# 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 . In the latter case, let be the exact field of constants.

Let be the set of infinite places of . If is a number field, the elements of correspond to embeddings of into up to complex conjugation. Define , and for let be a corresponding embedding. Then define for and if , or otherwise, and define . If is a function field, let , i.e. ; in this case, there exists an element 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 of , let be the valuation ring and its maximal idea, and denote the discrete valuation by . Then set and , . Define . In the number field case, let , and otherwise .

Denote the set of places of by . The divisor group of is , and for define . This is a homomorphism ; denote its kernel by . For , is a principal divisor; let the group of all these be denoted by . Then is the divisor class group of and its degree zero part.

The support of a divisor is the set . Consider the subgroups

then . Moreover, let .

The set is a Dedekind domain, whose maixmal ideals correspond to the places in . Moreover, the fractional ideal group is isomorphic to by , in case ; the inverse is given by the restriction of , to . The group of fractional principal ideals equals . The quotient is the ideal class group of . Putting all these things together, we get the following diagram with exact rows and columns:

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, .

#### A Geometry of Numbers in Global Fields.

Let and . Define

If and is the Riemann-Roch space of , then . 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

Then is a lattice by Dirichlet's Unit Theorem, and . We get , and is called the unit rank of . This 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 and . We say that is a minimum of if every with for all satisfies for all . Denote the set of all minima of by .

Using them, we can define reduced ideals:

Definition.

An ideal is said to be reduced if is a minimum. Write for the set of all reduced ideals of . For let .

The equivalence relation we need is defined by

for . We then get the following results:

Theorem.
1. We have that is a finite set.
2. In case for some , we get for if, and only if, .
3. We have that acts on by multiplication.
4. The map

is a bijection.

5. If and satisfies , then .

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 . Define and define

Then is well-defined and injective.

For , write

Define and

Then is the unrolled version of : if , is the projection, and , , then the following diagram commutes:

In particular, is the set 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 .

First, define for

where is the lexicographic order on .

Definition.

A tuple is said to be an -representation if is a smallest element of

with respect to . Denote the set of all -representations by .

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

Lemma (Uniqueness).

Let and such that

Then for all , i.e. .

Lemma (Reduction).

Let . Then there exists a smallest such that . If is minimal with respect to in that , then

and .

From that, we get the following result:

Theorem (Infrastructure, Part II).

Let . Then the map

is a bijection.

This allows to equip with a group operation. We will see that the group operation of 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 , then , for all and

where

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 is the discriminant of .

Therefore, -representations are small.

#### The Infrastructure and the Divisor Class Group.

Assume for a moment that , or that is a number field. Then we have a short exact sequence

and . This means that the divisor class group is covered by copies of , where the copies are indexed by the elements of the divisor class group. If and are in the same ideal class, and differ by a translation, i.e. they give essentially the same infrastructure; in fact, . Hence, one could get the idea to cover by , where ranges over the distinct ideal classes, i.e. by . It turns out that this is indeed the case, and the arithmetic on and turn out to be the same under the bijection we get.

In case is a function field and , we have in general (this is the case if, and only if, ), and does not needs to be surjective. It would be nice to change the above sequence to

in any case, but this is not possible with as it is; we have to replace it by something bigger. It turns out that the right replacement is , which is canonically isomorphic to in case . We then get the diagram

with exact rows.

The complete result is stated in the following theorem:

Theorem (Infrastructure, Part III).
1. Let be a number field. Then the map

where equals , is a bijection.

2. Let be a function field. Then the map

is a bijection.

Moreover, is a group homomorphism, where the group structure on is the one induced by the bijection .

Finally, we explicitly describe the group operation induced by this bijection on without using the bijection itself.

Theorem.

Let be the bijection from the previous theorem, and let , .

1. Set for . There exists a minimal with , and if is a smallest element of with respect to , we get

with .

2. Set for . There exists a minimal with , and if is a smallest element of with respect to , we get

with .

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.

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 and , consider

for . There exists a minimal such that . In case , contains exactly one -orbit, which gives a unique element . Otherwise, one has to add an order (lexicographic order as above) to chose an element. In any case, define ; then this gives a function resp. for any . 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 of the function field defined by over . The red arrows show , the blue arrows and the green arrows . The small black circles denote usual minima, the larger black circles denote elements of , and the shaded areas denote translates of an fundamental parallelepiped of :

The second example depicts the infrastructure of the function field defined by over , with the same notation: