Math 726: L-functions and modular forms Fall 2011

Lecture 14 : l-adic Galois representations

Instructor: Henri Darmon Notes written by: Maxime Turgeon

Let f ∈ S_{k}(Γ_{0}(N), χ) be a Hecke eigenform. In previous lectures, we saw that the L-function of such a modular form has a nice product expansion, namely,

L(f, s) = ∏_{p∤N}(1 − a_{p}(f)p^{−s} + χ(p)p^{k−1−2s})^{−1} ∏_{p|N}(1 − a_{p}(f)p^{−s})^{−1},

where a_{p}(f) is the p^{th} Fourier coefficient of f. The first factors are polynomials of degree two evaluated at p^{−s}, and the last factors, linear polynomials. We are thus led to a natural question:

Question 1. What is the connection between the Hecke L-functions and L-functions attached to two-dimensional representations of G_{ℚ}, the absolute Galois group of ℚ?

Before answering this question, let us look at an instructive example.

Example 2. Let f = E_{k} = c_{0} + q + ∑_{n=2}^{∞} σ_{k−1}(n)q^{n}, where, as usual, σ_{k−1}(n) = ∑_{d|n} d^{k−1}. Then E_{k} is a Hecke eigenform of level 1, and so its L-function has the following product expansion:

L(E_{k}, s) = ∑_{n=1}^{∞} σ_{k−1}(n)n^{−s}
= ∏_{p}(1 − σ_{k−1}(p)p^{−s} + p^{k−1−2s})^{−1}
= ∏_{p}(1 − (1 + p^{k−1})p^{−s} + p^{k−1−2s})^{−1}
= ∏_{p}(1 − p^{−s} + p^{k−1−s} + p^{k−1−2s})^{−1}
= ∏_{p}(1 − p^{−s})^{−1}(1 + p^{k−1−s})^{−1}
= ζ(s)ζ(s − k + 1),

where ζ(s) is the Riemann ζ-function. If L(E_{k}, s) were to correspond to the L-function of an Artin representation, then the first factor, namely ζ(s), corresponds to the trivial representation. However, if the second factor, namely ζ(s−k+1), was the L-function attached to some Galois representation representation (ρ^{?},V), then the image of the Frobenius element at the prime p would have to be

ρ^{?}(F rob_{p}) = p^{k−1}.

In particular, ρ^{?} cannot be a one-dimensional Artin representation. If it were the case, its image would lie in the roots of unity in ℂ^{×}, but ρ^{?}(F rob_{p}) clearly has infinite order. In order to define ρ^{?}, we need to include Galois representations with ℓ-adic rather than complex coefficients.

First, we fix a prime ℓ. Let μ_{ℓ^{n}} denote the set of ℓ^{n}-th roots of unity in \overline{ℚ}. Here is a key remark: for a prime p ≠ ℓ, the Frobenius element F rob_{p} acts on μ_{ℓ^{n}} by sending an element ζ to its p^{th} power ζ^{p}. That is, once you identitify μ_{ℓ^{n}} with the cyclic group ℤ/ℓ^{n}ℤ, it acts as multiplication by p modulo ℓ^{n}.

We thus define ℤ_{ℓ}(1) = \underleftarrow{lim} μ_{ℓ^{n}}, where the map μ_{ℓ^{n+1}} → μ_{ℓ^{n}} is the usual ℓ-power map. Note that ℤ_{ℓ}(1) is a free ℤ_{ℓ}-module isomorphic to ℤ_{ℓ} (via an identification of μ_{ℓ^{n}} with ℤ/ℓ^{n}ℤ), and that G_{ℚ} acts continuously on ℤ_{ℓ}(1). Moreover, F rob_{p} acts by multplication by p, for all p ≠ ℓ. We then convert ℤ_{ℓ}(1) into a ℚ_{ℓ}-vector space by defining ℚ_{ℓ}(1) := ℤ_{ℓ}(1) ⊗_{ℤ_{ℓ}} ℚ_{ℓ}, and we set ℚ_{ℓ}(r) := ℚ_{ℓ}(1)^{⊗r}. The Galois group G_{ℚ} thus acts naturally on ℚ_{ℓ}(r). We denote the corresponding representation by

ρ_{ℓ,r} : G_{ℚ} → ℚ_{ℓ}^{×}.

These representations satisfy the following properties:

Property 1.

1. (Rationality) Whenever p ≠ ℓ, ρ_{ℓ,r}(F rob_{p}) = p^{r} ∈ ℤ. Note however that not all elements of G_{ℓ} map into ℤ.

2. ρ_{ℓ,r} is part of a compatible system of Galois representations with coefficients in ℚ_{ℓ}, as ℓ varies. Namely, for all pair of primes ℓ_{1} ≠ ℓ_{2} non of which is equal to p , we have

ρ_{ℓ_{1},r}(F rob_{p}) = ρ_{ℓ_{2},r}(F rob_{p}).

Hence, these representations agree on Frobenius elements, but in spite of this, the representations ρ_{ℓ_{1},r} and ρ_{ℓ_{2},r} are different: ρ_{ℓ_{i},r} factors through ℚ(μ_{ℓ_{i}^{∞}}), which are linearly disjoint.

3. (Weight property) The representation ρ_{ℓ,r} has weight 2r, which means that ρ_{ℓ,r}(F rob_{p}) has eigenvalue p^{r} of absolute value |p^{r}| = p^{r}.

