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

By Jan H. Bruinier, Ken Ichi Ono, Robert C. Rhoades. 2008

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For integers k ≥ 2, we study two differential operators on harmonic weak Maass forms of weight 2 − k. The operator ξ2-k (resp. Dk-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 Dk-1.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/53cbc71d430dfe935a3bbb27c137b1ed96cbfd91

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In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

Eichler–Shimura theory for mock modular forms (2013, background): “1) Formula (1.16) gives an extension of the Petersson scalar product, one which works even when other “regularizations” fail.”