Galois Representations and Modular Forms Progress in Mathematics Talk, August 17, 1994 Professor Kenneth A. Ribet, UC Berkeley

This lecture is about:

• Elliptic curves and abelian varieties over Q

• Clasical modular forms in one complex variable

• Representations of Galois groups over Q

• Fermat’s Last Theorem

Concerning Fermat:

In January, I lectured at the AMS Winter Meeting. At that time, there was a “gap” in Andrew Wiles’s announced proof of Fermat’s Last Theorem.

To the best of my knowledge, there has been no significant progress since last winter.

The link between Fermat’s Last Theorem and the other topics stems from a construction of G. Frey and from subsequent work by Serre, Mazur, and myself.

The idea is to work by contradiction: You assume that a^{ℓ} + b^{ℓ} = c^{ℓ} and write down Frey’s equation

y^{2} = x(x − a^{ℓ})(x + b^{ℓ}).

Here, ℓ is a prime different from 2 and 3, and a, b, and c are non-zero relatively prime integers.

Frey’s equation defines an elliptic curve, a non-singular curve in the projective plane.

Theorem. Frey’s elliptic curve does not satisfy the Taniyama-Shimura conjecture.

The theorem means that Frey’s curve cannot be linked with modular forms. If you prove that all elliptic curves over Q are “modular,” then you achieve a contradiction and settle Fermat’s Last Theorem.

Modular elliptic curves

Suppose that A is an elliptic curve over Q. For all primes p, we can “reduce A mod p,” or at least attempt to. Usually we succeed, and get an elliptic curve over F_{p}. Occasionally, we fail and say that the elliptic curve has “bad reduction at p.”

For instance, under mild conditions on A and B, y^{2} = x(x − A)(x + B) has good reduction for all p |/ ABC, where A + B + C = 0.

For each good p, let t_{p} be the number of points on A over F_{p}, counting the extra “point at infinity.”

Set a_{p} := p+1 − t_{p}.

We get a sequence a_{2}, a_{3}, a_{5}, …, where the indices run through all primes of good reduction for A.

For example, consider

A : y^{2} + y = x^{3} − x^{2}.

Only p = 11 is bad. For small p:

\begin{matrix}p & 2 & 3 & 5 & 7 & 13 & 17 & 19 & 23 & 29 \\ a_{p} & −2 & −1 & 1 & −2 & 4 & −2 & 0 & −1 & 0 \end{matrix}

Incidentally, a_{p} ≡ p + 1 mod 5 for all p in this case; equivalently, 5|t_{p} for all p. The divisibility stems from the fact that A(Q) is a cyclic group of order 5. In this remark, we allude for the first time to the group structure of A.

When A is regarded as an algebraic group, the unique point at infinity O is the origin, and any three colinear points sum to O.

The curve A : y^{2}+y = x^{3}−x^{2} is modular because the sequence a_{p} occurs in the Fourier expansion of a cusp form of weight two on the group of integral 2 × 2 matrices which are upper-triangular modulo 11 and which have determinant 1.

Such cusp forms form a one-dimensional C-vector space, generated by a unique (“normalized”) element

f = ∑_{n=1}^{∞} c_{n}e^{2πinz}, c_{1} = 1.

For all p ≠ 11, we have a_{p} = c_{p}.

More generally, for each N ≥ 1, let Γ_{0}(N) be the group of 2 × 2 integral matrices which are upper-triangular modulo N and which have determinant 1. Consider the holomorphic f(z) on the upper half-plane which satisfy

f (\frac{az + b}{cz + d}) = (cz + d)^{2}f(z)

for all ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N) and which “vanish at infinity.” These form a finite dimensional

space S(N).

Each f(z) may be expanded as a Fourier series

∑_{n=1}^{∞} c_{n}q^{n}, q := e^{2πiz}.

Eigenforms are series ∑ c_{n}q^{n} which satisfy

∑_{n=1}^{∞} \frac{c_{n}}{n^{s}} = ∏_{p|/N}(1 − c_{p}p^{−s} + p^{1−2s})^{−1} · ∏_{p|N}(1 − c_{p}p^{−s})^{−1}.

The Taniyama-Shimura conjecture states that for each elliptic curve A over Q there is some eigenform f = ∑ c_{n}q^{n} (c_{n} ∈ Z) satisfying

a_{p} = c_{p} for all good primes p.

Every curve A has a discriminant Δ_{A} and a conductor N_{A}, both of which are divisible precisely by the bad primes. For the curve A : y^{2} + y = x^{3} − x^{2}, Δ_{A} = −11 and N_{A} = 11. For y^{2} + xy + y = x^{3} − x^{2} − 3002x + 63929, the discriminant is 6561000000 = 2^{6}3^{8}5^{6} and the conductor is 90.

