Differential modular forms
Copyright policy )Differential modular forms
Summary: Algebraic geometry can be enlarged by adjoining a new operation, the Fermat quotient operation [cf. \textit{A. Buium}, Invent. Math. 122, 309--340 (1995; Zbl 0841.14037)]. This new geometry is an arithmetic analogue of the Ritt-Kolchin ``differential algebraic geometry''. The aim of the present paper is to develop a theory of modular forms in this enlarged geometry, including a Hecke theory and a Fourier theory for such forms. A remarkable new phenomenon occurs in this setting: the existence of ``isogeny covariant'' forms. (Isogeny covariance is a property stronger than that of being a Hecke eigenform).
isogeny covariance, Fermat quotient operation, Hecke theory, Fourier coefficients of automorphic forms, \(p\)-adic theory, local fields, Hecke-Petersson operators, differential operators (one variable), modular forms, Fourier theory, Holomorphic modular forms of integral weight, differential algebraic geometry
isogeny covariance, Fermat quotient operation, Hecke theory, Fourier coefficients of automorphic forms, \(p\)-adic theory, local fields, Hecke-Petersson operators, differential operators (one variable), modular forms, Fourier theory, Holomorphic modular forms of integral weight, differential algebraic geometry
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