Some examples of abelian surfaces with everywhere good reduction

A celebrated result of Fontaine dated 1981 shows that there is no
abelian scheme over Z. In other words, there is no abelian variety over
Q with everywhere good reduction. This result is no longer true over
general number fields. In fact, long before Fontaine's result, Tate showed
that the discriminant of elliptic curve E: y^{^2} +xy + \epsilon^{^2} y= x^{^3},
where \epsilon =(5+\sqrt{29})/2, is a unit in Q(\sqrt{29}). There is
now a wealth of elliptic curves over quadratic fields with everywhere good
reduction in the literature. However, there is no known explicit examples
of (simple and non-CM) abelian surfaces with everywhere good reduction. In
this talk, we will present a strategy for looking for such surfaces and
give some examples. (This is joint work with Abhinav Kumar.)