How Wiles proved it

Andrew Wiles proved Fermat’s Last Theorem by proving something else: that every semistable elliptic curve over the rational numbers is modular. Earlier work by Gerhard Frey, Jean-Pierre Serre and Ken Ribet had shown that a counterexample to Fermat would produce a semistable elliptic curve that is not modular. The two results cannot both hold if a counterexample exists, so none does.

This page explains the ideas and the order they came in. It is a map of the proof, written for a reader with some undergraduate mathematics, and it leaves out nearly all of the technical work. The standard detailed account is the survey by Darmon, Diamond and Taylor.

The short version

• Frey (1985): a solution a^p + b^p = c^p gives the elliptic curve y² = x(x − a^p)(x + b^p), which looks too strange to be modular.
• Serre and Ribet (1985–1986): it is indeed not modular. Proved by Ribet in July 1986.
• Wiles (1993–1995): every semistable elliptic curve over the rationals is modular. The Frey curve is semistable. Contradiction.
• Breuil, Conrad, Diamond and Taylor (2001): every elliptic curve over the rationals is modular.

The logic in one chain

1. Suppose a^p + b^p = c^p for some prime p ≥ 5 and non-zero integers a, b, c. (Exponents 3 and 4 were settled long before.)
2. Build the Frey curve E from a, b and p. It is a semistable elliptic curve.
3. By Wiles’s theorem, E is modular.
4. By Ribet’s theorem, a modular E would give rise to a modular form of weight 2 and level 2.
5. There are no such forms. So the supposed solution cannot exist.

Steps 2 and 4 are the work of the 1980s. Step 3 is Wiles’s.

The two kinds of object

An elliptic curve over the rational numbers is a curve given by an equation such as y² = x³ + Ax + B with rational A and B and no singular points. For each prime p you can count the solutions of the equation modulo p. Writing the count, including one point at infinity, as p + 1 − a_p defines an integer a_p for each prime. The sequence of a_p is the curve’s arithmetic fingerprint.

A modular form is a function on the upper half of the complex plane with an extraordinary amount of symmetry. It has an expansion f = Σ a_nqⁿ, and for the special forms called eigenforms the coefficients a_n satisfy strict arithmetic rules. Each form has a weight and a level, both whole numbers.

A curve is modular if its fingerprint matches a modular form: there is an eigenform of weight 2 whose coefficients a_p are the curve’s a_p for all but finitely many primes. Nothing in the definitions suggests this should ever happen.

The modularity conjecture

In 1955, at a symposium in Tokyo and Nikko, Yutaka Taniyama posed problems suggesting that it happens for elliptic curves in general. Sharpened by Goro Shimura and made widely known through work of André Weil, this became the conjecture that every elliptic curve over the rational numbers is modular. It goes by several combinations of the three names. Since its proof it is called the modularity theorem.

For thirty years it had no known connection to Fermat.

Frey’s curve

The connection came from Gerhard Frey. In Ribet’s account, it “grew out of a 1985 Oberwolfach lecture”. Frey observed that a non-trivial solution a^p + b^p = c^p lets you write down the curve

y² = x(x − a^p)(x + b^p).

This Frey curve would have very unusual properties. Its discriminant is essentially (abc)^2p, a perfect p-th power many times over, which forces the points of order p on the curve to generate a number field with almost no ramification. Frey argued that such a curve should not be able to be modular. He gave an outline, not a proof, and expected experts on modular curves to complete it.

One property of the Frey curve matters later: after a harmless rearrangement of a, b and c it is semistable, a mild technical condition meaning that the curve degenerates in the gentlest possible way modulo each prime where it degenerates at all.

Serre’s epsilon and Ribet’s theorem

In August 1985 Jean-Pierre Serre, in a letter to Jean-François Mestre, identified exactly what Frey’s argument lacked. He formulated two conjectures about modular forms which, together with the modularity conjecture, would imply Fermat. Because the missing piece looked small, it was nicknamed the “epsilon conjecture”.

The statement concerns level lowering. Attached to the Frey curve E and the prime p is a Galois representation: a description of how the symmetries of the algebraic numbers act on the p² points of E whose order divides p. If E is modular, this representation comes from a modular form whose level is large, roughly the product of the primes dividing abc. The epsilon conjecture says that, because the representation is so mildly ramified, it must also come from a form of much smaller level. For the Frey curve the level drops all the way to 2.

Ken Ribet proved this in July 1986, and published it in Inventiones Mathematicae in 1990. The space of weight-2 cusp forms of level 2 is zero: the corresponding modular curve has genus 0. So a modular Frey curve leads to a non-existent form. From that moment:

modularity of semistable elliptic curves ⇒ Fermat’s Last Theorem.

Ribet recalled that this convinced mathematicians that Fermat’s claim must be true, since everyone expected the modularity conjecture to be proved one day. Almost nobody expected it soon.

Wiles’s strategy

Wiles began work on the conjecture in 1986 and kept at it for seven years. His approach has three ingredients.

Work with Galois representations, one prime at a time. For each prime ℓ, an elliptic curve E has a mod-ℓ representation (from its points of order ℓ) and an ℓ-adic representation (from its points of order ℓ, ℓ², ℓ³, … together). E is modular if its ℓ-adic representation comes from a modular form, and it is enough to show this for a single ℓ.

