Constructing Abelian Varieties over $\overline{\mbthbb{Q}}$ Not Isogenous to a Jacobian

Constructing Abelian Varieties over $\overline{\mbthbb{Q}}$ Not Isogenous to a Jacobian

Thu, 03/02/2011
16:00 		Jacob Tsimerman (Princeton University) Number Theory Seminar Add to calendar L3
Constructing Abelian Varieties over $ \overline{\mbthbb{Q}} $ Not Isogenous to a Jacobian
We discuss the following question of Nick Katz and Frans Oort: Given an Algebraically closed field K , is there an Abelian variety over K of dimension g which is not isogenous to a Jacobian? For K the complex numbers its easy to see that the answer is yes for g>3 using measure theory, but over a countable field like $ \overline{\mbthbb{Q}} $ new methods are required. Building on work of Chai-Oort, we show that, as expected, such Abelian varieties exist for $ K=\overline{\mbthbb{Q}} $ and g>3 . We will explain the proof as well as its connection to the Andre Oort conjecture.