Obtaining Infrastructures from Global Fields.

Basics on Global Fields.

Let be a global field, i.e. an algebraic number field or an algebraic function field with a finite constant field. In the first case, let be the roots of unity and $k = k^* \cup \{ 0 \}$ . In the latter case, let be the exact field of constants.

Let $S = \{ \frakp_1, \dots, \frakp_{n+1} \}$ be the set of infinite places of . If is a number field, the elements of correspond to embeddings of into $\C$ up to complex conjugation. Define $q := \exp(1)$ , and for $\frakp \in S$ let $\sigma : K \to \C$ be a corresponding embedding. Then define $\nu_\frakp(f) := -\log \abs{\sigma(f)}$ for $f \in K^*$ and $\deg \frakp := 1$ if $\sigma(K) \subseteq \R$ , or $\deg \frakp := 2$ otherwise, and define $\G_\frakp := \R$ . If is a function field, let $q := \abs{k}$ , i.e. $k = \F_q$ ; in this case, there exists an element $x \in K \setminus k$ whose poles are exactly the elements of , i.e. are the places of lying above the infinite place of . In all cases, is finite and non-empty.

For a non-archimedean place $\frakp$ of , let $\calO_\frakp$ be the valuation ring and $\frakm_\frakp$ its maximal idea, and denote the discrete valuation by $\nu_\frakp$ . Then set $\deg \frakp := \log_q \abs{\calO_\frakp / \frakm_\frakp}$ and $\abs{f}_\frakp := q^{-\nu_\frakp(f) \deg \frakp}$ , $f \in K$ . Define $\G_\frakp := \Z$ . In the number field case, let $\G := \R$ , and otherwise $\G := \Z$ .

Denote the set of places of by $\calP_K$ . The divisor group of is $\Div(K) := \coprod_{\frakp \in \calP} \G_\frakp$ , and for $D = \sum_{\frakp \in \calP_K} n_\frakp \frakp$ define $\deg D := \sum_{\frakp \in \calP_K} n_\frakp \deg \frakp$ . This is a homomorphism $\deg : \Div(K) \to \G$ ; denote its kernel by $\Div^0(K)$ . For $f \in K^*$ , $(f) := \sum_{\frakp \in \calP_K} \nu_\frakp(f) \frakp \in \Div^0(K)$ is a principal divisor; let the group of all these be denoted by $\Princ(K)$ . Then $\Pic(K) := \Div(K) / \Princ(K)$ is the divisor class group of and $\Pic^0(K) := \Div^0(K) / \Princ(K)$ its degree zero part.

The support of a divisor $D = \sum_{\frakp \in \calP_K} n_\frakp \frakp$ is the set $\support(D) = \{ \frakp \in \calP_K \mid n_\frakp \neq 0 \}$ . Consider the subgroups

$\Div_{fin}(K) :={} & \{ D \in \Div(K) \mid \support(D) \cap S = \emptyset \} \\ \text{and} \qquad \Div_\infty(K) :={} & \{ D \in \Div(K) \mid \support(D) \subseteq S \};$

then $\Div(K) = \Div_{fin}(K) \oplus \Div_\infty(K)$ . Moreover, let $\Div_\infty^0(K) := \Div^0(K) \cap \Div_\infty(K)$ .

The set $\calO := \calO_S := \{ f \in K \mid \nu_\frakp(f) \ge 0 \text{ for all } \frakp \in S \}$ is a Dedekind domain, whose maixmal ideals correspond to the places in $\calP_K \setminus S$ . Moreover, the fractional ideal group $\Id(\calO_S)$ is isomorphic to $\Div_{fin}(K)$ by $\divisor(\fraka) = \sum_{\frakp \not\in S} n_\frakp \frakp$ , in case $\fraka = \prod_{\frakp \not\in S} (\frakm_\frakp \cap \calO_S)^{-n_\frakp}$ ; the inverse is given by the restriction of $\ideal : \Div(K) \to \Id(\calO_S)$ , $\sum n_\frakp \frakp \mapsto \prod_{\frakp \not\in S} (\frakm_\frakp \cap \calO_S)^{-n_\frakp}$ to $\Div_{fin}(K)$ . The group of fractional principal ideals $\PId(\calO_S)$ equals $\ideal(\Princ(K))$ . The quotient $\Id(\calO_S) / \PId(\calO_S)$ is the ideal class group $\Pic(\calO_S)$ of $\calO_S$ . Putting all these things together, we get the following diagram with exact rows and columns:

