Differential characters of Drinfeld modules and
de Rham cohomology

Borger, James; Saha, Arnab

doi:10.2140/ant.2019.13.797

Cited by 8 publications

(6 citation statements)

References 18 publications

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“…Proof. The equal-characteristic analogue of this lemma is lemma 9.7 of [3]. It was proven using two results in equal-characteristic Dieudonné-Manin theory, namely (B.1.5) and (B.1.9) of [16], which are proved in pages 257-261.…”

Section: The F -Isocrystal and Hodge Sequence Of Amentioning

confidence: 97%

“…A natural question is whether the two determine each other, especially by some explicit linear-algebraic functor like the Fontaine functor mentioned above. In the analogous Drinfeld module setting of [3], the shtuka necessarily determines both, simply because it determines the Drinfeld module. But it would be interesting to go further and describe the functor in purely linear-algebraic terms, without a detour back through the Drinfeld module.…”

Section: Introductionmentioning

confidence: 99%

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Isocrystals associated to arithmetic jet spaces of abelian schemes

Borger

Saha

2019

Advances in Mathematics

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Using Buium's theory of arithmetic differential characters, we construct a filtered F -isocrystal H(A) K associated to an abelian scheme A over a p-adically complete discrete valuation ring with perfect residue field. As a filtered vector space, H(A) K admits a natural map to the usual de Rham cohomology of A, but the Frobenius operator comes from arithmetic differential theory and is not the same as the usual crystalline one. When A is an elliptic curve, we show that H(A) K has a natural integral model H(A), which implies an integral refinement of a result of Buium's on arithmetic differential characters. The weak admissibility of H(A) K depends on the invertibility of an arithmetic-differential modular parameter. Thus the Fontaine functor associates to suitably generic A a local Galois representation of an apparently new kind.

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Section: The F -Isocrystal and Hodge Sequence Of Amentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Isocrystals associated to arithmetic jet spaces of abelian schemes

Borger

Saha

2019

Advances in Mathematics

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“…The purpose of this short note is to make an observation which is a generalisation of a result shown in [3] for Drinfeld modules. Let B be a Dedekind domain and fix a maximal ideal p ∈ Spec B with k := B/p a finite field and let q = |k|.…”

Section: Introductionmentioning

confidence: 64%

On Frobenius and Fibers of Arithmetic Jet Spaces

Borger,

Saha

2017

Preprint

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In this article, given a scheme X we show the existence of canonical lifts of Frobenius maps in an inverse system of schemes obtained from the fiber product of the canonical prolongation sequence of arithmetic jet spaces J * X and a prolongation sequence S * over the scheme X. As a consequence, for any smooth group scheme E, if N n denote the kernel of the canonical projection map of the n-th jet space J n E → E, then the inverse system {N n }n is a prolongation sequence.

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“…Hence using the Fontaine functor, these isocrystals lead to Galois representations arising from δ-geometry. The equal characteristic analogue of the above construction was done in [5]. Delta geometric objects have also been used in the construction of prismatic cohomology by Bhatt and Scholze [2].…”

Section: Introductionmentioning

confidence: 99%

Kernel of Arithmetic Jet Spaces

Saha¹

2022

Preprint

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Fix a Dedekind domain O and a non-zero prime p in it along with a uniformizer π. In the first part of the paper, we construct m-shifted π-typical Witt vectors Wmn(B) for any O algebra B of length m + n + 1. They are a generalization of the usual π-typical Witt vectors. Along with it we construct a lift of Frobenius, called the lateral Frobenius F : Wmn(B) → W m(n−1) (B) and show that it satisfies a natural identity with the usual Frobenius map. Now given a group scheme G defined over Spec R, where R is an O-algebra with a fixed π-derivation δ on it, one naturally considers the n-th arithmetic jet space J n G whose points are the Witt ring valued points of G. This leads to a natural projection map of group schemes u : J m+n G → J m G. Let N mn G denote the kernel of u. Then in the case when G is an affine group scheme, we show that the lateral Frobenius F turns the system of schemes {N mn G} ∞ n=1 into a prolongation sequence. One of our main results then prove that for n ≥ 1, N mn G is naturally isomorphic to J n−1 (N m1 G) as group schemes. Hence this implies that for any π-formal group scheme Ĝ over Spf R, N mn Ĝ is isomorphic to J n−1 (N m1 G). As an application, if Ĝ is a smooth commutative π-formal group scheme of dimension d and R is of characteristic 0 whose ramification is bounded above by p − 2, then our result implies that J n G is a canonical extension of Ĝ by (W n−1 ) d where W n−1 is the π-formal group scheme Ân endowed with the group law of addition of Witt vectors. Our results also give a geometric characterization of G(π n+1 R) which is the subgroup of points of G(R) that reduces to identity under the modulo π n+1 map.

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Differential characters of Drinfeld modules and de Rham cohomology

Cited by 8 publications

References 18 publications

Isocrystals associated to arithmetic jet spaces of abelian schemes

Isocrystals associated to arithmetic jet spaces of abelian schemes

On Frobenius and Fibers of Arithmetic Jet Spaces

Kernel of Arithmetic Jet Spaces

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