With this simple but prototypical example in mind, we make the following definitions.

Definition 3.

• An ℓ-adic representation of G_{ℚ} is a finite dimensional ℚ_{ℓ}-vector space V_{ℓ} equipped with a continuous homomorphism

ρ_{V_{ℓ}} : G_{ℚ} → Aut_{ℚ_{ℓ}}(V_{ℓ}).

• The representation V_{ℓ} is said to be rational if there exists a finite set S of primes such that V_{ℓ} is unramified outside of S ∪ {ℓ} (that is, for all primes p ∉ S ∪ {ℓ}, the action of ρ_{V_{ℓ}} on the inertia group I_{p} is trivial), and furthermore, for all p ∉ S ∪ {ℓ}, the characteristic polynomial of ρ_{V_{ℓ}}(F rob_{p}) has integral coefficients.

• Finally, the representation ρ_{V_{ℓ}} is said to be of weight j if all the eigenvalues of ρ_{V_{ℓ}}(F rob_{p}) (when viewed as complex numbers) have absolute value p^{\frac{j}{2}} .

We are now ready to define the required generalisation of Artin representations. From now on, let ℙ be the set of all primes.

Definition 4. A compatible system of rational ℓ-adic representations is a collection {V_{ℓ}}_{ℓ∈ℙ} of rational ℓ-adic representations such that for all primes p ≠ ℓ outside a finite set of primes S independent of ℓ, the characteristic polynomial of ρ_{V_{ℓ}}(F rob_{p}) depends only on p and not on ℓ.

We can also ask for a slightly stronger requirement on the representations V_{ℓ}. Namely, for the primes p ∈ S (called the bad primes), we can still consider the action of ρ_{V_{ℓ}}(F rob_{p}) on V_{ℓ}^{I_{p}}. One can then ask that the characteristic polynomial

L_{p}(V_{ℓ}, x) = det((1 − xρ_{V_{ℓ}}(F rob_{p}))|_{V_{ℓ}^{Ip}})

should belong to ℤ[X], and should be independent of ℓ ≠ p.

These definitions may seem a little contrived at first, but by way of justification:

1. The representations {ℚ_{ℓ}(r)}_{ℓ∈ℙ} constructed earlier form a compatible system of ℓ-adic representations of G_{ℚ}. Note that in this case, the set S may be taken to be empty.

2. We can attach to a compatible system {V_{ℓ}}_{ℓ∈ℙ} an L-function, by setting

L({V_{ℓ}}_{ℓ∈ℙ}, s) = ∏_{p∈ℙ}L_{p}(V_{ℓ}, p^{−s})^{−1},

which is a function of the complex variable s. If {V_{ℓ}}_{ℓ∈ℙ} is of weight j, then this product converges on the right half-plane Re(s) > 1 + \frac{j}{2}. In particular, we get ζ(s − k + 1) = L({ℚ_{ℓ}(r)}_{ℓ∈ℙ}, s).

3. Compatible systems are abundant in nature. More precisely, let W be a smooth projective variety over ℚ, and consider the étale cohomology groups \{H_{ét}^{i}(W\overline{ℚ}, ℚ_{ℓ})\}_{ℓ∈ℙ}; these are in fact ℚ_{ℓ}-vector spaces. The functoriality of étale cohomology then implies that H_{ét}^{i}(W\overline{ℚ}, ℚ_{ℓ})\} inherits an action of G_{ℚ}, and these families are actually compatible systems of ℓ-adic representations of G_{ℚ}. Moreover, Deligne’s proof of the Weil conjectures implies that the weight is \frac{i}{2}. In this case, the set S of bad primes corresponds to the set of primes where W has bad reduction.

We are now ready to ask a more precise question than Question 1.

Question 5. Are Hecke eigenforms attached to compatible systems of two-dimensional representations of G_{ℚ}?

The anwser to this question is yes, after a slight extension of the notion of compatible systems of ℓ-adic representation, which we now describe. For any number field K (that is, a finite extension of ℚ), we define a compatible system of K-rational representations as follows: this is a collection {V_{λ}}_{λ∈Spec(O_{K})} of K_{λ}-vector spaces (where K_{λ} is the completion of K at the prime λ) such that

ρ_{V_{λ}} : G_{ℚ} → Aut_{K_{λ}}(V_{λ})

is continuous and K-rational for all λ ∈ Spec(O_{K}), and such that this family is a compatible system in the obvious sense.

Theorem 6. Let f ∈ S_{k}(Γ_{0}, χ) be a (normalised) Hecke eigenform with Fourier coefficients in K (recall that the Fourier coefficients of f are all algebraic numbers lying in some totally real finite extension of ℚ). Then there exists a compatible system {V_{f,λ}}_{λ∈Spec(O_{K})} of λ-adic K-rational representations of G_{ℚ} such that

L({V_{f,λ}}_{λ∈Spec(O_{K})}, s) = L(f, s).

This theorem is due to various people:

• Eichler and Shimura in the case of weight 2 and completed by Igusa, who gave the precise description of the Euler factors at the bad primes;

• Deligne in the case of a general weight k > 2;

• Deligne and Serre in the case of weight 1.

In the next few lectures, we are going to discuss these results, and also try to give some idea of the constructions involved, especially in the weight 2 case.