The ring of differential Fourier expansions

Buium, Alexandru; Saha, Arnab

doi:10.1016/j.jnt.2011.12.007

Cited by 13 publications

(22 citation statements)

References 18 publications

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“…This seems to be a quite general principle with various incarnations throughout the theory (cf. [15,16,25]) and illustrates, again, how a limitation of classical algebraic geometry can be overcome by passing to δ-geometry.…”

Section: Motivations Of the Theorymentioning

confidence: 98%

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Differential calculus with integers

Buium¹

2015

Arithmetic and Geometry

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Abstract. Ordinary differential equations have an arithmetic analogue in which functions are replaced by numbers and the derivation operator is replaced by a Fermat quotient operator. In this survey we explain the main motivations, constructions, results, applications, and open problems of the theory.The main purpose of these notes is to show how one can develop an arithmetic analogue of differential calculus in which differentiable functions x(t) are replaced by integer numbers n and the derivation operator x → dx dt is replaced by the Fermat quotient operator n → n−n p p , where p is a prime integer. The Lie-Cartan geometric theory of differential equations (in which solutions are smooth maps) is then replaced by a theory of "arithmetic differential equations" (in which solutions are integral points of algebraic varieties). In particular the differential invariants of groups in the Lie-Cartan theory are replaced by "arithmetic differential invariants" of correspondences between algebraic varieties. A number of applications to diophantine geometry over number fields and to classical modular forms will be explained.This program was initiated in [11] and pursued, in particular, in [12]- [35]; for an exposition of some of these ideas we refer to the monograph [16]. We shall restrict ourselves here to the ordinary differential case. For the partial differential case we refer to [20,21,22,7]. Throughout these notes we assume familiarity with the basic concepts of algebraic geometry and differential geometry; some of the standard material is being reviewed, however, for the sake of introducing notation, and "setting the stage". The notes are organized as follows. The first section presents some classical background, the main concepts of the theory, a discussion of the main motivations, and a comparison with other theories. The second section presents a sample of the main results. The third section presents a list of open problems.Acknowledgement. The author is indebted to HIM for support during part of the semester on Algebra and Geometry in Spring 2013. These notes are partially based on lectures given at the IHES in Fall

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Section: Motivations Of the Theorymentioning

confidence: 98%

“…Cf. [15,2,16,26,17,25]. Let X 1 (N ) be the complete modular curve over R of level N > 4 and let L 1 (N ) → X 1 (N ) be the line bundle with the property that the sections of its various powers are the classical modular forms on Γ 1 (N ) of various weights.…”

Section: δ-Invariants Of Correspondencesmentioning

confidence: 99%

Differential calculus with integers

Buium¹

2015

Arithmetic and Geometry

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“…of δ π -modular forms of level Np, order n, and weight 0 to the ring S n+1 ♥ ⊗ R p R π where S n+1 ♥ is the ring of Igusa δ p -modular forms of level N [10]. We start by reviewing the rings S n ♥ .…”

Section: Link With Igusa Differential Modular Functionsmentioning

confidence: 99%

“…The ring S ∞ ♥ can be referred to as the ring of Igusa δ p -modular functions of level N (or the ring of the δ p -Igusa curve of level N [10] …”

Section: Link With Igusa Differential Modular Functionsmentioning

confidence: 99%

Differential modular forms attached to newforms mod p

Buium

2015

Journal of Number Theory

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“…For instance, the computations in section 10 show the structure constants of H(E) do involve the higher π-derivatives of the structure constants of the Drinfeld module. The phenomenon of π-differential invariants depending on higher π-derivatives of modular parameters in the mixed-characteristic setting can be found in [BoSa2], [Bui2], [BuSa1], [BuSa2], [BuSa3], [BuSa4].…”

Section: Introductionmentioning

confidence: 99%

Differential characters of Drinfeld modules and de Rham cohomology

Borger

Saha

2019

Alg. Number Th.

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We introduce differential characters of Drinfeld modules. These are function-field analogues of Buium's p-adic differential characters of elliptic curves and of Manin's differential characters of elliptic curves in differential algebra, both of which have had notable Diophantine applications. We determine the structure of the group of differential characters. This shows the existence of a family of interesting differential modular functions on the moduli of Drinfeld modules. It also leads to a canonical F -crystal equipped with a map to the de Rham cohomology of the Drinfeld module. This F -crystal is of a differential-algebraic nature, and the relation to the classical cohomological realizations is presently not clear.

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The ring of differential Fourier expansions

Cited by 13 publications

References 18 publications

Differential calculus with integers

Differential calculus with integers

Differential modular forms attached to newforms mod p

Differential characters of Drinfeld modules and de Rham cohomology

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