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Finite quotients of Z[C_n]-lattices and Tamagawa numbers of semistable abelian varieties
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 -invariant points on a quotient of -lattices
for varying subgroups of and integers . In
particular, we give a simple formula for the change of Tamagawa numbers in
totally ramified extensions (corresponding to varying ) 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 -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 , under some
simplifying hypotheses.Comment: Two new lemmas are added. The first describes permutation
representations, and the second describes the dependence of the B-group on
the maximal fixpoint-free invariant sublattice. Contact details and
bibliographic details have been update
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