Differential modular forms

Buium, Alexandru

doi:10.1515/crll.2000.024

Cited by 50 publications

(227 citation statements)

References 15 publications

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“…A d-modular form is a d-modular function with a weight. This de¢nition of d-modular forms introduced in Buium's paper [1] represents an extension of the de¢nitions given by Deligne and Katz [2]. A special collection these d-modular forms possess the following property that does not have a classical analog.…”

Section: Introductionmentioning

confidence: 83%

“…The theorem gives an explicit expression for f def of p-derivations and hence by Proposition 4.6 in [1] an explicit formula for the reduction modulo p of the unique d-modular form of weight w ÀpÀ1;À1 .…”

Section: 3mentioning

confidence: 94%

“…This homomorphism from M 1 3 Z p q ^induces a homomorphism M 1 0 3 F p q ^that takes the reduction of F modulo p denoted F to the reduction of F q modulo p denoted F q . Buium proves in Remark 7.5 [1] that the unique isogeny covariant d-modular form of order one and weight w ÀpÀ1;À1 is in the kernel of this homomorphism. We also know that E pÀ1 q 1 modulo p. As an immediate consequence COROLLARY 1.4.…”

Section: Introductionmentioning

confidence: 89%

“…The key to proving Theorem 1.3 is the fact that the unique isogeny covariant d-modular form of weight w ÀpÀ1;À1 is congruent modulo p to a speci¢c`deformation class' of an elliptic curve called f def in Buium's paper,`Differential modular forms' [1]. This`deformation class' is a p-derivation analog of the Kodaira^Spencer class for a family of elliptic curves for usual derivations.…”

Section: Introductionmentioning

confidence: 99%

“…Differential modular forms, introduced in A. Buium's paper entitled`Differential modular forms' [1], involve either arithmetic analogs of derivations called p-derivations or usual derivations. In providing the relevent de¢nitions we de¢ne everything for the case of p-derivations and then note later in the introduction how the de¢nitions vary from those for p-derivations in the case of usual derivations.…”

Section: Introductionmentioning

confidence: 99%

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Hurlburt

2001

Compositio Mathematica

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Abstract. An interesting theory arises when the classical theory of modular forms is expanded to include differential analogs of modular forms. One of the main motivations for expanding the theory of modular forms is the existence of differential modular forms with a remarkable property, called isogeny covariance, that classical modular forms cannot possess. Among isogeny covariant differential modular forms there exists a particular modular form that plays a central role in the theory. The main result presented in the article will be the explicit computation modulo p of this fundamental isogeny covariant differential modular form.

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Section: Introductionmentioning

confidence: 83%

Section: 3mentioning

confidence: 94%

Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hurlburt

2001

Compositio Mathematica

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Arithmetic Levi–Cività connection

Buium

2019

Sel. Math. New Ser.

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This paper is part of a series of papers where an arithmetic analogue of classical differential geometry is being developed. In this arithmetic differential geometry functions are replaced by integer numbers, derivations are replaced by Fermat quotient operators, and connections (respectively curvature) are replaced by certain adelic (respectively global) objects attached to symmetric matrices with integral coefficients. Previous papers were devoted to an arithmetic analogue of the Chern connection. The present paper is devoted to an arithmetic analogue of the Levi-Civita connection.

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On differential modular forms and some analytic relations between Eisenstein series

Movasati

2007

Ramanujan J

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In the present article we define the algebra of differential modular forms and we prove that it is generated by Eisenstein series of weight 2, 4 and 6. We define Hecke operators on them, find some analytic relations between these Eisenstein series and obtain them in a natural way as coefficients of a family of elliptic curves. The fact that a complex manifold over the moduli of polarized Hodge structures in the case h 10 = h 01 = 1 has an algebraic structure with an action of an algebraic group plays a basic role in all of the proofs.

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Differential modular forms

Cited by 50 publications

References 15 publications

Untitled

Untitled

Arithmetic Levi–Cività connection

On differential modular forms and some analytic relations between Eisenstein series

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