Number Theory Seminar

The Shafarevich conjecture is concerned with finiteness results for families of g-dimensional principally polarized abelian varieties over a base B. Faltings settled the case of B=O_{K,S}. In the case where B is a curve over a finite field, finiteness can never be true as one may always compose with Frobenius. In this setting, to get a theorem one must consider families up to p-power isogenies. We formulate an analogous statement for Exceptional Shimura varieties S, and describe ongoing work to prove it. This is joint work with Ben Bakker and Jacob Tsimerman.