Heegner points and Mordell-Weil groups of elliptic curves over large fields

Abstract

Let E/Q be an elliptic curve defined over Q of conductor N and let Gal((Q) over bar /Q) be the absolute Galois group of an algebraic closure Q of Q. For an automorphism sigma epsilon. Gal((Q) over bar /Q), we let (Q) over bar sigma s be the fixed sub field of Q under s. We prove that for every s. Gal((Q) over bar /Q), the Mordell-Weil group of E over the maximal Galois extension of Q contained in (Q) over bar sigma has in finite rank, so the rank of E((Q) over bar sigma) is in finite. Our approach uses the modularity of E/Q and a collection of algebraic points on E - the so-called Heegner points - arising from the theory of complex multiplication. In particular, we show that for some integer r and for a prime p prime to rN, the rank of E over all the ring class fields of a conductor of the form rp(n) is unbounded, as n goes to infinity