Eichler-Shimura Relations

Abstract

The well-known classical Eichler-Shimura relation for modular curves asserts that the Hecke operator T_p is equal, as an algebraic correspondence over the special fiber, to the sum of Frobenius and Verschiebung. Blasius and Rogawski proposed a generalization of this result for general Shimura varieties with good reduction at p, and conjectured that the Frobenius satisfies a certain Hecke polynomial. We provide a proof of this result for a large class of Shimura varieties of abelian type, and describe how for a Hodge type Shimura variety a general proof will follow assuming a non-trivial action of certain cohomological correspondences. Moreover, we extend this result to partial Frobenii in some cases.