**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.