Stark - Heegner points Course and Student Project description Arizona Winter School 2011

is the Hecke L-series attached to the eigenform f . Hecke’s theory shows that L(f, s) has an Euler product expansion identical to (2), and also that it admits an integral representation as a Mellin transform of f . This extends L(f, s) analytically to the whole complex plane and shows that it satisfies a functional equation relating its values at s and 2− s. The modularity of E thus implies that L(E, s), which a priori is only defined on the right half-plane {s ∈ C,Re(s) > 3/2} of absolute convergence for (2), enjoys a similar analytic continuation and functional equation. This fact is of great importance for the theory of elliptic curves. For example, the Birch and Swinnerton-Dyer conjecture equates the rank of the Mordell-Weil group E(Q) to the order of vanishing of L(E, s) at s = 1: rank(E(Q)) ? = ran(E/Q) := ords=1(L(E, s)). (4)