Monday, December 12 2016
11:30 - 12:30

#### Chai and Oort have asked the following question: For any algebraicallyclosed field \$k\$, and for \$g \geq 4\$, does there exist an abelian varietyover \$k\$ of dimension \$g\$ not isogenous to a Jacobian? The answer incharacteristic 0 is now known to be yes. We present a heuristic whichsuggests that for certain \$g \geq 4\$, the answer in characteristic \$p\$ isno. We will also construct a proper subvariety of \$X(1)^n\$ which intersectsevery isogeny class, thereby answering a related question, also asked byChai and Oort. This is joint work with Jacob Tsimerman.

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