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