The Shimura-taniyama Conjecture

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 Γ0(N) be the group of matrices in SL2(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 Γ0(N) is an analytic function f onH satisfying the relation f ( az+b cz+d ) = (cz+d)f(z), for all ( a b c d ) ∈ Γ0(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 with no constant term: f(z) = ∑∞ n=1 λnq . The Dirichlet series L(f, s) = ∑ λnn −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) ∫∞ 0 f(iy)y s−1dy. The space of cusp forms of weight 2 on Γ0(N) is a finite-dimensional vector space and is preserved by the involution WN defined by WN(f)(z) = Nz 2f(−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 Γ0(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