Elliptic Curves over Real Quadratic Fields are Modular

We prove that all elliptic curves defined over real quadratic fields are modular.Comment: 38 pages. Magma scripts available as ancillary files with this arXiv versio

SIMPLE THINGS WE DON’T KNOW

Abstract. This is a quite faithful rendering of a Colloquio De Giorgi I had the honor to give at Scuola Normale Superiore on March 21, 2012. The idea was to explain some open problems in arithmetic algebraic geometry which are simple to state but which remain shrouded in mystery. 1. An interactive game: dimension zero Suppose I give you an integer N ≥ 2, and tell you that I am thinking of a monic integer polynomial f(X) ∈ Z[X] whose discriminant ∆(f) divides some power of N. I tell you further, for every prime number p not 1 dividing N, the number np(f): = #{x ∈ Fp|f(x) = 0 in Fp} of its solutions in the prime field Fp: = Z/pZ. You must then tell me the degree of the polynomial f. In this “infinite ” version, where I tell you the np(f) for every good prime, your task is simple; the degree of f is simply the largest of th