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Growth of $\unicode[STIX]{x0428}$ in towers for isogenous curves

Part of: Algebraic number theory: global fields Arithmetic algebraic geometry Diophantine equations

Published online by Cambridge University Press:  30 June 2015

Tim Dokchitser  and
Vladimir Dokchitser
Tim Dokchitser
Affiliation:
Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK email tim.dokchitser@bristol.ac.uk
Vladimir Dokchitser
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK email v.dokchitser@warwick.ac.uk
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Abstract

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We study the growth of $\unicode[STIX]{x0428}$ and $p^{\infty }$-Selmer groups for isogenous abelian varieties in towers of number fields, with an emphasis on elliptic curves. The growth types are usually exponential, as in the ‘positive ${\it\mu}$-invariant’ setting in the Iwasawa theory of elliptic curves. The towers we consider are $p$-adic and $l$-adic Lie extensions for $l\neq p$, in particular cyclotomic and other $\mathbb{Z}_{l}$-extensions.

Keywords

elliptic curvesSelmer groupTate–Shafarevich groupp-adic Lie extensionsisogenyIwasawa theory𝜇-invariantabelian varieties

MSC classification

Primary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11G05: Elliptic curves over global fields
Secondary: 11G10: Abelian varieties of dimension $> 1$ 11G07: Elliptic curves over local fields 11R23: Iwasawa theory 11D68: Rational numbers as sums of fractions
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 11 , November 2015 , pp. 1981 - 2005
Copyright
© The Authors 2015 

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