Title : The rank of elliptic curves over large fields


Abstract : In this talk, we consider an elliptic curve E over a number field K and the Mordell-Weil group of E over the fixed fields under elements of the absolute Galois group G over K. First, we show that if all 2-torsion points of E are K-rational, then for every element g of G, the rank of E over the fixed field under g is infinite. And then we show that if K is totally imaginary and E has a K-rational point which is not 2-torsion, then we get the same result. We also show that the rank of E over the fixed field under any complex conjugate automorphism of G is infinite.