Update and Overview on Fermat’s Last Theorem AMS-MAA Invited Hour Talk, January 13, 1994 Professor Kenneth A. Ribet, UC Berkeley

Elliptic Curves

Elliptic curves over Q are non-singular plane cubic curves such as

\{ \begin{matrix}y^{2} + y = x^{3} − x^{2}, \\ y^{2} = x^{3} − x, \\ y^{2} = x(x − 3)(x + 32), \\ y^{2} + xy = x^{3} + x^{2} − 23x + 33 \end{matrix}

which are given by equations with rational coordinates.

We view them as projective curves with a distinguished point O. This point is the “point at infinity,” the one point on the curve not visible in the affine plane.

We can easily view the set of real points of an elliptic curve [there was a picture of an elliptic curve here]:

The complex points form a complex torus — a doughnut.

Elliptic curves are groups — give me two points on an elliptic curve and I’ll hand you a third. This is because (in general) straight lines intersect an elliptic curve in three points.

The origin of the group law is the point that we know about — the point at infinity. In other words, O = 0.

The process of adding points together is called the “chord and tangent” process. If we add a point to itself, we have to draw a tangent instead of a chord, or secant line.

Points mod p

In number theory, it is often easier to study equations by replacing “=” by “≡ mod p,” where p is a varying prime. If you take an equation like y^{2} = x^{3} − x and reduce it mod p, you get a cubic curve, but it might be singular. If not, then the original curve has good reduction at p and we get an elliptic curve over the field F_{p} with p elements.

For example, the equation

y^{2} + y = x^{3} − x^{2}

defines an elliptic curve over F_{2}, so the elliptic curve y^{2} + y = x^{3} − x^{2} has good reduction at 2.

An important activity has been to count the number of points on an elliptic curve over F_{p}. For example, Gauss found a formula for the number of points on y^{2} = x^{3} − x over F_{p}.

There are at most p^{2} + 1 points on a curve a priori because there are p values each for x and y and we have to add one point at infinity. A theorem of Hasse states that the number of points differs from p + 1 by a number smaller than 2\sqrt{p} in absolute value.

It is traditional to define

a_{p} := p+1 − #E(F_{p}).

Here the notation suggests correctly that p varies and that E is a fixed elliptic curve over Q.

For example, take p = 2 and look at

y^{2} + y = x^{3} − x.

There are five points on the curve and so

a_{2} = −2.

The Taniyama-Shimura Conjecture

The elliptic curve y^{2} +y = x^{3} −x^{2} has good reduction at all p ≠ 11. It is a fact that the a_{p} for this curve are the pth coefficients of the formal series ∑ c_{n}q^{n} obtained by developing the product expansion

q ∏_{n=1}^{∞}(1 − q^{n})^{2}(1 − q^{11n})^{2}.

The series ∑ c_{n}e^{2πinz} is a weight-two cusp form on a congruence subgroup of SL(2, Z) called Γ_{0}(11). Here z is the “variable,” a complex number with positive imaginary part.

The Taniyama-Shimura Conjecture says that all elliptic curves over Q are like this. In other words, for each E over Q there is to be a weight-two cusp form f = ∑ c_{n}e^{2πinz} on an appropriate congruence subgroup of SL(2, Z) with the property that a_{p}(E) = c_{p} for all good primes p. The previous example is thus supposed to be typical, except that there is usually no cute product expansion giving the coefficients.

This amazing conjecture links the algebraic world of elliptic curves with the analytic world of modular forms.

If E satisfies the conjecture, we say that it is “modular” — it is a modular elliptic curve.

To an elliptic curve over Q we can attach a rational number — its j-invariant. Every rational number is the j-invariant of some elliptic curve over Q. Two elliptic curve with the same j-invariant are isomorphic over C, but not necessarily over Q. Nevertheless, if E and E′ have the same j-invariant, they are either both modular or both non-modular.

In fact, the modularity of an elliptic curve can be seen over C. This leads to a description involving hyperbolic geometry. . ..

Before Wiles’s work, we could exhibit only finite sets of numbers which were the j-invariants of modular elliptic curves over Q. For example, Shimura proved in 1971 that elliptic curves with complex multiplication are modular. One year ago, proving that all elliptic curves are modular seemed to be completely out of reach.

In June, Wiles announced a proof that two disjoint classes of elliptic curves over Q were modular. Both classes correspond to infinite numbers of j-invariants.

One of the classes has an application to Fermat’s Last Theorem; this class is the one occluded by the famous “gap”:

In view of the speculation on the status of my work on the Taniyama-Shimura conjecture and Fermat’s Last Theorem I will give a brief account of the situation. During the review process a number of problems emerged, most of which have been resolved, but one in particular I have not yet settled. The key reduction of (most cases of ) the Taniyama-Shimura conjecture to the calculation of the Selmer group is correct. However the final calculation of a precise upper bound for the Selmer group in the semistable case (of the symmetric square representation associated to a modular form) is not yet complete as it stands. I believe that I will be able to finish this in the near future using the ideas explained in my Cambridge lectures.

