Delta Characters and Filtered Isocrystals
Given an abelian scheme $A$ over a $p$-adic ring $R$, Borger and Saha constructed a filtered module $\{\mathbf{H}_δ(A) \supset \mathbf{X}_{\mathrm{prim}}(A)\supset \{0\}\}$ with a semilinear operator $\mathfrak{f}^*$ on $\mathbf{H}_δ(A)$ using the theory of arithmetic jet spaces. The above object admits a canonical map $Φ$ to the Hodge sequence $\{\mathbf{H}^1_{\mathrm{dR}}(A) \supset H^0(A,Ω_A)\supset \{0\}\}$ of $A$ in the category of filtered modules. As a result, by restricting $Φ$, we obtain a natural $R$-linear map $Υ: \mathbf{X}_{\mathrm{prim}}(A) \rightarrow H^0(A,Ω_A)$. In this paper, we show that the map $Υ$ is an isomorphism of vector spaces over $K$, the field of fractions of $R$. As a consequence, we will show that for all abelian schemes $A$, the operator $\mathfrak{f}^*$ on $\mathbf{H}_δ(A)_K$ is a bijection, and our object $\{\mathbf{H}_δ(A)_K \supset \mathbf{X}_{\mathrm{prim}}(A)_K\supset \{0\}\}$ becomes a filtered isocrystal. In fact, the above results admit a generalization to the setting of semi-abelian schemes. The elements of $\mathbf{X}_{\mathrm{prim}}(A)$ are represented by primitive additive characters of the arithmetic jet spaces attached to $A$. Hence, our isomorphism given by $Υ$ provides an interesting character-theoretic interpretation of $H^0(A,Ω_A)$ in terms of primitive delta characters. As a result, to any $1$-form $ω$, the above isomorphism associates a canonical numerical invariant that depends on deformation theoretic data of $A$. Furthermore, we also extend a comparison theorem between $\mathbf{H}_δ(A)_K$ and the first crystalline cohomology $\mathbf{H}_{\mathrm{cris}}^1(A)_K$ in the general case when the elliptic curve $A$ is defined over the ring of integers of a $p$-adic field $K$ that is a finite extension of $\mathbb{Q}_p$.