L. Alexander Betts, Vladimir Dokchitser
Published 2014-05-13, updated 2014-12-24Version 2
We investigate the behaviour of Tamagawa numbers of semistable principally polarised abelian varieties in extensions of local fields. In view of the Raynaud parametrisation, this translates into a purely algebraic problem concerning the number of $H$-invariant points on a quotient of $C_n$-lattices $\Lambda/e\Lambda'$ for varying subgroups $H$ of $C_n$ and integers $e$. In particular, we give a simple formula for the change of Tamagawa numbers in totally ramified extensions (corresponding to varying $e$) 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 $y^2=f(x)$, under some simplifying hypotheses.
Comments: A new appendix by A. Morgan and V. Dokchitser deals with hyperelliptic curves of genus 2 given by C: y^2 = f(x) over p-adic fields, assuming that p is odd and f(x) has no triple roots modulo p. It explains how to read off both the Tamagawa number of the Jacobian of C from the defining equation, and the action of Frobenius on the dual character group of the toric part of its reduction
Campana points, Vojta's conjecture, and level structures on semistable abelian varieties
Hasse norm principle for $M_{11}$ extensions
Functions which are PN on infiitely many extensions of Fp, p odd
![QR code for arXiv:1405.3151 [math.NT]](http://oss.arxitics.com/images/favicon.svg)