Selmer groups for elliptic curves in Z_l^d-extensions of function fields of characteristic p

Details Selmer groups for elliptic curves in Z_l^d-extensions of function fields of characteristic p Selmer groups for elliptic curves in Z_l^d-extensions of function fields of characteristic p

2007-07-08

arxiv.org/abs/0707.1143v3

Mathematics

Number Theory

Final version to appear in Annales de l'Institut Fourier

Scientific paper

Let $F$ be a function field of characteristic $p>0$, $\F/F$ a Galois extension with $Gal(\F/F)\simeq \Z_l^d$ (for some prime $l\neq p$) and $E/F$ a non-isotrivial elliptic curve. We study the behaviour of Selmer groups $Sel_E(L)_r$ ($r$ any prime) as $L$ varies through the subextensions of $\F$ via appropriate versions of Mazur's Control Theorem. As a consequence we prove that $Sel_E(\F)_r$ is a cofinitely generated (in some cases cotorsion) $\Z_r[[Gal(\F/F)]]$-module.

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