In more precise form, the T-S conjecture requires eigenform f for A to come from S(N_{A}).

In the opposite direction, for each eigenform f ∈ S(N) with integral coefficients, one can manufacture an A = A_{f} whose sequence (a_{p}) is derived from the coefficients of f. The conductor of A_{f} is N, whenever f is a newform in the sense that it doesn’t really belong to S(M) for some smaller M|N. The T-S conjecture states that the construction can be reversed:

Curves of conductor N T-S Conjecture

⟷ Newforms in S(N) with Z coeffs.

Since there is a construction in one direction, it is tempting to try to prove the conjecture by “counting.” Although we know the dimension of S(N), it seems impossible to know in advance the number of eigenforms with integral coefficients. Also, the elliptic curves have to be counted up to isogeny.

When N ≤ 10, S(N) = 0 and it is possible to check that there are no elliptic curves of conductor N. Similarly, when N = 11, there is exactly one newform, and one can find three elliptic curves, all isogenous, which correspond to it.

To verify the conjecture for N = 11 is to prove that there are no further elliptic curves of conductor 11. This was done in the mid-1970s by four authors who sweated to do it. Nowadays, a technique of Faltings and Serre makes fast work of this.

One possible way of making the counting easier is to include all newforms, not just those with integral coefficients. Instead of elliptic curves, one has to take abelian varieties of arbitrary dimension d ≥ 1. If f = ∑ c_{n}q^{n} is an eigenform, then the c_{n} are all algebraic integers, and the field they generate

E = Q(…, c_{n}, …)

has finite degree over Q.

Shimura associates to f an abelian variety A_{f} over Q whose dimension is the degree of E over Q. The abelian variety A_{f} has extra structure: an action of E up to isogeny.

The T-S conjecture admits a natural generalization

Newforms ⟷ Abelian vars. with extra structure.

This seems to “help,” because all newforms get considered. The new complication is that A_{f} corresponds not only to f, but also to the conjugates of f.

For example, take N = 23. Then S(N) has dimension two. There are two newforms, with coefficients in Z[\frac{1+\sqrt{5}}{2}]. Changing the sign of \sqrt{5} moves between the two forms. Up to isogeny, there is exactly one abelian variety in the correspondence. Known as J_{0}(23), it is a simple two dimensional abelian variety.

Serre’s Conjecture

An outgrowth of Frey’s elliptic curve construction was a close analysis of the conjectured relation between elliptic curves and modular forms. Serre, who was one of the analyzers, was led to make a new conjecture linking

Newforms mod ℓ ⟷ Galois representations mod ℓ.

The new conjecture — actually a series of conjectures — has to do with two-dimensional representations of the Galois group of Q: it relates them to modular forms.

Two dimensional representations of the type envisaged arise naturally from elliptic curves over Q and more generally from the abelian varieties with “extra structure” which appeared above. Modular forms give rise to such abelian varieties, and thereby to representations. The conjecture proposes a suitable converse.

In 1987, Serre conjectured a series of refined statements with striking consequences: the Taniyama-Shimura conjecture, Fermat’s Last Theorem, and a host of other results.

My work showing that T-S implies Fermat, and subsequent work of other authors, reduced the “refined” Serre conjecture to a single “coarse” statement which proposes a necessary and sufficient condition for irreducible representations to be modular.

Since Serre’s conjecture does not involve elliptic curves, we can say that it is different in spirit from the T-S conjecture. Hence it might turn out to shed light on the T-S conjecture and other related conjectures.

To introduce the representations which come into play, we review the case of elliptic curves. If A is an elliptic curve over C, we can write it as a complex torus C/L. The lattice L ≈ Z^{2} appears canonically as H_{1}(A(C), Z).

In studies of elliptic curves, the construction A ↦ L seems to play a central role.

In the 1940s and 1950s, A. Weil studied elliptic curves and abelian varieties over finite fields. He needed a replacement for L which did not rely on the topology of C. He noticed that L/nL = \frac{1}{n}L/L has the definition

A[n] := { P ∈ A(C) | nP = O }.

The collection of these groups can give the same information as L.

In practice, one studies L/nL when n is a prime → ∞, or when n = ℓ^{ν} runs through the powers of a single prime.

Weil used the definition of A[n] with C replaced by the algebraic closure of a finite field and n prime to the characteristic of this field.

However, mathematicians including Mumford, Serre, Taniyama and Tate realized that the construction

A ↦ A[n]

is fruitful in characteristic 0, for example when A is defined over Q.

