Modularity theorem

History

Yutaka Taniyama stated a preliminary (slightly incorrect) version of the conjecture at the 1955 international symposium on algebraic number theory in Tokyo and Nikkō as the twelfth of his set of 36 unsolved problems. Goro Shimura and Taniyama worked on improving its rigor until 1957. André Weil rediscovered the conjecture, and showed in 1967 that it would follow from the (conjectured) functional equations for some twisted L-series of the elliptic curve; this was the first serious evidence that the conjecture might be true. Weil also showed that the conductor of the elliptic curve should be the level of the corresponding modular form. The Taniyama–Shimura–Weil conjecture became a part of the Langlands program.

The conjecture attracted considerable interest when Gerhard Frey suggested in 1986 that it implies FLT. He did this by attempting to show that any counterexample to FLT would imply the existence of at least one non-modular elliptic curve. This argument was moved closer to its goal in 1987 when Jean-Pierre Serre identified a missing link (now known as the epsilon conjecture or Ribet's theorem) in Frey's original work, followed two years later by Ken Ribet's completion of a proof of the epsilon conjecture.

Even after gaining serious attention, the Taniyama–Shimura–Weil conjecture was seen by contemporary mathematicians as extraordinarily difficult to prove or perhaps even inaccessible to proof. For example, Wiles's Ph.D. supervisor John Coates states that it seemed "impossible to actually prove", and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible".

With Ribet’s proof of the epsilon conjecture, Andrew Wiles saw an opportunity: Fermat’s Last Theorem was a respectable research project because it was now a corollary of the TSW conjecture. He had expertise in Iwasawa theory; maybe there was a path from Iwasawa theory to Taniyama–Shimura–Weil.

In 1995, Andrew Wiles, with some help from Richard Taylor, proved the Taniyama–Shimura–Weil conjecture for all semistable elliptic curves. Wiles used this to prove FLT, and the full Taniyama–Shimura–Weil conjecture was finally proved by Diamond, Conrad, Diamond & Taylor; and Breuil, Conrad, Diamond & Taylor; building on Wiles's work, they incrementally chipped away at the remaining cases until the full result was proved in 1999. Once fully proven, the conjecture became known as the modularity theorem.

Several theorems in number theory similar to FLT follow from the modularity theorem. For example: no cube can be written as a sum of two coprime nth powers, n ≥ 3.

In 2025, modularity was extended to over 10% of abelian surfaces by Boxer, Calegari, Gee and Pilloni.

Generalizations

The modularity theorem is a special case of more general conjectures due to Robert Langlands. The Langlands program seeks to attach an automorphic form or automorphic representation (a suitable generalization of a modular form) to more general objects of arithmetic algebraic geometry, such as to every elliptic curve over a number field. Most cases of these extended conjectures have not yet been proved.

In 2013, Freitas, Le Hung, and Siksek proved that elliptic curves defined over real quadratic fields are modular.

Example

For example, the elliptic curve y² + y = x³ − x, with discriminant (and conductor) 37, is associated to the form

${\displaystyle f(z)=q-2q^2-3q^3+2q^4-2q^5+6q^6+\cdots ,q=e^{2\pi iz}}$

For prime numbers l not equal to 37, one can verify the property about the coefficients. Thus, for l = 3, there are 6 solutions of the equation modulo 3: (0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1); thus a(3) = 3 − 6 = −3.

The conjecture, going back to the 1950s, was completely proven by 1999 using the ideas of Andrew Wiles, who proved it in 1994 for a large family of elliptic curves.

There are several formulations of the conjecture. Showing that they are equivalent was a main challenge of number theory in the second half of the 20th century. The modularity of an elliptic curve E of conductor N can be expressed also by saying that there is a non-constant rational map defined over ℚ, from the modular curve X₀(N) to E. In particular, the points of E can be parametrized by modular functions.

For example, a modular parametrization of the curve y² − y = x³ − x is given by

${\displaystyle \begin{array}{cc}x(z)& =q^{-2}+2q^{-1}+5+9q+18q^2+29q^3+51q^4+\cdots \\ y(z)& =q^{-3}+3q^{-2}+9q^{-1}+21+46q+92q^2+180q^3+\cdots \end{array}}$

where, as above, q = e^2πiz. The functions x(z) and y(z) are modular of weight 0 and level 37; in other words they are meromorphic, defined on the upper half-plane Im(z) > 0 and satisfy

${\displaystyle x\left(\frac{az+b}{cz+d}\right)=x(z)}$

and likewise for y(z), for all integers a, b, c, d with ad − bc = 1 and 37 | c.

Another formulation depends on the comparison of Galois representations attached on the one hand to elliptic curves, and on the other hand to modular forms. The latter formulation has been used in the proof of the conjecture. Dealing with the level of the forms (and the connection to the conductor of the curve) is particularly delicate.

The most spectacular application of the conjecture is the proof of Fermat's Last Theorem (FLT). Suppose that for a prime p ≥ 5, the Fermat equation

${\displaystyle a^p+b^p=c^p}$

has a solution with non-zero integers, hence a counter-example to FLT. Then as Yves Hellegouarch was the first to notice, the elliptic curve

${\displaystyle y^2=x(x-a^p)(x+b^p)}$

of discriminant

${\displaystyle \mathrm{\Delta }=\frac{1}{256}(abc{)}^{2p}}$

cannot be modular. Thus, the proof of the Taniyama–Shimura–Weil conjecture for this family of elliptic curves (called Hellegouarch–Frey curves) implies FLT. The proof of the link between these two statements, based on an idea of Gerhard Frey (1985), is difficult and technical. It was established by Kenneth Ribet in 1987.