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Variation of Tamagawa numbers of semistable abelian varieties in field extensions

Published online by Cambridge University Press:  16 May 2018

L. ALEXANDER BETTS
Affiliation:
King's College London, Strand, London, WC2R 2LS, United Kingdom. e-mails: alexander.betts@kcl.ac.uk, vladimir.dokchitser@kcl.ac.uk, adam.morgan@kcl.ac.uk
VLADIMIR DOKCHITSER
Affiliation:
King's College London, Strand, London, WC2R 2LS, United Kingdom. e-mails: alexander.betts@kcl.ac.uk, vladimir.dokchitser@kcl.ac.uk, adam.morgan@kcl.ac.uk
V. DOKCHITSER
Affiliation:
King's College London, Strand, London, WC2R 2LS, United Kingdom. e-mails: alexander.betts@kcl.ac.uk, vladimir.dokchitser@kcl.ac.uk, adam.morgan@kcl.ac.uk
A. MORGAN
Affiliation:
King's College London, Strand, London, WC2R 2LS, United Kingdom. e-mails: alexander.betts@kcl.ac.uk, vladimir.dokchitser@kcl.ac.uk, adam.morgan@kcl.ac.uk

Abstract

We investigate the behaviour of Tamagawa numbers of semistable principally polarised abelian varieties in extensions of local fields. In particular, we give a simple formula for the change of Tamagawa numbers in totally ramified extensions and one that computes Tamagawa numbers up to rational squares in general extensions. As an application, we extend some of the existing results on the p-parity conjecture for Selmer groups of abelian varieties by allowing more general local behaviour. We also give a complete classification of the behaviour of Tamagawa numbers for semistable 2-dimensional principally polarised abelian varieties that is similar to the well-known one for elliptic curves. The appendix explains how to use this classification for Jacobians of genus 2 hyperelliptic curves given by equations of the form y2 = f(x), under some simplifying hypotheses.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2018 

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Footnotes

*

Appendix by V. Dokchitser and A. Morgan

References

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