Modularity theorem

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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. It was proved, for all elliptic curves over the rationals whose conductor (see definition below) was not a multiple of 27, in fundamental work of Andrew Wiles and Richard Taylor. The result had previously been called the Taniyama–Shimura–Weil conjecture, or related names. The great interest in the theorem was that it was already known to imply Fermat's Last Theorem, a celebrated unsolved problem on diophantine equations.

The remaining cases of the modularity theorem (of elliptic curve not with semistable reduction) were subsequently settled by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor.

The modularity theorem is a special case of more general conjectures due to Robert Langlands. In the Langlands programme, to every elliptic curve over a number field one can associate an automorphic form or automorphic representation, a suitable generalization of a modular form. These extended conjectures have not yet been established: the method of the proof of the modularity theorem does not simply apply to them.

[edit] Statement

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.

From another point of view, given an elliptic curve E over Q we may define a corresponding L-series. The L-series is a Dirichlet series which we may write

$L(s, E) = \sum_{n=1}^\infty \frac{c_n}{n^s}.$

We can take the same coefficients, and use them to define a function in powers of q

$f(q, E) = \sum_{n=1}^\infty c_n q^n.$

If we make the substitution q = exp(2πiτ), then the series becomes a Fourier series, and so the coefficients are sometimes called "q-series coefficients", but other times "Fourier coefficients". The function obtained in this way, remarkably, is 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).

[edit] History

An incorrect version of this theorem was first conjectured by Yutaka Taniyama in September 1955. With Goro Shimura he improved its rigor until 1957. Taniyama died in 1958. 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. Ken Ribet later proved this result. In 1995, Andrew Wiles, with the partial help of Richard Taylor, proved the modularity theorem for semistable elliptic curves, which was strong enough to yield a proof of Fermat's Last Theorem.

The full modularity theorem 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.

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.

[edit] References

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, Disscusses 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.