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Claudia Alfes-Neumann

Publications and source records attributed to Claudia Alfes-Neumann.

11 recordsLinked to original sources
arXiv2024

A classification of polyharmonic Maaß forms via quiver representations

We give a classification of the Harish-Chandra modules generated by the pullback to~$\SL{2}(\RR)$ of \emph{poly}harmonic Maaß forms for congruence subgroups of~$\SL{2}(\ZZ)$ with exponential growth allowed at the cusps. This extends results of Bringmann--Kudla in the harmonic case. While in the harmonic setting there are nine cases, our classification comprises ten; A new case arises in weights $k > 1$. To obtain the classification we introduce quiver representations into the topic and show that those associated with polyharmonic Maaß forms are cyclic, indecomposable representations of the two-cyclic or the Gelfand quiver. A classification of these transfers to a classification of polyharmonic weak Maaß forms. To realize all possible cases of Harish-Chandra modules we develop a theory of weight shifts for Taylor coefficients of vector-valued spectral families. We provide a comprehensive computer implementation of this theory, which allows us to provide explicit examples.

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arXiv2023

On Kleinian mock modular forms

We give an explicit and computationally efficient construction of harmonic weak Maass forms which map to weight $2$ newforms under the $ξ$-operator. Our work uses a new non-analytic completion of the Kleinian $ζ$-function from the theory of Abelian functions.

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arXiv2023

On Jacobi--Weierstrass mock modular forms

We construct harmonic weak Maass forms that map to cusp forms of weight $k\geq 2$ with rational coefficients under the $ξ$-operator. This generalizes work of the first author, Griffin, Ono, and Rolen, who constructed distinguished preimages under this differential operator of weight $2$ newforms associated to rational elliptic curves using the classical Weierstrass theory of elliptic functions. We extend this theory and construct a vector-valued Jacobi--Weierstrass $ζ$-function which is a generalization of the classical Weierstrass $ζ$-function.

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arXiv2022

Harmonic weak Maass forms and periods II

In this paper we investigate the Fourier coefficients of harmonic Maass forms of negative half-integral weight. We relate the algebraicity of these coefficients to the algebraicity of the coefficients of certain canonical meromorphic modular forms of positive even weight with poles at Heegner divisors. Moreover, we give an explicit formula for the coefficients of harmonic Maass forms in terms of periods of certain meromorphic modular forms with algebraic coefficients.

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arXiv2022

A classification of harmonic weak Maaß forms of half-integral weight

We classify Harish-Chandra modules generated by the pullback to the metaplectic group of harmonic weak Maaß forms with exponential growth allowed at the cusps. This extends work by Schulze-Pillot and parallels recent work by Bringmann-Kudla, who investigated the case of integral weights. We realize each of our cases via a regularized theta lift of an integral weight harmonic weak Maaß form. Harish-Chandra modules in both integral and half-integral weight that occur need not be irreducible. Therefore, our display of the role that the theta lifting takes in this picture, we hope, contributes to an initial understanding of a theta correspondence for extensions of Harish-Chandra modules.

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arXiv2020

On the rationality of cycle integrals of meromorphic modular forms

We derive finite rational formulas for the traces of cycle integrals of certain meromorphic modular forms. Moreover, we prove the modularity of a completion of the generating function of such traces. The theoretical framework for these results is an extension of the Shintani theta lift to meromorphic modular forms of positive even weight.

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arXiv2020

Traces of reciprocal singular moduli

We show that the generating series of traces of reciprocal singular moduli is a mixed mock modular form of weight $3/2$ whose shadow is given by a linear combination of products of unary and binary theta functions. To prove these results, we extend the Kudla-Millson theta lift of Bruinier and Funke to meromorphic modular functions.

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arXiv2017

Shintani theta lifts of harmonic Maass forms

We define a regularized Shintani theta lift which maps weight $2k+2$ ($k \in \Z, k \geq 0$) harmonic Maass forms for congruence subgroups to (sesqui-)harmonic Maass forms of weight $3/2+k$ for the Weil representation of an even lattice of signature $(1,2)$. We show that its Fourier coefficients are given by traces of CM values and regularized cycle integrals of the input harmonic Maass form. Further, the Shintani theta lift is related via the $ξ$-operator to the Millson theta lift studied in our earlier work. We use this connection to construct $ξ$-preimages of Zagier's weight $1/2$ generating series of singular moduli and of some of Ramanujan's mock theta functions.

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arXiv2016

On a theta lift related to the Shintani lift

We study a certain theta lift which maps weight $-2k$ to weight $1/2-k$ harmonic weak Maass forms for $k \in \mathbb{Z}, k \geq 0$, and which is closely related to the classical Shintani lift from weight $2k+2$ to weight $k+3/2$ cusp forms. We compute the Fourier expansion of the theta lift and show that it involves twisted traces of CM values and geodesic cycle integrals of the input function. As an application, we obtain a criterion for the non-vanishing of the central $L$-value of an integral weight newform $G$ in terms of the holomorphicity of the theta lift of a certain harmonic weak Maass form associated to $G$. Moreover, we derive interesting identities between cycle integrals of different kinds of modular forms.

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