# Log smooth reduction of tamely ramified abelian varieties

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Alberto Bellardini

created by risa on 12 Nov 2015

19 nov 2015
-- 14:00

Aula 211, Dip. Matematica, Università "Roma Tre", Roma

**Abstract.**

Let L be complete discrete valuation field with perfect residue field
k and ring of integers O. A classical theorem of Serre and Tate says
that an abelian variety A over L admits a model over O which is an
abelian scheme if and only if the inertia subgroup of the absolute
Galois group of L acts trivially on the first l-adic e'tale cohomology
group of A, for some l prime number different from the characteristic
of k. In this talk I want to discuss a logarithmic version of this
result. Namely if the wild inertia subgroup of the absolute Galois
group of L acts trivially on the first l-adic e'tale cohomology group
of A, for some l prime number different from the characteristic of k,
then A admits a projective, log smooth model over O. During the talk I
will recall and give examples about some basic facts on this subject,
especially about log smoothness. I am not going to assume familiarity
with logarithmic geometry. This is a joint work with A. Smeets.