Room 326

Abelian varieties isogenous to Jacobians

Ananth Shankar

Harvard University

Chai and Oort have asked the following question: For any algebraically
closed field $k$, and for $g \geq 4$, does there exist an abelian variety
over $k$ of dimension $g$ not isogenous to a Jacobian? The answer in
characteristic 0 is now known to be yes. We present a heuristic which
suggests that for certain $g \geq 4$, the answer in characteristic $p$ is
no. We will also construct a proper subvariety of $X(1)^n$ which intersects
every isogeny class, thereby answering a related question, also asked by
Chai and Oort. This is joint work with Jacob Tsimerman.