The finite quotient groups of étale fundamental groups of algebraic curves in positive characteristic are precisely determined, but without explicit construction of quotient maps, by well-known results of Raynaud, Harbater and Pop, previously known as Abhyankar's conjecture. Katz, Rojas León and Tiep have been studying the constructive side of this problem using certain "easy to remember" local systems. In this talk, I will discuss the main results and methods of this project in the case of a specific type of local systems called hypergeometric sheaves.

EO strata are subvarieties in the moduli space of g-dimensional abelian varieties in characterstic p which classify points with given isomorphism type of p-torson subgroups. We are interested in how automorphism groups of points vary in supersingular EO strata. We show that when g is even and p>3, there is an open dense of the maximal supersingular EO stratum in which every point has automorphism group {\pm 1}, and prove Oor's conjecture in this case. This is joint work in progress with Valentijn Karemaker.