Isocrystals associated to arithmetic jet spaces of abelian schemes

2017/12/24 by Borger, James, Saha, Arnab · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1712.09346

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.

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• Differential Characters and D-Group Schemes

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