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We investigate the behaviour of Tamagawa numbers of semistable principallypolarised abelian varieties in extensions of local fields. In view of theRaynaud parametrisation, this translates into a purely algebraic problemconcerning 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 $e$. Inparticular, we give a simple formula for the change of Tamagawa numbers intotally ramified extensions (corresponding to varying $e$) and one thatcomputes Tamagawa numbers up to rational squares in general extensions.  As an application, we extend some of the existing results on the $p$-parityconjecture for Selmer groups of abelian varieties by allowing more generallocal behaviour. We also give a complete classification of the behaviour ofTamagawa numbers for semistable 2-dimensional principally polarised abelianvarieties, that is similar to the well-known one for elliptic curves. Theappendix explains how to use this classification for Jacobians of genus 2hyperelliptic curves given by equations of the form $y^2=f(x)$, under somesimplifying hypotheses.

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