In this case, A[n] acquires an action of the Galois group Gal(\overline{Q}/Q). The points in A[n], which have algebraic coordinates, are permuted around by Gal(\overline{Q}/Q); it acts on A[n] by group automorphisms:

ρ_{A,n}: Gal(\overline{Q}/Q) → Aut(A[n]).

Concretely, the image of ρ_{A,n} is a finite Galois group Gal(K/Q) embedded in the matrix group

Aut(A[n]) = GL(2, Z/nZ).

The arithmetic of A emerges from the ρ_{A,n} because the numbers a_{p} mod n (for good primes p |/ N) can be seen as matrix traces of distinguished (“Frobenius”) elements σ_{p} in the image of ρ_{A,n}.

Qualitatively: if you know the ρ_{A,n} well enough for lots of n (e.g., whenever n = ℓ is prime), then you know the numbers a_{p} (because you know them mod n for many n) and you can decide whether or not A is a modular elliptic curve.

Galois representations are attached to newforms f (on groups somewhat more general than Γ_{0}(N) and of weights k ≥ 2). If f = ∑ c_{n}q^{n} is such a newform, and if E is the field generated by the a_{n}, then there’s a representation ρ_{f,λ} for each prime (ideal) λ in the ring of integers of E:

ρ_{f,λ}: Gal(\overline{Q}/Q) → GL(2, F_{λ}),

where F_{λ} is the residue field of λ. The ρ_{f,λ} can be made from “division points” of abelian varieties with extra structure, so they are like ρ_{A,ℓ}’s.

The connection between f and the ρ_{f,λ}s is given by the congruence

tr (ρ_{f,λ}(σ_{p})) ≡ a_{p} mod λ,

which is valid for all p not dividing the product of the level of f with the prime number associated to λ.

• The determinant of ρ_{f,λ} is easy to calculate. This determinant takes the value −1 on “complex conjugation” c ∈ Gal(\overline{Q}/Q).

• The representation ρ_{f,λ} is typically irreducible, so we stick to this case.

Serre’s conjecture, in its coarse form, states that an irreducible

ρ: Gal(\overline{Q}/Q) → GL(2, F)

over a finite field is isomorphic to a ρ_{f,λ} if and only if ρ(c) = −1.

The coarse form has been shown equivalent to a refined form of the conjecture — it’s this which implies Fermat, Taniyama-Shimura and more.

The conjecture doesn’t involve any “geometry” in its statement — no abelian varieties, no elliptic curves.

I think that Serre’s conjecture is fundamental, and may ultimately be the way to prove that elliptic curves are modular. But for the moment, no one seems to know how to manufacture a form f, given a representation ρ.

We’ve surely made “Progress in Mathematics” with Serre’s conjecture, but the big payoff is not yet in sight.

This brings us to Wiles’s idea for proving T-S. This idea is “orthogonal” to the Serre conjecture method. Start with an elliptic curve A over Q. Because the finite group GL(2, F_{3}) is solvable, it is possible to show that ρ_{A,3} is of the form ρ_{f,λ}. This is already a highly non-trivial theorem of Langlands and Tunnell.

Wiles tries to show that all ρ_{A,3^{ν}} come from modular forms by a sort of bootstrapping process. Using the language of deformations, Wiles produces a surjection of Z_{3}-algebras φ: R → T and shows that the ρ_{A,3^{ν}} are modular if φ is an isomorphism.

Here T is a subtly defined Hecke ring and R is a universal deformation ring that classifies modular-looking liftings of ρ_{A,3}. The definitions of T and R are due to Wiles and to Ramakrishna, respectively.

Wiles has exhibited several interrelated criteria, each practical enough to prove that φ is an isomorphism in many cases.

Using his criteria, Wiles has proved the T-S conjecture for several infinite classes of elliptic curves — this is a first.

One criterion involves a Selmer group for the symmetric squares of the representations ρ_{A,n}. This group is analogous to classical ideal class groups. Wiles believed that he could verify this criterion by adapting the technique of M. Flach’s recent Cambridge PhD thesis.

It now appears that the construction of Flach does not generalize as easily as Wiles expected. On the other hand, some people feel that Flach’s thesis alone contains information that has yet to be “mined.”

Wiles gave the last plenary talk at the ICM in Zurich last week. According to reports, Wiles likened our current state of knowledge of the symmetric square Selmer group to the state of knowledge early in the last century when Jacobi studied the class group C_{k} of an imaginary quadratic field k. Jacobi wrote down an integer h_{k} which he conjectured to be the order of the group. He could prove only that h_{k} annihilated all elements of C_{k}. Six years later, Dirichlet proved that h_{k} is indeed the order of C_{k}.