Lassina Dembélé (University of Warwick):

On the existence of abelian surfaces with everywhere good reduction

Abstract: A famous result of Fontaine (and Abrashkin) states that there is no abelian variety over the rationals with everywhere good reduction. Fontaine's proof of this result relies on the non-existence of certain finite flat group schemes. His technique has been refined by several people (including Schoof, Brumer and Calegari) to prove non-existence of semi-stable abelian varieties over various fields. But one has to expect that such non-existence results are the exception rather than the norm. Indeed, as the base field varies, we must hope to find more abelian varieties with everywhere good reduction.

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