$\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] & \\ 0 \ar[r] & \calO_S^* / k^* \ar[r] \ar[d] & \Div^0_\infty(K) \ar[r] \ar[d] & T \ar[r] \ar[d] & 0 \\ 0 \ar[r] & K^* / k^* \ar[r] \ar[d] & \Div^0(K) \ar[r] \ar[d] & \Pic^0(K) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & K^* / \calO_S^* \ar[r] \ar[d] & \Id(\calO_S) \ar[r] \ar[d] & \Pic(\calO_S) \ar[r] \ar[d] & 0 \\ & 0 & H \ar@{=}[r] \ar[d] & H \ar[d] & \\ & & 0 & 0 & }$

Here, and are essentially defined by the diagram, i.e. are the kernels and cokernels of the respective maps. In the number field case, , and in the function field case, $H \cong (\deg \frakp \mid \frakp \in \calP_K) / (\deg \frakp \mid \frakp \in S)$ .

A Geometry of Numbers in Global Fields.

Let $\fraka \in \Id(\calO_S)$ and $t_1, \dots, t_{n+1} \in \G$ . Define

$B(\fraka, (t_1, \dots, t_{n+1})) := \{ f \in \fraka \mid \forall i : \abs{f}_{\frakp_i} \le q^{t_i \deg \frakp_i} \}.$

If $D := \divisor(\fraka) + \sum_{i=1}^{n+1} t_i \frakp_i \in \Div(K)$ and is the Riemann-Roch space of , then $L(D) = B(\fraka, (t_1, \dots, t_{n+1}))$ . In particular, the set is finite and invariant under multiplication by elements of ; in case is a function field, is a finite-dimensional -vector space, whose dimension is described by the Riemann-Roch theorem. In the number field case, we can make statements on with Minkowski's Lattice Point Theorem.

Consider the map

$\Psi : K^* \to \G^n, \quad f \mapsto (-\nu_{\frakp_1}(f), \dots, -\nu_{\frakp_n}(f)).$

Then $\Lambda := \Psi(\calO^*) \cong \Z^n$ is a lattice by Dirichlet's Unit Theorem, and $\ker \Psi|_{\calO^*} = k^*$ . We get $\calO^* \cong k^* \times \Z^n$ , and is called the unit rank of $\calO_S$ . This $\Lambda$ will be the lattice for our -dimensional infrastructure.

Reduced Ideals.

The elements of will be principal reduced fractional ideals, modulo an equivalence relation. We begin by defining minima, which are similar to the ones introduced in the previous post for lattices.

Definition.

Let $\fraka \in \Id(\calO_S)$ and $\mu \in \fraka \setminus \{ 0 \}$ . We say that $\mu$ is a minimum of $\fraka$ if every $f \in \fraka \setminus \{ 0 \}$ with $\abs{f}_{\frakp_i} \le \abs{\mu}_{\frakp_i}$ for all satisfies $\abs{f}_{\frakp_i} = \abs{\mu}_{\frakp_i}$ for all . Denote the set of all minima of $\fraka$ by $\calC(\fraka)$ .

Using them, we can define reduced ideals:

Definition.

An ideal $\fraka \in \Id(\calO_S)$ is said to be reduced if $1 \in \fraka$ is a minimum. Write $\Red_S(K)$ for the set of all reduced ideals of $\calO_S$ . For $\frakb \in \Id(\calO_S)$ let $\Red_S(\frakb) := \{ \fraka \in \Red_S(K) \mid \exists f \in K^* : f \fraka = \frakb \}$ .

