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Finding large Selmer rank via an arithmetic theory of local constants | | Barry Mazur
; Karl Rubin
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; | | Date: |
5 Dec 2005 | | Subject: | Number Theory | | Abstract: | 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 $F$. | | Source: | arXiv, math/0512085 | | Services: | Forum | Review | PDF | Favorites | |
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