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In mathematics, the modularity theorem establishes an important connection between elliptic curves over the field of rational numbers and modular forms, certain analytic functions introduced in 19th century mathematics. In 1995, Andrew Wiles proved the modularity theorem for all semistable elliptic curves over the rationals, with some help from Richard Taylor. The case of the remaining (non-semistable) curves was subsequently settled jointly by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor by 2001. Before its proof the statement (see below) was known as the Taniyama–Shimura–Weil conjecture and by several related names.

The modularity theorem is a special case of more general conjectures due to Robert Langlands. The Langlands programme 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.

The theorem states that any elliptic curve over Q can be obtained via a rational map with integer coefficients from the classical modular curve

X₀(N)

for some integer N; this is a curve with integer coefficients with an explicit definition. This mapping is called a modular parametrization of level N. If N is the smallest integer for which such a parametrization can be found (which by the modularity theorem itself is now known to be a number called the conductor), then the parametrization may be defined in terms of a mapping generated by a particular kind of modular form of weight two and level N, a normalized newform with integer q-expansion, followed if need be by an isogeny.

The modularity theorem implies a closely related analytic statement: to an elliptic curve E over Q we may attach a corresponding L-series. The L-series is a Dirichlet series, commonly written

The generating function of the coefficients a_n is then

If we make the substitution q = exp(2πiτ), we see that we have written the Fourier expansion of a function f(τ,E) of the complex variable τ, so the coefficients of the q-series, are also thought of as the Fourier coefficients of f. The function obtained in this way is, remarkably, a cusp form of weight two and level N and is also an eigenform (an eigenvector of all Hecke operators); this is the Hasse–Weil conjecture, which follows from the modularity theorem.

Some modular forms of level two, in turn, correspond to holomorphic differentials for an elliptic curve. The Jacobian of the modular curve can (up to isogeny) be written as a product of irreducible abelian varieties, corresponding to Hecke eigenforms of weight 2. The 1-dimensional factors are elliptic curves (there can also be higher dimensional factors, so not all Hecke eigenforms correspond to rational elliptic curves). The curve we obtain by finding the corresponding cusp form, and then constructing a curve from it, is isogenous to the original curve (but not in general isomorphic to it).

An incorrect version of this theorem was first conjectured by Yutaka Taniyama in September 1955. Together with Goro Shimura he worked on improving its rigor until 1957. The conjecture was rediscovered by André Weil in 1967, who showed 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. In the 1970s it became associated with the Langlands program of unifying conjectures in mathematics.

It attracted considerable interest in the 1980s when Gerhard Frey suggested that the Taniyama–Shimura–Weil conjecture implies Fermat's last theorem. He did this by attempting to show that any counterexample to Fermat's last theorem would give rise to a non-modular elliptic curve. However, it wasn't complete. In order to imply Fermat's last theorem from the Taniyama–Shimura–Weil conjecture, he needed a little bit more. In mathematics, a little bit more is commonly denoted as ε, or epsilon, and this little bit more which was needed to link Taniyama-Shimura-Weil to Fermat's last theorem was identified by Jean-Pierre Serre and became known as the epsilon conjecture. In the summer of 1986, Ken Ribet proved the epsilon conjecture, thereby proving that the Taniyama–Shimura–Weil conjecture together with the epsilon conjecture, implied Fermat's Last Theorem. In 1995, Andrew Wiles, with the partial help of Richard Taylor, proved the Taniyama–Shimura–Weil conjecture for a class of elliptic curves called semistable elliptic curves, which was strong enough to yield a proof of Fermat's Last Theorem.

The full Taniyama–Shimura–Weil conjecture was finally proved in 1999 by Breuil, Conrad, Diamond, and Taylor who, building on Wiles' work, incrementally chipped away at the remaining cases until the full result was proved. The now fully proved conjecture became known as the modularity theorem.

Several theorems in number theory similar to Fermat's last theorem follow from the modularity theorem. For example: no cube can be written as a sum of two coprime n-th powers, n ≥ 3. (The case n = 3 was already known by Euler.)

In March 1996 Wiles shared the Wolf Prize with Robert Langlands.

• Henri Darmon: A Proof of the Full Shimura-Taniyama-Weil Conjecture Is Announced, Notices of the American Mathematical Society, Vol. 46 (1999), No. 11. Contains a gentle introduction to the theorem and an outline of the proof.
• Christophe Breuil, Brian Conrad, Fred Diamond, Richard Taylor: On the modularity of elliptic curves over Q: Wild 3-adic exercises, Journal of the American Mathematical Society 14 (2001), pp. 843–939. Contains the proof of the modularity theorem.
• Barry Mazur, Number theory as gadfly- American Mathematical Monthly, 98 (7), August-September 1991, pp. 593–610, Discusses the Taniyama-Shimura conjecture 3 years before it was proven for infinitely many cases.
• Weil, André Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann. 168 1967 149-156.
• Wiles, Andrew Modular elliptic curves and Fermat's last theorem. Ann. of Math. (2) 141 (1995), no. 3, 443--551.
• Taylor, Richard; Wiles, Andrew Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2) 141 (1995), no. 3, 553--572
• Wiles, Andrew Modular forms, elliptic curves, and Fermat's last theorem. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 243--245, Birkhäuser, Basel, 1995.