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

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

2004-11-24

arxiv.org/abs/math/0411534v2

Mathematics

Number Theory

18 pages. submitted, only change of the title

Scientific paper

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 algebraic points on $E$ -- the so-called {\em Heegner points} -- arising from the theory of complex multiplication.

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