Аннотация:
It is well known that in dimension
$g\ge 4$ there exist complex abelian varieties not isogenous to any
Jacobian. A question of Katz and Oort asked whether one can find
such examples over the field of algebraic numbers. This was answered
affirmatively by Oort-Chai under the André-Oort conjecture, and by
Tsimerman unconditionally. They gave examples within Complex
Multiplication. In joint work with Masser, by means of a completely
different method, we proved that in a sense the general abelian
variety over $\overline{\mathbb{Q}}$ is indeed not isogenous to any Jacobian. I
shall illustrate the basic principles of the proofs.