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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 ,
VLADIMIR DOKCHITSER ,
V. DOKCHITSER  and
A. MORGAN
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
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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
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 166 , Issue 3 , May 2019 , pp. 487 - 521
Copyright
Copyright © Cambridge Philosophical Society 2018 

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Footnotes

*

Appendix by V. Dokchitser and A. Morgan

References

REFERENCES

[1] Bhargava, M., Gross, B. H. and Wang, X.. Pencils of quadrics and the arithmetic of hyperelliptic curves. J. Amer. Math. Soc. 30 (2017), 451–493.10.1090/jams/863Google Scholar
[2] Coates, J., Fukaya, T., Kato, K. and Sujatha, R.. Root numbers, Selmer groups and non-commutative Iwasawa theory. J. Algebr. Geom. 19, no. 1, (2010), 19–97.10.1090/S1056-3911-09-00504-9Google Scholar
[3] Dokchitser, T. and Dokchitser, V.. Regulator constants and the parity conjecture. Invent. Math. 178, no. 1 (2009), 23–73.10.1007/s00222-009-0193-7Google Scholar
[4] Dokchitser, T. and Dokchitser, V.. Root numbers and parity of ranks of elliptic curves. J. Reine Angew. Math. 658 (2011), 39–64.Google Scholar
[5] Dokchitser, T. and Dokchitser, V.. Growth of III in towers for isogenous curves. Compositio Math. 151 (2015), 1981–2005.10.1112/S0010437X15007423Google Scholar
[6] Dokchitser, T., Dokchitser, V., Maistret, C. and Morgan, A.. Arithmetic of hyperelliptic curves over local fields, preprint (2017).Google Scholar
[7] Grothendieck, A.. Modèles de Néron et monodromie, LNM 288, Séminaire de Géométrie 7, Exposé IX (Springer-Verlag, 1973).Google Scholar
[8] Halle, L. H. and Nicaise, J.. The Néron component series of an abelian variety. Math. Ann. 348, no. 3 (2010) 749–778.10.1007/s00208-010-0495-5Google Scholar
[9] Lorenzini, D.. On the group of components of a Néron model. J. Reine Angew. Math. 445 (1993), 109–160.Google Scholar
[10] Morgan, A.. Parity of 2-Selmer ranks of hyperelliptic curves over quadratic extensions, arxiv: 1504.01960.Google Scholar
[11] Namikawa, Y. and Ueno, K.. The complete classification of fibres in pencils of curves of genus two. Manuscripta Math. 9 (1973), 143–186.10.1007/BF01297652Google Scholar
[12] Nekovář, J.. On the parity of ranks of Selmer groups IV. Compositio Math. 145 (2009), 1351–1359.10.1112/S0010437X09003959Google Scholar
[13] Raynaud, M.. Variétés abéliennes et géométrie rigide, Actes du congrès international de Nice 1970, tome 1, 473–477.Google Scholar
[14] Rohrlich, D.. The vanishing of certain Rankin–Selberg convolutions. In: Automorphic Forms and Analytic Number Theory, 123–133. Les publications CRM, Montreal (1990).Google Scholar
[15] Rohrlich, D.. Galois theory, elliptic curves and root numbers. Compositio Math. 100 (1996) 311–349.Google Scholar
[16] Samuel, P.. Algebraic Theory of Numbers (Silberger translation) (Kershaw 1971).Google Scholar