz" alt="f = \min \{ f \in \R \mid f \ge 0, \; r + f \in d(X) \}" title="f = \min \{ f \in \R \mid f \ge 0, \; r + f \in d(X) \}"></span>, or chose <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/M.ji_f0zsVnTLyaKegBkRJLOeqrrt6brNnapOQ.svgz" alt="f" title="f"></span> such that <span class="inline-formula"><img class="img-inline-formula img-formula" width="228" height="18" src="https://math.fontein.de/formulae/HeddQzJpBGsxun5KHYhrlqZ6Sk2SWK0FQKsdDw.svgz" alt="\abs{f} = \min\{ \abs{f} \mid r + f \in d(X) \}" title="\abs{f} = \min\{ \abs{f} \mid r + f \in d(X) \}"></span>, with some additional condition to rule out ties. Any other arbitrary choice of <span class="inline-formula"><img class="img-inline-formula img-formula" width="26" height="12" src="https://math.fontein.de/formulae/x9uDaMcRWmFnhjgyf0CLBlJpQRUKxXbuUM0.1Q.svgz" alt="red" title="red"></span> is also possible, as long as <span class="inline-formula"><img class="img-inline-formula img-formula" width="104" height="15" src="https://math.fontein.de/formulae/3TNT2dOT2EXA3eU7nQ52xNGV4zlFqL_Gje1h0Q.svgz" alt="red \circ d = \id_X" title="red \circ d = \id_X"></span> is satisfied. We will later see that our definition of <span class="inline-formula"><img class="img-inline-formula img-formula" width="26" height="12" src="https://math.fontein.de/formulae/x9uDaMcRWmFnhjgyf0CLBlJpQRUKxXbuUM0.1Q.svgz" alt="red" title="red"></span> is exactly the one we obtain in a canonical way if we obtain infrastructures from global fields of unit rank one. We call such maps <em>reduction maps</em>.
</p>
</div>
</div>
</div>