On the action of Hecke operators on Drinfeld modular forms

Kirti N Joshi, Aleksandar Petrov

Research output: Contribution to journalArticle

Abstract

In this paper we determine the explicit structure of the semisimple part of the Hecke algebra that acts on Drinfeld modular forms of full level modulo T. We show that modulo T the Hecke algebra has a non-zero semisimple part. In contrast, a well-known theorem of Serre asserts that for classical modular forms the action of Tℓ for any odd prime ℓ is nilpotent modulo 2. After proving the result for Drinfeld modular forms modulo T, we use computations of the Hecke action modulo T to show that certain powers of the Drinfeld modular form h cannot be eigenforms. Finally, we pose a question a positive answer to which will mean that the Hecke algebra that acts on Drinfeld modular forms of full level is not smooth for large weights, which again contrasts the classical situation.

Original languageEnglish (US)
Pages (from-to)186-200
Number of pages15
JournalJournal of Number Theory
Volume137
DOIs
StatePublished - Apr 2014

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Hecke Operators
Modular Forms
Modulo
Hecke Algebra
Semisimple
Odd
Theorem

Keywords

  • Drinfeld modular forms
  • Reduction of Drinfeld modular forms modulo T

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

On the action of Hecke operators on Drinfeld modular forms. / Joshi, Kirti N; Petrov, Aleksandar.

In: Journal of Number Theory, Vol. 137, 04.2014, p. 186-200.

Research output: Contribution to journalArticle

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