Finding large Selmer rank via an arithmetic theory of local constants

Details Finding large Selmer rank via an arithmetic theory of local constants Finding large Selmer rank via an arithmetic theory of local constants

2005-12-05

arxiv.org/abs/math/0512085v2

Mathematics

Number Theory

Revised and improved. To appear in Annals of Mathematics

Scientific paper

We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose $K/k$ is a quadratic extension of number fields, $E$ is an elliptic curve defined over $k$, and $p$ is an odd prime. Let $F$ denote the maximal abelian $p$-extension of $K$ that is unramified at all primes where $E$ has bad reduction and that is Galois over $k$ with dihedral Galois group (i.e., the generator $c$ of $Gal(K/k)$ acts on $Gal(F/K)$ by -1). We prove (under mild hypotheses on $p$) that if the rank of the pro-$p$ Selmer group $S_p(E/K)$ is odd, then the rank of $S_p(E/L)$ is at least $[L:K]$ for every finite extension $L$ of $K$ in $F$.

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