Isocrystals associated to arithmetic jet spaces of abelian schemes

Abstract

Using Buium's theory of arithmetic differential characters, we construct a filtered F-isocrystal H(A)K associated to an abelian scheme A over a p-adically complete discrete valuation ring with perfect residue field. As a filtered vector space, H(A)K admits a natural map to the usual de Rham cohomology of A, but the Frobenius operator comes from arithmetic differential theory and is not the same as the usual crystalline one. When A is an elliptic curve, we show that H(A)K has a natural integral model H(A), which implies an integral refinement of a result of Buium's on arithmetic differential characters. The weak admissibility of H(A)K depends on the invertibility of an arithmetic-differential modular parameter. Thus the Fontaine functor associates to suitably generic A a local Galois representation of an apparently new kind.