### Abelian varieties isogenous to Jacobians

- Series
- Algebra Seminar
- Time
- Friday, April 28, 2017 - 11:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- 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.