#### 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.

Done