The equivalence relation we need is defined by

$\fraka \sim_S \fraka' :\Leftrightarrow \exists f \in K^* : \fraka = f \fraka' \wedge \forall \frakp \in S : \abs{f}_\frakp = 1$

for $\fraka, \fraka' \in \Id(\calO_S)$ . We then get the following results:

Theorem.

We have that $\Red_S(K)$ is a finite set.
In case $\deg \frakp = 1$ for some $\frakp \in S$ , we get $\fraka \sim_S \fraka'$ for $\fraka, \fraka' \in \Red(K)$ if, and only if, $\fraka = \fraka'$ .
We have that $\calO^*$ acts on $\calC(\fraka)$ by multiplication.
The map

$\calC(\fraka) / \calO^* \to \Red(\fraka), \quad \mu \calO^* \mapsto \frac{1}{\mu} \fraka$

is a bijection.
If $\fraka \in \Red(K)$ and $\frakb \in \Id(\calO)$ satisfies $\fraka \sim_S \frakb$ , then $\frakb \in \Red(\fraka)$ .

The proofs of these and the following results or hints to the proofs can be found here. We next construct the map :

Theorem (Infrastructure, Part I).

Fix an ideal $\fraka \in \Id(\calO)$ . Define $X_\fraka := \Red(\fraka)/_{\sim_S}$ and define

$d_\fraka : X \to \G^n / \Lambda, \quad [\tfrac{1}{\mu} \fraka]_{\sim_S} \mapsto \Psi(\mu) + \Lambda.$

Then $d_\fraka$ is well-defined and injective.

For $a, a' \in K^*$ , write

$a \sim_S a' :\Longleftrightarrow \forall \frakp \in S : \abs{a}_\frakp = \abs{a'}_\frakp.$

Define $\hat{X} := \calC(\fraka)/_{\sim_S}$ and

$\hat{d} : \hat{X} \to \G^n, \quad [\mu]_\sim \mapsto \Psi(\mu).$

Then $(\hat{X}, \hat{d})$ is the unrolled version of : if $\pi : \G^n \to \G^n / \Lambda$ , $x \mapsto x + \Lambda$ is the projection, and $\psi : \hat{X} \to X$ , $[\mu]_\sim \mapsto [\frac{1}{\mu} \fraka]_\sim$ , then the following diagram commutes:

$\xymatrix{ \hat{X} \ar[d]_{\psi} \ar[r]^{\hat{d}} & \G^n \ar[d]^{\pi} \\ X \ar[r]_{d} & \G^n/\Lambda }$

In particular, $\hat{d}(\hat{X})$ is the set $\hat{X}$ from the previous post.

The Reduction Map, -Representations, and the Infrastructure.

We proceed by defining -representations, as giving these is equivalent to give a reduction map. Fix an ideal $\fraka \in \Id(\calO_S)$ .

First, define for $f, f' \in K^*$

$f \le_S f' :\Longleftrightarrow (\abs{f}_{\frakp_{n+1}}, \abs{f}_{\frakp_1}, \dots, \abs{f}_{\frakp_n}) \le_{\ell ex} (\abs{f'}_{\frakp_{n+1}}, \abs{f'}_{\frakp_1}, \dots, \abs{f'}_{\frakp_n}),$

where $\le_{\ell ex}$ is the lexicographic order on $\R^{n+1}$ .

Definition.

A tuple $([\frakb]_{\sim_S}, (t_1, \dots, t_n)) \in \Red_S(\fraka)/_{\sim_S} \times \G^n$ is said to be an -representation if is a smallest element of

$B(\frakb, (t_1, \dots, t_n, 0)) \setminus \{ 0 \}$

with respect to $\le$ . Denote the set of all -representations by $\fRep(\fraka)$ .

