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Let $E/\bbq$ be an elliptic curve defined over $\bbq$ with conductor $N$ and$\gq$ the absolute Galois group of an algebraic closure $\bar{\bbq}$ of $\bbq$.We prove that for every $\sigma\in \gq$, the Mordell-Weil group $E(\oqs)$ of$E$ over the fixed subfield of $\bar{\bbq}$ under $\sigma$ has infinite rank.Our approach uses the modularity of $E/\bbq$ and a collection of algebraicpoints on $E$ -- the so-called {\em Heegner points} -- arising from the theoryof complex multiplication.

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