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 an $ L$-function $ L(E,s)$. This $ L$-function is yet another generalization of the Riemann Zeta function, that is different from the $ L$-functions attached to complex representations $ \Gal (\overline{\mathbf{Q}}/\mathbf{Q})\to \GL _n(\mathbf{C})$, which we encountered before in Section 9.5.

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

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

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

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

sage: E = EllipticCurve([1,1]); E
Elliptic Curve defined by y...
... Subgroup isomorphic to Z/2 + Z/2 associated ...
sage: T.gens()

$\displaystyle \left(\frac{20}{9661} b^{5} + \frac{147}{9661} b^{4} + \frac{700}...
...c{1315}{9661} b^{2} + \frac{5368}{28983} b + \frac{4004}{28983} : 0 : 1\right),$

$\displaystyle \left(\frac{10}{9661} b^{5} + \frac{147}{19322} b^{4} + \frac{350...
...{1315}{19322} b^{2} - \frac{23615}{57966} b + \frac{2002}{28983} : 0 : 1\right)$

We continue to assume that $ E$ is an elliptic curve over a number field $ K$. 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

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

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

$\displaystyle 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 mod $ p$ Galois representation

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

This representation $ \overline{\rho}_{E,p}$ is continuous if $ \GL _2(\mathbf{F}_p)$ is endowed with the discrete topology, because the field

$\displaystyle K(E[p]) = K(\{a,b : (a,b) \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. Unfortunately, this approach is doomed in general, since $ \GL _2(\mathbf{F}_p)$ frequently does not embed in $ \GL _2(\mathbf{C})$. The following Sage session shows that for $ p=5,7$, there are no 2-dimensional irreducible representations of $ \GL _2(\mathbf{F}_p)$, so $ \GL _2(\mathbf{F}_p)$ does not embed in $ \GL _2(\mathbf{C})$. (The notation in the output below is [degree of rep, number of times it occurs].)
sage: gap(GL(2,GF(2))).CharacterTable().CharacterDegrees()
[ ...
[ [ 1, 6 ], [ 6, 21 ], [ 7, 6 ], [ 8, 15 ] ]

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

$\displaystyle \overline{\rho}_{E,p^m}: G_K \to \Aut (E[p^m]) \approx \GL _2(\mathbf{Z}/p^m\mathbf{Z}).

We combine together all of these representations (for all $ m\geq 1$) using the inverse limit. Recall that the $ p$-adic numbers are

$\displaystyle \mathbf{Z}_p = \varprojlim \mathbf{Z}/p^m\mathbf{Z},

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

$\displaystyle \rho_{E,p}: G_K \to \Aut (\varprojlim E[p^m]) \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

$\displaystyle 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$--see [ST68]). 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

$\displaystyle 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}$.

William Stein 2012-09-24