A modularity lifting theorem. Wiles’s central result has the shape: if the mod-ℓ representation of E is modular and irreducible, then, under suitable conditions, the ℓ-adic representation is modular too. The proof compares two rings. One, R, parametrises all the well-behaved ways of deforming the mod-ℓ representation. The other, T, is a Hecke algebra that parametrises the deformations that come from modular forms. There is a natural map from R onto T, and the theorem amounts to showing that it is an isomorphism, a statement now abbreviated R = T.

A place to start: the prime 3. Lifting needs a mod-ℓ representation already known to be modular. For ℓ = 3 this is available because the group GL₂(F₃) is solvable, which brings the representation within range of a theorem of Robert Langlands and Jerrold Tunnell. So if the mod-3 representation of E is irreducible, E is modular.

The 3–5 switch. If the mod-3 representation of E is reducible, Wiles moves to ℓ = 5. He shows there is a second semistable curve E′ with the same mod-5 representation as E and an irreducible mod-3 representation. Then E′ is modular by the argument above, so its mod-5 representation is modular, so E’s mod-5 representation is modular, and the lifting theorem at ℓ = 5 finishes the job. The case where both the mod-3 and mod-5 representations are reducible does not occur for semistable curves.

The announcement, the gap and the fix

Wiles presented the work in three lectures at the Isaac Newton Institute in Cambridge on June 21–23, 1993, announcing at the end of the third that Fermat’s Last Theorem followed.

During refereeing a gap emerged. Proving R = T required an upper bound on the size of a Selmer group, and the 1993 manuscript obtained it by constructing an Euler system, extending a method of Matthias Flach. The construction did not hold up. On December 4, 1993, Wiles acknowledged that the calculation of the bound “is not yet complete as it stands”.

He worked on the repair through 1994, joined by his former student Richard Taylor. On September 19, 1994, he saw a way around the Euler system altogether. In Ribet’s description, the new method came from an observation Wiles had made early on: the required bound would follow if certain Hecke algebras could be shown to be complete intersection rings. That statement is the main theorem of the joint paper by Taylor and Wiles.

The two manuscripts were released in October 1994 and published together in May 1995 in the Annals of Mathematics, volume 141:

• Andrew Wiles, “Modular elliptic curves and Fermat’s Last Theorem”, pages 443–551.
• Richard Taylor and Andrew Wiles, “Ring-theoretic properties of certain Hecke algebras”, pages 553–572.

Together they run to 129 pages.

The full modularity theorem

Wiles proved modularity for semistable curves, which was all Fermat needed. Fred Diamond soon weakened the semistability condition. In 2001 Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor removed it completely in “On the modularity of elliptic curves over Q: wild 3-adic exercises”, published in the Journal of the American Mathematical Society, volume 14, pages 843–939. Every elliptic curve over the rational numbers is modular.

The method outlived the problem. Modularity lifting, the R = T idea and the Taylor–Wiles patching argument became standard tools of number theory. The Abel Prize citation in 2016 honoured Wiles’s proof for “opening a new era in number theory”.

What this page leaves out

Almost everything that makes the proof hard: the deformation theory of Galois representations, the commutative algebra criterion behind R = T, the Taylor–Wiles system of auxiliary primes, and the arithmetic of modular curves that underlies Ribet’s theorem. Each is a subject in itself. For the events around the mathematics, see the history of Fermat’s Last Theorem. For the effort to have every step checked by computer, see Formalizing Fermat’s Last Theorem in Lean.

Frequently asked questions

How did Andrew Wiles prove Fermat's Last Theorem?

By contradiction, through elliptic curves. A solution to Fermat's equation would define an elliptic curve, the Frey curve. Ribet had shown that this curve could not be modular. Wiles proved that every semistable elliptic curve over the rationals, the Frey curve included, is modular. So no solution can exist.

What is the Taniyama-Shimura conjecture?

It is the statement, now a theorem called the modularity theorem, that every elliptic curve over the rational numbers is modular: the numbers of points on the curve modulo each prime are encoded in the coefficients of a modular form. Wiles proved it for semistable curves, and Breuil, Conrad, Diamond and Taylor completed it in 2001.

What was the gap in Wiles's 1993 proof?

The 1993 argument needed an upper bound on the size of a certain Selmer group, and the method used to get it, an Euler system construction, did not work as claimed. In 1994 Wiles and Richard Taylor obtained the bound a different way, by proving that certain Hecke algebras are complete intersection rings.

What did Ribet's theorem show?

Ken Ribet proved in July 1986 that the Galois representation attached to a Frey curve, if the curve were modular, would have to come from a modular form of weight 2 and level 2. No such form exists. So the modularity of semistable elliptic curves implies Fermat's Last Theorem.

Did Wiles prove Fermat's Last Theorem alone?

The decisive work was his, done largely in isolation between 1986 and 1993. The final proof depends on a second paper written jointly with Richard Taylor, and on earlier results of Frey, Serre, Ribet, Mazur, Langlands, Tunnell and many others.