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Journal Article

Isocrystals associated to arithmetic jet spaces of abelian schemes


Saha,  Arnab
Max Planck Institute for Mathematics, Max Planck Society;

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Borger, J., & Saha, A. (2019). Isocrystals associated to arithmetic jet spaces of abelian schemes. Advances in Mathematics, 351, 388-428. doi:10.1016/j.aim.2019.04.065.

Cite as: https://hdl.handle.net/21.11116/0000-0004-8F42-1
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.