# Galois Representations Attached to Elliptic Curves

Let  be an elliptic curve over a number field . In this section we attach representations of to , and use them to define an -function . This -function is yet another generalization of the Riemann Zeta function, that is different from the -functions attached to complex representations , which we encountered before in Section 9.5.

Fix an integer . The group structure on  is defined by algebraic formulas with coefficients that are elements of , so the subgroup

is invariant under the action of . We thus obtain a homomorphism

We continue to assume that is an elliptic curve over a number field . For any positive integer , the group is isomorphic as an abstract abelian group to . There are various related ways to see why this is true. One is to use the Weierstrass -theory to parametrize by the the complex numbers, i.e., to find an isomorphism , where is a lattice in and the isomorphism is given by with respect to an appropriate choice of coordinates on . It is then an easy exercise to verify that .

Another way to understand is to use that is isomorphic to the quotient

of homology groups and that the homology of a curve of genus  is isomorphic to . Then

If is a prime, then upon chosing a basis for the two-dimensional -vector space , we obtain an isomorphism . We thus obtain a mod  Galois representation

This representation is continuous if is endowed with the discrete topology, because the field

is a Galois extension of  of finite degree.

In order to attach an -function to , one could try to embed into and use the construction of Artin -functions from Section 9.5. Unfortunately, this approach is doomed in general, since frequently does not embed in . The following Sage session shows that for , there are no 2-dimensional irreducible representations of , so does not embed in . (The notation in the output below is [degree of rep, number of times it occurs].)

Instead of using the complex numbers, we use the -adic numbers, as follows. For each power of , we have defined a homomorphism

We combine together all of these representations (for all ) using the inverse limit. Recall that the -adic numbers are

which is the set of all compatible choices of integers modulo for all . We obtain a (continuous) homomorphism

where is the ring of -adic integers. The composition of this homomorphism with the reduction map is the representation , which we defined above, which is why we denoted it by . We next try to mimic the construction of from Section 9.5 in the context of a -adic Galois representation .

Definition 10.2.1 (Tate module)   The -adic Tate module of is

Let be the fixed field of . The image of is infinite, so is an infinite extension of . Fortunately, one can prove that  is ramified at only finitely many primes (the primes of bad reduction for and --see [ST68]). If  is a prime of , let be a choice of decomposition group for some prime  of  lying over , and let be the inertia group. We haven't defined inertia and decomposition groups for infinite Galois extensions, but the definitions are almost the same: choose a prime of over , and let be the subgroup of that leaves  invariant. Then the submodule of inertia invariants is a module for and the characteristic polynomial of on is well defined (since inertia acts trivially). Let be the polynomial obtained by reversing the coefficients of . One can prove that and that , for does not depend on the choice of . Define for using a different prime , so the definition of does not depend on the choice of .

Definition 10.2.2   The -series of is

A prime  of is a prime of good reduction for  if there is an equation for such that is an elliptic curve over .

If and is a prime of good reduction for , then one can show that that where and is the reduction of a local minimal model for  modulo . (There is a similar statement for .)

One can prove using fairly general techniques that the product expression for defines a holomorphic function in some right half plane of  , i.e., the product converges for all  with Re, for some real number .

Conjecture 10.2.3   The function extends to a holomorphic function on all  .

Subsections
William Stein 2012-09-24