One quickly sees that this is well-defined. We have two auxilliary results:

Lemma (Uniqueness).

Let $A = ([\frakb]_{\sim_S}, (t_1, \dots, t_n)) \in \fRep(\fraka)$ and $f \in K^*$ such that

$B = ([\tfrac{1}{f} \frakb]_{\sim_S}, (t_1 + \nu_{\frakp_1}(f), \dots, t_n + \nu_{\frakp_n}(f))) \in \fRep(\fraka).$

Then $\abs{f}_\frakp = 1$ for all $\frakp \in S$ , i.e. .

Lemma (Reduction).

Let $v = (v_1, \dots, v_n) \in \G^n$ . Then there exists a smallest $\ell \in \G$ such that $B_\ell := B(\fraka, (v_1, \dots, v_n, \ell)) \setminus \{ 0 \} \neq \emptyset$ . If $\mu$ is minimal with respect to $\le$ in that $B_\ell$ , then

$([\tfrac{1}{\mu} \fraka]_{\sim_S}, (v_1 + \nu_{\frakp_1}(\mu), \dots, v_n + \nu_{\frakp_n}(\mu))) \in \fRep(\fraka)$

and $\Phi(\mu) + (v_1 + \nu_{\frakp_1}(\mu), \dots, v_n + \nu_{\frakp_n}(\mu)) + \Lambda = v + \Lambda$ .

From that, we get the following result:

Theorem (Infrastructure, Part II).

Let $\fraka \in \Id(\calO)$ . Then the map

$\Phi :{} & \fRep(\fraka) \to \G^n / \Lambda \\ & ([\tfrac{1}{\mu} \fraka]_{\sim_S}, (t_1, \dots, t_n)) \mapsto \Psi(\mu) + (t_1, \dots, t_n) + \Lambda$

is a bijection.

This allows to equip $\fRep(\fraka)$ with a group operation. We will see that the group operation of $\fRep(\calO_S)$ 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.

Before ending this section, we want to state a result which shows that -representations are small.

Theorem.

Let $([\frakb]_{\sim_S}, (t_1, \dots, t_n)) \in \fRep(\fraka)$ , then $\divisor(\frakb) \ge 0$ , $t_i \ge 0$ for all and

$\deg \divisor(\fraka) + \sum_{i=1}^n t_i \deg \frakp_i \le \kappa,$

where

$\kappa := \begin{cases} g + \deg \frakp_{n+1} - 1 & \text{if } K \text{ is a function field} \\ s \log \tfrac{2}{\pi} + \tfrac{1}{2} \log \abs{\Delta} & \text{if } K \text{ is a number field;} \end{cases}$

here, is the genus of in case is a function field, and in case is a number field, denotes the number of places of degree two and $\Delta$ is the discriminant of $\calO_S$ .

Therefore, -representations are small.

The Infrastructure and the Divisor Class Group.

Assume for a moment that $\deg \frakp_{n+1} = 1$ , or that is a number field. Then we have a short exact sequence

$\xymatrix{ 0 \ar[r] & T \ar[r] & \Pic^0(K) \ar[r] & \Pic(\calO_S) \ar[r] & 0, }$

and $T \cong \G^n / \Lambda \cong \fRep(\fraka)$ . This means that the divisor class group $\Pic^0(K)$ is covered by copies of $\G^n/\Lambda$ , where the copies are indexed by the elements of the divisor class group. If $\fraka$ and $\fraka'$ are in the same ideal class, $X_\fraka$ and $X_{\fraka'}$ differ by a translation, i.e. they give essentially the same infrastructure; in fact, $\fRep(\fraka) = \fRep(\fraka')$ . Hence, one could get the idea to cover $\Pic^0(K)$ by $\fRep(\fraka)$ , where $\fraka$ ranges over the distinct ideal classes, i.e. by $\fRep(K) := \bigcup_{\fraka \in \Id(\calO_S)} \fRep(\fraka)$ . It turns out that this is indeed the case, and the arithmetic on $\fRep(\calO_S)$ and $\Pic^0(K)$ turn out to be the same under the bijection we get.

In case is a function field and $\deg \frakp_{n+1} > 1$ , we have $T \not\cong \G^n / \Lambda$ in general (this is the case if, and only if, $\deg \frakp_{n+1} = \gcd(\deg \frakp_1, \dots, \frakp_n, \frakp_{n+1})$ ), and $\Pic^0(K) \to \Pic(\calO_S)$ does not needs to be surjective. It would be nice to change the above sequence to

$\xymatrix{ 0 \ar[r] & \G^n/\Lambda \ar[r] & \Pic^0(K) \ar[r] & \Pic(\calO_S) \ar[r] & 0 }$

in any case, but this is not possible with $\Pic^0(K)$ as it is; we have to replace it by something bigger. It turns out that the right replacement is $\Pic(K) / \ggen{[\frakp_{n+1}]}$ , which is canonically isomorphic to $\Pic^0(K)$ in case $\deg \frakp_{n+1} = \gcd(\deg \frakp \mid \frakp \in \calP_K)$ . We then get the diagram

$\xymatrix{ 0 \ar[r] & T \ar[r] \ar@{^(->}[d] & \Pic^0(K) \ar@{^(->}[d] \ar[r] & \Pic(\calO_S) \ar@{=}[d] & \\ 0 \ar[r] & \G^n/\Lambda \ar[r] & \Pic(K) / \ggen{[\frakp_{n+1}]} \ar[r] & \Pic(\calO_K) \ar[r] & 0 }$

with exact rows.

The complete result is stated in the following theorem:

Theorem (Infrastructure, Part III).

Let be a number field. Then the map

$\Phi :{} & \fRep(K) \to \Pic^0(K), \\ & ([\frakb]_{\sim_S}, (t_1, \dots, t_n)) \mapsto \biggl[ \divisor(\frakb) + \sum_{i=1}^n t_i \frakp_i - \frac{\dots}{\deg \frakp_{n+1}} \frakp_{n+1} \biggr],$

where $\dots$ equals $\deg \divisor(\frakb) + \sum_{i=1}^n t_i \deg \frakp_i$ , is a bijection.
Let be a function field. Then the map

$\Phi :{} & \fRep(K) \to \Pic(K) / \ggen{[\frakp_{n+1}]}, \\ & ([\frakb]_{\sim_S}, (t_1, \dots, t_n)) \mapsto \biggl[ \divisor(\frakb) + \sum_{i=1}^n t_i \frakp_i \biggr] + \ggen{[\frakp_{n+1}]}$

is a bijection.

Moreover, $\Phi|_{\fRep(\calO_S)}$ is a group homomorphism, where the group structure on $\fRep(\calO_S)$ is the one induced by the bijection $\fRep(\calO_S) \to \G^n/\Lambda$ .

Finally, we explicitly describe the group operation induced by this bijection on $\fRep(K)$ without using the bijection itself.

Theorem.

Let $\Phi$ be the bijection from the previous theorem, and let $A = ([\fraka]_{\sim_S}, (t_1, \dots, t_n))$ , $A' = ([\fraka']_{\sim_S}, (t'_1, \dots, t'_n)) \in \fRep(K)$ .

