#### Alladi Ramakrishnan Hall

#### 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