Galois Representations Attached to Elliptic Curves

Let $E$ be an elliptic curve over a number field $K$. In this section we attach representations of $G_K = \Gal (\overline{K}/K)$ to $E$, and use them to define the $L$-function $L(E,s)$.

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

$$E[n] = \{R \in E(\overline{K}) : nR = \O\}$$

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

$$\overline{\rho}_{E,n} : G_K \to \Aut (E[n]).$$

It is a fact, which we will not prove in this book, that for any positive integer $n$, the group $E[n]$ is isomorphic as an abstract abelian group to $(\mathbf{Z}/n\mathbf{Z})^2$. There are various related ways to see why this is true. One is to use the Weierstrass $\wp$-theory to parametrize $E(\mathbf{C})$ by the the complex numbers, i.e., to find an isomorphism $\mathbf{C}/\Lambda \cong E(\mathbf{C})$, where $\Lambda$ is a lattice in $\mathbf{C}$ and the isomorphism is given by $z\mapsto (\wp(z),\wp'(z))$ with respect to an appropriate choice of coordinates on $E(\mathbf{C})$. It is then an easy exercise to verify that $(\mathbf{C}/\Lambda)[n]\cong (\mathbf{Z}/n\mathbf{Z})^2$.

Another way to understand $E[n]$ is to use that $E(\mathbf{C})_{\tor }$ is isomorphic to the quotient

$$\H_1(E(\mathbf{C}),\mathbf{Q})/\H_1(E(\mathbf{C}),\mathbf{Z})$$

of homology groups and that the homology of a curve of genus $g$ is isomorphic to $\mathbf{Z}^{2g}$. Then

$$E[n]\cong (\mathbf{Q}/\mathbf{Z})^2[n] = (\mathbf{Z}/n\mathbf{Z})^2.$$

If $n=p$ is a prime, then upon chosing a basis for the two-dimensional $\mathbf{F}_p$-vector space $E[p]$, we obtain an isomorphism $\Aut (E[p]) \cong \GL _2(\mathbf{F}_p)$. We thus obtain a two-dimensional representation

$$\overline{\rho}_{E,p} : G_K \to \GL _2(\mathbf{F}_p),$$

which is continuous if $\GL _2(\mathbf{F}_p)$ has the discrete topology, because the field

$$K(E[p]) = \{x,y : (x,y) \in E[p]\}$$

is a Galois extension of $K$ of finite degree.

In order to attach an $L$-function to $E$, one could try to embed $\GL _2(\mathbf{F}_p)$ into $\GL _2(\mathbf{C})$ and use the construction of Artin $L$-functions from Section 9.5, but this approach depends on the choice of $p$, and does not ``capture the essence'' of $E$, in that there can be many elliptic curves with exactly the same mod $p$ representation (though I think for $p\geq 23$ there are conjecturally only finitely many). Instead, we pass to a $p$-adic limit as follows. For each power $p^n$ of $p$, we have a Galois representation $\rho_{E,p^n}$. The inverse limit of these representations is a continuous homomorphism

$$\rho_{E,p}: G_K \to \Aut (\varprojlim E[p^n]) \cong \GL _2(\mathbf{Z}_p),$$

where $\mathbf{Z}_p$ is the ring of $p$-adic integers. The composition of this homomorphism with the reduction map $\GL _2(\mathbf{Z}_p) \to \GL _2(\mathbf{F}_p)$ is the representation $\overline{\rho}_{E,p}$, which we defined above, which is why we denoted it by $\overline{\rho}_{E,p}$. We next try to mimic the construction of $L(\rho,s)$ from Section 9.5 in the context of a $p$-adic Galois representation $\rho_{E,p}$.

Definition 10.2.1 (Tate module)   The $p$-adic Tate module of $E$ is

$$T_p(E) = \varprojlim E[p^n].$$

Let $M$ be the fixed field of $\ker(\rho_{E,p})$. The image of $\rho_{E,p}$ is infinite, so $M$ is an infinite extension of $K$. Fortunately, one can prove that $M$ is ramified at only finitely many primes (the primes of bad reduction for $E$ and $p$). If $\ell$ is a prime of $K$, let $D_{\ell}$ be a choice of decomposition group for some prime  $\mathfrak{p}$ of $M$ lying over $\ell$, and let $I_{\ell}$ 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 $\O_M$ over $\ell$, and let $D_{\ell}$ be the subgroup of $\Gal (M/K)$ that leaves  $\mathfrak{p}$ invariant. Then the submodule $T_p(E)^{I_{\ell}}$ of inertia invariants is a module for $D_{\ell}$ and the characteristic polynomial $F_{\ell}(x)$ of $\Frob _{\ell}$ on $T_p(E)^{I_{\ell}}$ is well defined (since inertia acts trivially). Let $R_{\ell}(x)$ be the polynomial obtained by reversing the coefficients of $F_{\ell}(x)$. One can prove that $R_{\ell}(x) \in \mathbf{Z}[x]$ and that $R_{\ell}(x)$, for $\ell\neq p$ does not depend on the choice of $p$. Define $R_{\ell}(x)$ for $\ell=p$ using a different prime $q\neq p$, so the definition of $R_{\ell}(x)$ does not depend on the choice of $p$.

Definition 10.2.2   The $L$-series of $E$ is

$$L(E,s) = \prod_{\ell} \frac{1}{R_\ell(\ell^{-s})}.$$

A prime  $\mathfrak{p}$ of $\O_K$ is a prime of good reduction for $E$ if there is an equation for $E$ such that $E \mod \mathfrak{p}$ is an elliptic curve over $\O_K/\mathfrak{p}$.

If $K=\mathbf{Q}$ and $\ell$ is a prime of good reduction for $E$, then one can show that that $R_{\ell}(\ell^{-s}) = 1 - a_\ell \ell^{-s} + \ell^{1-2s},$ where $a_{\ell} = \ell + 1 - \char93 \tilde{E}(\mathbf{F}_\ell)$ and $\tilde{E}$ is the reduction of a local minimal model for $E$ modulo $\ell$. (There is a similar statement for $K\neq \mathbf{Q}$.)

One can prove using fairly general techniques that the product expression for $L(E,s)$ defines a holomorphic function in some right half plane of  $\mathbf{C}$, i.e., the product converges for all $s$ with Re$(s)>\alpha$, for some real number $\alpha$.

Conjecture 10.2.3   The function $L(E,s)$ extends to a holomorphic function on all  $\mathbf{C}$.