Modularity of generating series of winding numbers

@article{Bruinier2017ModularityOG,
title={Modularity of generating series of winding numbers},
author={J. Bruinier and Jens Funke and {\"O}. Imamoḡlu and Yingkun Li},
journal={Research in the Mathematical Sciences},
year={2017},
volume={5},
pages={1-23}
}
The Shimura correspondence connects modular forms of integral weights and half-integral weights. One of the directions is realized by the Shintani lift, where the inputs are holomorphic differentials and the outputs are holomorphic modular forms of half-integral weight. In this article, we generalize this lift to differentials of the third kind. As an application, we obtain a modularity result concerning the generating series of winding numbers of closed geodesics on the modular curve.
On the rationality of cycle integrals of meromorphic modular forms
• Mathematics
• 2018
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.Expand
Shintani Lifts of Nearly Holomorphic Modular Forms.
• Mathematics
• 2020
In this paper, we compute the Fourier expansion of the Shintani lift of nearly holomorphic modular forms. As an application, we deduce modularity properties of generating series of cycle integrals ofExpand
Traces of reciprocal singular moduli
• Mathematics
• 2019
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 binaryExpand
Meromorphic modular forms with rational cycle integrals
• Mathematics
• 2019
We study rationality properties of geodesic cycle integrals of meromorphic modular forms associated to positive definite binary quadratic forms. In particular, we obtain finite rational formulas forExpand
J ul 2 01 9 MEROMORPHIC MODULAR FORMS WITH RATIONAL CYCLE INTEGRALS
We study rationality properties of geodesic cycle integrals of meromorphic modular forms associated to positive definite binary quadratic forms. In particular, we obtain finite rational formulas forExpand
Shintani theta lifts of harmonic Maass forms
• Mathematics
• 2017
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 theExpand

References

SHOWING 1-10 OF 37 REFERENCES
Traces of CM values of modular functions
• Mathematics
• 2004
Abstract Zagier proved that the traces of singular moduli, i.e., the sums of the values of the classical j-invariant over quadratic irrationalities, are the Fourier coefficients of a modular form ofExpand
On construction of holomorphic cusp forms of half integral weight
In [10], G.Shimura gave a method of constructing holomorphic cusp forms of even integral weight from given forms of half integral weight. In this paper, we try to present an inverse construction. ToExpand
Heegner Divisors and Nonholomorphic Modular Forms
We consider an embedded modular curve in a locally symmetric space M attached to an orthogonal group of signature (p, 2) and associate to it a nonholomorphic elliptic modular form by integrating aExpand
Intersection numbers of cycles on locally symmetric spaces and fourier coefficients of holomorphic modular forms in several complex variables
• Mathematics
• 1990
Using the theta correspondence we construct liftings from the cohomology with compact supports of locally symmetric spaces associated to O(p, q) (resp. U(p, q)) of degreenq (resp. Hodge typenq, nq)Expand
Cycles in hyperbolic manifolds of non-compact type and Fourier coefficients of Siegel modular forms
• Mathematics
• 2001
Abstract: Using the theta correspondence, we study a lift from (not necessarily rapidly decreasing) closed differential (p−n)-forms on a non-compact arithmetic quotient of hyperbolic p-space toExpand
Mock modular forms and geometric theta functions for indefinite quadratic forms
• Mathematics
• 2017
Theta functions for indefinite quadratic forms are an important tool to construct modular forms and Mock modular forms. In this note, we recall the representation-theoretic background in theExpand
On two geometric theta lifts
• Mathematics
• 2002
The theta correspondence has been an important tool in studying cycles in locally symmetric spaces of orthogonal type. In this paper we establish for the orthogonal group O(p,2) an adjointness resultExpand
Twisted traces of CM values of weak Maass forms
• Mathematics
• 2013
Abstract We show that the twisted traces of CM values of weak Maass forms of weight 0 are Fourier coefficients of vector valued weak Maass forms of weight 3/2. These results generalize work by ZagierExpand
On some incomplete theta integrals
• Mathematics
• Compositio Mathematica
• 2019
In this paper we construct indefinite theta series for lattices of arbitrary signature $(p,q)$ as ‘incomplete’ theta integrals, that is, by integrating the theta forms constructed by the secondExpand
Automorphic forms with singularities on Grassmannians
We construct some families of automorphic forms on Grassmannians which have singularities along smaller sub Grassmannians, using Harvey and Moore's extension of the Howe (or theta) correspondence toExpand