Hyperdifferential properties of Drinfeld quasi-modular forms

2007/03/28 by Vincent Bosser, Bosser, Vincent, Federico Pellarin +1
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.math/0703842

arxiv created 2007/03/28 · openalex publication_date 2007/03/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This article is divided in two parts. In the first part we endow a certain ring of ``Drinfeld quasi-modular forms'' for \GL₂(\FF_q[T]) (where q is a power of a prime) with a system of "divided derivatives" (or hyperderivations). This ring contains Drinfeld modular forms as defined by Gekeler in \citeGe, and the hyperdifferential ring obtained should be considered as a close analogue in positive characteristic of famous Ramanujan's differential system relating to the first derivatives of the classical Eisenstein series of weights 2, 4 and 6. In the second part of this article we prove that, when q\not=2,3, if \cal P is a non-zero hyperdifferential prime ideal, then it contains the Poincaré series h=P_q+1,1 of \citeGe. This last result is the analogue of a crucial property proved by Nesterenko \citeNes in characteristic zero in order to establish a multiplicity estimate.

Related

• On hyperderivatives of single-cuspidal Drinfeld modular forms with A-expansions
• On Drinfeld modular forms of higher rank and quasi-periodic functions
• Drinfeld modular forms of arbitrary rank, Part III: Examples
• On certain families of Drinfeld quasi-modular forms
• Drinfeld Quasi-Modular Forms of Higher Level