Heegner Points and Mordell-weil Groups of Elliptic Curves over Large Fields

Let E/Q be an elliptic curve defined over Q of conductor N and let Gal(Q/Q) be the absolute Galois group of an algebraic closure Q of Q. For an automorphism σ ∈ Gal(Q/Q), we let Q be the fixed subfield of Q under σ. We prove that for every σ ∈ Gal(Q/Q), the Mordell-Weil group of E over the maximal Galois extension of Q contained in Q σ has infinite rank, so the rank of E(Q σ ) is infinite. 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 rpn is unbounded, as n goes to infinity. This paper is motivated by the following conjecture of M. Larsen [8]: Conjecture. Let K be a number field and E/K an elliptic curve over K. Then, for every σ ∈ Gal(K/K), the Mordell-Weil group E(K) of E over K = {x ∈ K | σ(x) = x} has infinite rank. In [3] and [4], we have proved this conjecture in certain cases: For a number field K and an elliptic curve E/K over K, • if E/K has a K-rational point P such that 2P = O and 3P = O, or • if 2-torsion points of E/K are K-rational, then for every automorphism σ ∈ Gal(K/K), the rank of the Mordell-Weil group E(K σ ) is infinite. Recall that a field L is said to be PAC (pseudo algebraically closed) if every absolutely irreducible nonempty variety defined over L has an L-rational point. We note that if K is a countable separably Hilbertian field, then M. Jarden has proven in [6, Theorem 2.7] that for almost all σ ∈ Gal(K/K) in the sense of Haar measure on Gal(K/K), the maximal Galois extension contained in K σ is a PAC field with the absolute Galois group isomorphic to the free profinite group on countably many generators, so the maximal Galois extension of K in K σ is smaller than K σ for almost all σ. In [3], we have shown that under the first assumption above, the Mordell-Weil group of E over the maximal Galois extension of K in K σ for every σ ∈ Gal(K/K) has infinite rank, and under the second assumption, we have shown a stronger result in [4] that the rank of E over the maximal abelian extension of K in K σ (hence, over the maximal Galois extension of K in K σ ) is infinite. Received by the editors August 4, 2004 and, in revised form, April 25, 2006. 2000 Mathematics Subject Classification. Primary 11G05. c ©2007 American Mathematical Society