Delta Characters and Filtered Isocrystals
Abstract
Given an abelian scheme A over a p-adic ring R, Borger and Saha constructed a filtered module {Hδ(A) ⊃ Xprim(A)⊃ {0}} with a semilinear operator f* on Hδ(A) using the theory of arithmetic jet spaces. The above object admits a canonical map Φ to the Hodge sequence {H1dR(A) ⊃ H0(A,ΩA)⊃ {0}} of A in the category of filtered modules. As a result, by restricting Φ, we obtain a natural R-linear map Υ: Xprim(A) → H0(A,ΩA). 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 f* on Hδ(A)K is a bijection, and our object {Hδ(A)K ⊃ Xprim(A)K⊃ {0}} becomes a filtered isocrystal. In fact, the above results admit a generalization to the setting of semi-abelian schemes. The elements of Xprim(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 H0(A,ΩA) 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 Hδ(A)K and the first crystalline cohomology Hcris1(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 ℚp.
Related
- Delta Characters and Ranks of Minimal Subisocrystals of Abelian Schemes
- Overholonomicity of Overconvergent F-isocrystals over Laurent Series Fields
- Potential isotriviality of isocrystals and proper covers
- On hyperbolic PDEs, filtered feedback control laws, and fractal-like stability crossing curves
- Besov and Triebel--Lizorkin Spaces on Filtered Lie Groups, II: Pointwise Multiplication, Localization, Differences
BibTeX
@misc{gurney2026delta,
title = {{Delta Characters and Filtered Isocrystals}},
author = {Lance Gurney and Sudip Pandit and Arnab Saha},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28365},
eprint = {2609.28365},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28365},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Delta Characters and Filtered Isocrystals
AU - Lance Gurney
AU - Sudip Pandit
AU - Arnab Saha
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28365
UR - https://arxiv.org/abs/2609.28365
AB - Given an abelian scheme A over a p-adic ring R, Borger and Saha constructed a filtered module {H_δ(A) ⊃ X_prim(A)⊃ {0}} with a semilinear operator f^* on H_δ(A) using the theory of arithmetic jet spaces. The above object admits a canonical map Φ to the Hodge sequence {H^1_dR(A) ⊃ H^0(A,Ω_A)⊃ {0}} of A in the category of filtered modules. As a result, by restricting Φ, we obtain a natural R-linear map Υ: X_prim(A) → 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 f^* on H_δ(A)_K is a bijection, and our object {H_δ(A)_K ⊃ X_prim(A)_K⊃ {0}} becomes a filtered isocrystal. In fact, the above results admit a generalization to the setting of semi-abelian schemes. The elements of X_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 H_δ(A)_K and the first crystalline cohomology H_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 ℚ_p.
ER -