Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues

Author： Bruinier Jan  

Publisher： Springer Publishing Company

ISSN： 0025-5831

Source： Mathematische Annalen, Vol.342, Iss.3, 2008-11, pp. : 673-693

Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.

Previous Menu Next

Abstract

For integers k ≥ 2, we study two differential operators on harmonic weak Maass forms of weight 2 − k. The operator ξ2-k (resp. D k-1) defines a map to the space of weight k cusp forms (resp. weakly holomorphic modular forms). We leverage these operators to study coefficients of harmonic weak Maass forms. Although generic harmonic weak Maass forms are expected to have transcendental coefficients, we show that those forms which are “dual” under ξ2-k to newforms with vanishing Hecke eigenvalues (such as CM forms) have algebraic coefficients. Using regularized inner products, we also characterize the image of D k-1.

Related content

The Distribution of the Eigenvalues of Hecke Operators

By Golubeva E.  

Journal of Mathematical Sciences, Vol. 133, Iss. 6, 2006-03 ,pp. : 1622-1626 (5)

Springer Publishing Company

Access to resources Recommend Favorite

Hecke Operators on Linear Ordinary Differential Equations

By Lee M.H.  

Acta Applicandae Mathematicae, Vol. 59, Iss. 2, 1999-11 ,pp. : 203-213 (11)

Springer Publishing Company

Access to resources Recommend Favorite

On Hecke eigenvalues of newforms

By Kohnen Winfried  

Mathematische Annalen, Vol. 329, Iss. 4, 2004-08 ,pp. : 623-628 (6)

Springer Publishing Company

Access to resources Recommend Favorite

A Weak Gordon Type Condition for Absence of Eigenvalues of One-dimensional Schrödinger Operators

By Seifert Christian   Vogt Hendrik  

Integral Equations and Operator Theory, Vol. 78, Iss. 3, 2014-03 ,pp. : 383-405 (23)

Springer Publishing Company

Access to resources Recommend Favorite

On general divisor problems involving Hecke eigenvalues

By Lü Guangshi  

Acta Mathematica Hungarica, Vol. 135, Iss. 1-2, 2012-04 ,pp. : 148-159 (12)

Springer Publishing Company

Access to resources Recommend Favorite