Finite quotients of Z[C_n]-lattices and Tamagawa numbers of semistable abelian varieties

TL;DR

This paper studies Tamagawa numbers of semistable abelian varieties over local fields, providing formulas for their change in extensions, extending p-parity results, and classifying their behavior for 2-dimensional cases.

Contribution

It offers new algebraic formulas for Tamagawa number variation, extends p-parity conjecture results, and classifies Tamagawa behavior for 2D semistable abelian varieties.

Findings

Formulas for Tamagawa number changes in ramified extensions

Extension of p-parity conjecture results

Complete classification for 2D abelian varieties

Abstract

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.