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Galois Theory for the Selmer Group of an Abelian Variety

Published online by Cambridge University Press:  04 December 2007

Ralph Greenberg
Ralph Greenberg
Affiliation:
Department of Mathematics, University of Washington, Seattle, WA 98195-4350, U.S.A. e-mail: [email protected]
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Abstract

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This paper concerns the Galois theoretic behavior of the p-primary subgroup SelA(F)p of the Selmer group for an Abelian variety A defined over a number field F in an extension K/F such that the Galois group G(K/F) is a p-adic Lie group. Here p is any prime such that A has potentially good, ordinary reduction at all primes of F lying above p. The principal results concern the kernel and the cokernel of the natural map sK/F′ SelA(F′)p →  SelA(K)pG(K/F′) where F′ is any finite extension of F contained in K. Under various hypotheses on the extension K/F, it is proved that the kernel and cokernel are finite. More precise results about their structure are also obtained. The results are generalizations of theorems of B. Mazurand M. Harris.

Keywords

Abelian varietyGalois theorySelmer group
Type
Research Article
Information
Compositio Mathematica , Volume 136 , Issue 3 , May 2003 , pp. 255 - 297
Copyright
© 2003 Kluwer Academic Publishers