Descending Rational Points on Elliptic Curves to Smaller Fields

Abstract In this paper, we study the Mordell-Weil group of an elliptic curve as a Galois module. We consider an elliptic curve E defined over a number field K whose Mordell-Weil rank over a Galois extension F is 1, 2 or 3. We show that E acquires a point (points) of infinite order over a field whose Galois group is one of C n × C m ( n = 1, 2, 3, 4, 6, m = 1, 2), D n × C m ( n = 2, 3, 4, 6, m = 1, 2), A 4 × C m ( m = 1, 2), S 4 × C m ( m = 1, 2). Next, we consider the case where E has complex multiplication by the ring of integers of an imaginary quadratic field contained in K. Suppose that the -rank over a Galois extension F is 1 or 2. If ≠ and and h (class number of ) is odd, we show that E acquires positive -rank over a cyclic extension of K or over a field whose Galois group is one of SL 2 ( /3 ), an extension of SL 2 ( /3 ) by /2 , or a central extension by the dihedral group. Finally, we discuss the relation of the above results to the vanishing of L -functions.