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GENERALIZED EXPLICIT DESCENT AND ITS APPLICATION TO CURVES OF GENUS 3

Part of: Curves Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  17 February 2016

NILS BRUIN ,
BJORN POONEN  and
MICHAEL STOLL
NILS BRUIN
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada; [email protected]
BJORN POONEN
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA; [email protected]
MICHAEL STOLL
Affiliation:
Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany; [email protected]

Abstract

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We introduce a common generalization of essentially all known methods for explicit computation of Selmer groups, which are used to bound the ranks of abelian varieties over global fields. We also simplify and extend the proofs relating what is computed to the cohomologically defined Selmer groups. Selmer group computations have been practical for many Jacobians of curves over  $\mathbb{Q}$ of genus up to 2 since the 1990s, but our approach is the first to be practical for general curves of genus 3. We show that our approach succeeds on some genus 3 examples defined by polynomials with small coefficients.

MSC classification

Primary: 11G30: Curves of arbitrary genus or genus $ne 1$ over global fields
Secondary: 11G10: Abelian varieties of dimension $> 1$ 14G25: Global ground fields 14H45: Special curves and curves of low genus
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 4 , 2016 , e6
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2016

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