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. On the other

hand, a result of More-Bailly states that in every genus $g$, there is a curve of genus $g$ with everywhere

good reduction over ${Z}$. So, as the base field varies, we must hope to find more abelian varieties

with trivial conductor.

In this talk, I will some existence results for abelian surfaces with everywhere good reduction over

real quadratic fields.

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