Isocrystals associated to arithmetic jet spaces of abelian schemes

TL;DR

This paper constructs a new filtered F-isocrystal associated with abelian schemes using arithmetic differential characters, revealing novel Frobenius structures and Galois representations in the p-adic setting.

Contribution

It introduces a new filtered F-isocrystal linked to abelian schemes via arithmetic differential characters, with an integral model and implications for Galois representations.

Findings

Constructed a filtered F-isocrystal ${f H}(A)_K$ for abelian schemes.

Established an integral model ${f H}(A)$ for elliptic curves.

Connected the isocrystal's properties to Galois representations.

Abstract

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