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