Stephen Patrikis (Ohio State)

Let be a Shimura datum, and let be a compact open subgroup of . One hopes that under mild assumptions on and , the points of the Shimura variety parametrize a family of motives; unlike in abelian type (moduli of abelian varieties, etc.), in non-abelian type this problem remains completely mysterious. I will discuss joint work with Christian Klevdal showing that for “superrigid,” including all non-abelian type, Shimura varieties the points (over number fields, say) at least yield compatible systems of l-adic representations, which should be the l-adic realizations of the conjectural motives. Time permitting, I will discuss some work in progress (with Jake Huryn, Kiran Kedlaya, and Klevdal) on a crystalline analogue.