Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues
TLDR
In this paper, two differential operators on harmonic weak Maass forms of weight 2−k were studied for integers k ≤ 2 and they were shown to have algebraic coefficients for CM forms with vanishing Hecke eigenvalues.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.read more
Citations
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Journal ArticleDOI
Heegner divisors, l-functions and harmonic weak maass forms
Jan Hendrik Bruinier,Ken Ono +1 more
TL;DR: In this paper, it was shown that the twisted Heegner divisor can also serve as a generator for central values and derivatives of quadratic twists of weight 2 modular L-functions.
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Heegner divisors, $L$-functions and harmonic weak Maass forms
Jan Hendrik Bruinier,Ken Ono +1 more
TL;DR: In this article, it was shown that the twisted Heegner divisor can also serve as a generator for central values and derivatives of quadratic twists of weight 2 modular $L$-functions.
Journal ArticleDOI
Eichler–Shimura theory for mock modular forms
TL;DR: In this paper, the authors derived an Eichler-Shimura theory for weakly holomorphic modular forms and mock modular forms, as well as a correspondence between mock modular periods and classical periods, and a Haberland-type formula which expresses Petersson's inner product and a related antisymmetric inner product in terms of periods.
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p-Adic coupling of mock modular forms and shadows.
TL;DR: The modular solution to the cubic “arithmetic-geometric mean” is considered, and it relies on the definition of an algebraic “regularized mock modular form” based on the coefficients of shadows and mock modular forms.
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Locally Harmonic Maass Forms and the Kernel of the Shintani Lift
TL;DR: In this paper, the authors define a new type of modular object and construct explicit examples of such functions, which are closely related to cusp forms constructed by Zagier [37] which played an important role in the construction of a kernel function for the Shimura and Shintani lifts between half-integral and integral weight Cusp forms.
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