Delta Characters and Ranks of Minimal Subisocrystals of Abelian Schemes

Abstract

Using the framework of delta characters on arithmetic jet spaces, we construct a canonical sequence of filtered isocrystals associated to an abelian scheme $A$ of relative dimension $g$ over a $p$-adically complete discrete valuation ring. These isocrystals map injectively via a canonical morphism into the first (rational) crystalline cohomology group $\mathbf{H}^1_{\mathrm{cris}}(A)$ of the abelian scheme $A$, directly generalizing a prior construction due to Borger and Saha. Pandit showed that the canonical morphism is a map of isocrystals. This then implies that the image of this canonical embedding coincides with the minimal subisocrystal $F_{\mathrm{cris}}(Ω_A)$ containing the space of invariant differential forms $H^0(A,Ω_A)$. Berthelot and Ogus showed that the dimension of $F_{\mathrm{cris}}(Ω_A)$ is $g$ if and only if $A$ has an endomorphism that lifts the absolute Frobenius on the special fiber of $A$. As a primary application, for a general abelian scheme $A$, we show that the minimal subisocrystal $F_{\mathrm{cris}}(Ω_A)$ can be viewed as a module obtained from primitive delta characters of $A$ via appropriate twists by the canonical Frobenius morphism $ϕ$ induced from the theory of arithmetic jet spaces. As a consequence, we show that the rank of $F_{\mathrm{cris}}(Ω_A)$ depends on numerical invariants arising from the theory of delta characters of $A$. More precisely, the dimension of $F_{\mathrm{cris}}(Ω_A)$ is given by the sum of the exact orders of the primitive delta characters associated to the abelian scheme $A$.