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.
In this talk, I will present a search method for finding abelian surfaces with everywhere good reduction over real quadratic fields. One main feature of this approach is that it allows for the determination of abelian surfaces with trivial endomorphism rings in some cases.