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Published online by Cambridge University Press:  17 February 2016

Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada;
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA;
Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany;


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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.

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