Isocrystals associated to arithmetic jet spaces of abelian schemes

Abstract

Using Buium's theory of arithmetic differential characters, we construct a filtered $F$-isocrystal $\mathbf{H}(A{)}_K$ associated to an abelian scheme $A$ over a $p$-adically complete discrete valuation ring with perfect residue field. As a filtered vector space, $\mathbf{H}(A{)}_K$ 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 $\mathbf{H}(A{)}_K$ has a natural integral model $\mathbf{H}(A)$, which implies an integral refinement of a result of Buium's on arithmetic differential characters. The weak admissibility of $\mathbf{H}(A{)}_K$ 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.

arXiv commentFinal version, to appear in Advances in Mathematics. arXiv admin note: text overlap with arXiv:1703.05677