THE SHIMURA-TANIYAMA CONJECTURE - Also re-

ferred to in the literature as the Shimura-Taniyama-Weil conjecture, the Taniyama-Shimura conjecture, the Taniyama-Weil conjecture, or the modularity conjecture, it postulates a deep connection between elliptic curves over the rational numbers and modular forms. It has now been almost completely proved thanks to the fundamental work of A. Wiles and R. Taylor [W], [TW], and its further refinements [Di], [CDT].

Let Γ_{0}(N) be the group of matrices in SL_{2}(Z) which are upper triangular modulo a given positive integer N. It acts as a discrete group of Mobius transformations on the Poincaré upper half-plane H := {z ∈ C|Im(z) > 0}. A cusp form of weight 2 for Γ_{0}(N) is an analytic function

f on H satisfying the relation f (\frac{az+b}{cz+d}) = (cz+d)^{2}f(z), for all ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈

Γ_{0}(N), together with suitable growth conditions on the boundary of H. (Cf. Modular forms.) The function f is periodic of period 1, and it can be written as a Fourier series in q = e^{2πiz} with no constant term: f(z) = ∑_{n=1}^{∞} λ_{n}q^{n}. The Dirichlet series L(f, s) = ∑ λ_{n}n^{−s} is called the L-function attached to f. (Cf. L-functions.) It is essentially the Mellin transform of f: Λ(f, s) := Γ(s)L(f, s) = (2π)^{s} ∫_{0}^{∞} f(iy)y^{s−1}dy. The space of cusp forms of weight 2 on Γ_{0}(N) is a finite-dimensional vector space and is preserved by the involution W_{N} defined by W_{N}(f)(z) = Nz^{2}f(\frac{−1}{Nz}). Hecke showed that if f lies in one of the two eigenspaces for this involution (with eigenvalue w = ±1) then L(f, s) satisfies the functional equation: Λ(f, s) = −wΛ(f, 2 − s), and that L(f, s) has an analytic continuation to all of C.

Let E be an elliptic curve over the rationals, and let L(E, s) denote its Hasse-Weil L-series. The curve E is said to be modular if there exists a cusp form f of weight 2 on Γ_{0}(N) for some N such that L(E, s) = L(f, s). The Shimura-Taniyama conjecture asserts that every elliptic curve over Q is modular. Thus it gives a framework for proving the analytic continuation and functional equation for L(E, s). It is prototypical of a general relationship between the L-functions attached to arithmetic objects and those attached to automorphic forms, as described in the far-reaching Langlands program.

Weil’s refinement of the conjecture predicts that the integer N is equal to the arithmetic conductor of E. Thanks to the ideas introduced by Wiles (cf. [CDT]) one now knows that E is modular, if 27 does not divide the conductor of E. Wiles’ proof proceeds by viewing the Shimura-Taniyama conjecture in a wider framework which predicts the modularity of the (twodimensional) Galois representations arising from the cohomology of varieties over Q.

The modularity of E can also be formulated as the statement that E is a quotient of the modular curve X_{0}(N) over Q; this curve represents the solution to the moduli problem of classifying pairs (A, C) consisting of an elliptic curve A with a distinguished cyclic subgroup C of order N. Alternately, if E is modular, then there is a (non-constant) complex analytic uniformisation H/Γ_{0}(N) ⟶ E(C).

The importance of the Shimura-Taniyama conjecture is manifold. Firstly it gives the analytic continuation of L(E, s) for a large class of elliptic curves. The L-function itself plays a key role in the study of E, most notably through the celebrated Birch and Swinnerton-Dyer conjecture. Secondly, the modular curve X_{0}(N) is endowed with a natural collection of algebraic points arising from the theory of complex multiplication, and the existence of a modular parametrisation allows the construction of points on E defined over abelian extensions of certain imaginary quadratic fields. This fact was exploited by Gross-Zagier and Kolyvagin to give strong evidence for the Birch and Swinnerton-Dyer conjecture for E, under the assumption that E is modular.

The Shimura-Taniyama conjecture admits various generalizations. Replacing Q by an arbitrary number field K, it predicts that an elliptic curve E over K is associated to an automorphic form on GL_{2}(K). When K is totally real, such an E is often uniformized by a Shimura curve attached to a suitable quaternion algebra over K with exactly one split place at infinity (when K is of odd degree, or when E has at least one prime of multiplicative reduction.) In the context of function fields over finite fields, the Shimura-Taniyama conjecture admits an analogue which was established earlier by Drinfeld using methods different from those of Wiles.

References

[CDT] CONRAD, B., DIAMOND, F. AND TAYLOR, R.: ‘Modularity of certain potentially Barsotti-Tate Galois representations’. J. Amer.

Math. Soc., to appear.

[Di] DIAMOND, F.: ‘On deformation rings and Hecke rings’. Ann. of Math. (2) 144 (1996), no. 1, 137–166.

[TW] TAYLOR, R., AND WILES, A.: ‘Ring-theoretic properties of certain Hecke algebras’, Annals of Mathematics. (2) 141 (1995), no. 3, 553– 572.

[W] WILES, A.: ‘Modular elliptic curves and Fermat’s last theorem’. Annals of Mathematics. (2) 141 (1995), no. 3, 443–551.

H. Darmon