The fact that a lot of work remains to be done on the manuscript makes it still unsuitable for release as a preprint . In my course in Princeton beginning in February I will give a full account of this work.

Andrew Wiles.

According to Karl Rubin and Alice Silverberg, the second class includes the infinite set of j-invariants obtained by plugging into the formula

(\frac{12(1 + 18t^{2} − 27t^{4})}{(1 − 18t^{2} − 27t^{4})})^{3}

all rational numbers t with no 3 in their denominators.

The idea of Wiles is that being modular is a contagious disease. The elliptic curves in the Rubin-Silverberg example contracted their modularity from the curve y^{2} = x^{3} − x.

Taniyama-Shimura ⟹ Fermat

In 1985, G. Frey initiated the construction of elliptic curves whose coefficients depend on the a, b and c occurring in a Fermat counterexample a^{ℓ} + b^{ℓ} = c^{ℓ}. (Here a, b and c are non-zero and relatively prime — and ℓ is a prime ≥ 5.)

Frey proposed that

y^{2} = x(x − a^{ℓ})(x + b^{ℓ})

could not be modular.

Serre laid out a specific strategy for showing this, and in 1987 I proved the conjectures of Serre which justified this strategy.

Frey’s curves belong to the class of semistable elliptic curves. The curve for (a, b, c) has bad reduction when two of the roots of x(x − a^{ℓ})(x + b^{ℓ}) = 0 mod p are the same. This happens precisely when p divides abc. The semistability stems from the fact that the three roots can never all be equal.

In June, Wiles announced that he could prove that all semistable elliptic curves over Q are modular. The proof of this statement depends on an inequality involving Selmer groups, which was to be obtained by the methods of Kolyvagin and Flach.

The “gap” occurs in the application of those methods to Wiles’s problem.

Elements of Wiles’s Work

Wiles fixes a prime p; we will take it to be 3. Let E be a semistable elliptic curve and consider the group of 3-division points

E[3] = { P ∈ E(C) | P + P + P = 0 },

where “+” is the group operation. By thinking about the complex torus E(C), we realize that E[3] is the direct sum of two cyclic groups of order 3: E[3] ≈ (Z/3Z)^{2}.

Now E[3] is stable by all automorphisms of C (which act on the coordinates of points), so its points have algebraic coordinates. Conjugation is compatible with addition, so conjugations are described by 2 × 2 matrices with entries in F_{3}.

But E[3] is modular, according to a hard theorem of Langlands and Tunnell (Saito-Shintani lifting). This means that there is an f = ∑ c_{n}q^{n} for which the a_{p} (with p ≠ 3) are congruent to the c_{p} mod 3 (or, actually, modulo an ideal dividing 3 in a ring of algebraic integers).

The contagion metaphor refers to the principle that the modularity of E[3] should propagate out to the modularity of E. (In the process, f changes.) This doesn’t work if E[3] is reducible in the sense that it has a non-zero cyclic subgroup stable under conjugation. In this case, Wiles has an ingenious argument for switching from 3 to 5. (Ingenuity is required because there is no Langlands-Tunnell theorem for 5.)

Deformations

Let G be the Galois group of the field of algebraic complex numbers. The “conjugation” on E[3] is a continuous homomorphism ρ: G → GL(2, F_{3}). It comes equipped with a lift

ρ_{E}: G → GL(2, Z_{3})

gotten by considering all 3-power division points instead of 3-division points. To prove that E is modular is to prove that ρ_{E} is modular.

That E[3] is already modular implies that there is a second lift

ρ_{f}: G → GL(2, O)

coming from the modular form f. Here O is the integer ring of a 3-adic field. We can think: O = Z_{3} without loss of generality.

The lifts of ρ are called “deformations.” A conjecture of Mazur states that all deformations with suitable local properties should be modular. Using commutative algebra, Wiles reduces Mazur’s conjecture to what has been termed a “count,” a single inequality. We want to prove something along the lines:

#(lifts) ≤^{?} #(forms congruent to f).

We’ll call this inequality “(∗)”.

In mathematical language, the left-hand side turns into a Selmer group; this is a subgroup of the cohomology group

H^{1}(G, Ad(ρ_{f}) ⊗_{Z_{3}} Q_{3}/Z_{3}).

The right-hand side can be expressed in a number of ways. Through work of H. Hida, it can be related to the “algebraic part” of the value at s = 2 of the symmetric square L-function made from f.

The “fundamental inequality” (∗) then strongly resembles an equality which was conjectured by Bloch and Kato (in a very general context) several years ago.

There are many possible inequalities (∗), obtained as we tighten or relax the local conditions that we impose on lifts. Wiles proves that (∗) for a given problem implies the analogous inequality for all looser problems. One can also show that f may be chosen so that ρ_{f} obeys all the constraints suggested by ρ. This allows Wiles to assume that the deformation problem is as tight as possible (“minimal problem”).

Even if the “gap” does not get filled in the coming weeks, it appears likely that the Taniyama-Shimura Conjecture for E will have been reduced to a single numerical inequality: (∗) for the minimal deformation problem.