Set $B_\ell := B(\fraka \fraka', (t_1 + t'_1, \dots, t_n + t'_n, \ell)) \setminus \{ 0 \}$ for $\ell \in \G$ . There exists a minimal $\ell$ with $B_\ell \neq \emptyset$ , and if $\mu$ is a smallest element of $B_\ell$ with respect to $\le$ , we get

$B := ([\tfrac{1}{\mu} \fraka \fraka']_{\sim_S}, (t_i + t'_i + \nu_{\frakp_i}(\mu))_{i=1,\dots,n}) \in \fRep(K)$

with $\Phi(A) + \Phi(A') = \Phi(B)$ .
Set $B_\ell := B(\fraka^{-1}, (-t_1, \dots, -t_n, \ell)) \setminus \{ 0 \}$ for $\ell \in \G$ . There exists a minimal $\ell$ with $B_\ell \neq \emptyset$ , and if $\mu$ is a smallest element of $B_\ell$ with respect to $\le$ , we get

$C := ([\tfrac{1}{\mu} \fraka^{-1}]_{\sim_S}, (-t_i + \nu_{\frakp_i}(\mu))_{i=1,\dots,n}) \in \fRep(K)$

with $-\Phi(A) = \Phi(C)$ .

This shows that the -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, “Real and Imaginary Quadratic Representations of Hyperelliptic Function Fields”. The first relation between the infrastructure of number fields and the Arakelov divisor class group was described by R. Schoof in his paper Computing Arakelov class groups.

What about... Baby Steps?

As I mentioned, there is no known construction for baby steps in general -dimensional infrastructures, but there exists a construction for infrastructures obtained from global fields. I want to describe this construction here.

For $i \in \{ 1, \dots, n + 1 \}$ and $\fraka \in \Red(K)$ , consider

$B_\ell := \biggl\{ f \in \fraka \;\biggm|\begin{matrix} \abs{f}_{\frakp_j} \le 1 \text{ for all } j \neq i, \\ \exists j' : \abs{f}_{\frakp_{j'}} < 1, \; \abs{f}_{\frakp_i} \le \ell \end{matrix} \biggr\} \setminus \{ 0 \}$

for $\ell > 0$ . There exists a minimal $\ell$ such that $B_\ell \neq \emptyset$ . In case $\deg \frakp_i = 1$ , $B_\ell$ contains exactly one -orbit, which gives a unique element $\mu \in B_\ell$ . Otherwise, one has to add an order (lexicographic order as $\le$ above) to chose an element. In any case, define $\bs_i([\fraka]_{\sim_S}) := [\frac{1}{\mu} \fraka]_{\sim_S}$ ; then this gives a function $\Red(K) \to \Red(K)$ resp. $\Red(\frakb) \to \Red(\frakb)$ for any $\frakb \in \Id(\calO)$ . Opposed to the one-dimensional case, this function neither has to be injective nor surjective, as examples below will show.

We begin with a “small” example: the infrastructure $(X_{\calO_S}, d_{\calO_S})$ of the function field defined by over $\F_7$ . The red arrows show $\bs_1$ , the blue arrows $\bs_2$ and the green arrows $\bs_3$ . The small black circles denote usual minima, the larger black circles denote elements of $\Lambda$ , and the shaded areas denote translates of an fundamental parallelepiped of $\Lambda$ :

The second example depicts the infrastructure $(X_{\calO_S}, d_{\calO_S})$ of the function field defined by over $\F_7$ , with the same notation:

Felix' Math Place

Focussed on, but not limited to Computational Number Theory

Obtaining Infrastructures from Global Fields.

Basics on Global Fields.

A Geometry of Numbers in Global Fields.

Reduced Ideals.

The Reduction Map, -Representations, and the Infrastructure.

The Infrastructure and the Divisor Class Group.

What about... Baby Steps?

Comments.

About Me.

About This Blog.

Overview Pages.

Recent Posts.

Archives.

Impressum.

Data Protection.

Felix' Math Place

Focussed on, but not limited to Computational Number Theory

Obtaining Infrastructures from Global Fields.

Basics on Global Fields.

A Geometry of Numbers in Global Fields.

Reduced Ideals.

The Reduction Map, -Representations, and the Infrastructure.

The Infrastructure and the Divisor Class Group.

What about... Baby Steps?

Comments.

About Me.

About This Blog.

Overview Pages.

Categories.

Recent Posts.

Archives.

Tags.

Impressum.

Data Protection.