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The Shafarevich conjecture is concerned with finiteness results for families of g-dimensional principally polarized abelian varieties over a base B. Famously, Faltings settled the arithmetic 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, we show that one may recover the theorem by considering families up to p-power isogenies.

We formulate an analogous statement for Exceptional Shimura varieties S, and describe a proof in the genericaly ordinary setting. This is joint work with Ben Bakker and Ananth Shankar.