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 H1cris(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 Fcris(ΩA) containing the space of invariant differential forms H0(A,ΩA). Berthelot and Ogus showed that the dimension of Fcris(ΩA) 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 Fcris(ΩA) 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 Fcris(ΩA) depends on numerical invariants arising from the theory of delta characters of A. More precisely, the dimension of Fcris(ΩA) is given by the sum of the exact orders of the primitive delta characters associated to the abelian scheme A.
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BibTeX
@misc{mishra2026delta,
title = {{Delta Characters and Ranks of Minimal Subisocrystals of Abelian Schemes}},
author = {Rajat Kumar Mishra and Arnab Saha},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33948},
eprint = {2610.33948},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33948},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Delta Characters and Ranks of Minimal Subisocrystals of Abelian Schemes
AU - Rajat Kumar Mishra
AU - Arnab Saha
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33948
UR - https://arxiv.org/abs/2610.33948
AB - 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 H^1_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_cris(Ω_A) containing the space of invariant differential forms H^0(A,Ω_A). Berthelot and Ogus showed that the dimension of F_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_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_cris(Ω_A) depends on numerical invariants arising from the theory of delta characters of A. More precisely, the dimension of F_cris(Ω_A) is given by the sum of the exact orders of the primitive delta characters associated to the abelian scheme A.
ER -