Delta characters and crystalline cohomology

Abstract

The first part of the paper develops the theory of m-shifted π-typical Witt vectors which can be viewed as subobjects of the usual π-typical Witt vectors. We show that the shifted Witt vectors admit a delta structure that satisfy a canonical identity with the delta structure of the usual π-typical Witt vectors. Using this theory, we prove that the generalized kernels of arithmetic jet spaces are jet spaces of the kernel at the first level. This also allows us to interpret the arithmetic Picard-Fuchs operator geometrically. For a π-formal group scheme G, by a previous construction, one attaches a canonical filtered isocrystal Hδ(G) associated to the arithmetic jet spaces of G. In the second half of our paper, we show that Hδ(A) is of finite rank if A is an abelian scheme. As an application, for an elliptic curve A defined over Zp, we show that our canonical filtered isocrystal Hδ(A)⊗Qp is weakly admissible. In particular, if A does not admit a lift of Frobenius, we show that that Hδ(A)⊗Qp is canonically isomorphic to the first crystalline cohomology H1cris(A)⊗Qp in the category of filtered isocrystals. On the other hand, if A admits a lift of Frobenius, then Hδ(A)⊗Qp is isomorphic to the sub-isocrystal H0(A,ΩA)⊗Qp of H1cris(A)⊗Qp. The above result can be viewed as a character rheoretic interpretation of the crystalline cohomology. The difference between the integral structures of Hδ(A) and H1cris(A) is measured by a delta modular form f1 constructed by Buium.