  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.
