Regularized theta liftings and periods of modular functions

Bruinier, Jan Hendrik; Funke, Jens; Imamoḡlu, Özlem

doi:10.1515/crelle-2013-0035

Cited by 35 publications

(83 citation statements)

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“…sinh(2πy) + 2 sinh(2π/y)) dy y , so the values of Tr 1,1 (j 1 ) in[2] and (1.7) agree.The modular traces Tr d,D (j m ) for m > 1 are also related to the coefficients a(D, d). With the modular trace now defined when dD is a square, we obtain [3, Theorem 3] with the condition 'dD not a square' removed.…”

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confidence: 71%

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Periods of the j -function along infinite geodesics and mock modular forms

Andersen

2015

Bulletin of the London Mathematical Society

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Abstract. Zagier's well-known work on traces of singular moduli relates the coefficients of certain weakly holomorphic modular forms of weight 1/2 to traces of values of the modular j-function at imaginary quadratic points. A real quadratic analogue was recently studied by Duke, Imamoḡlu, and Tóth. They showed that the coefficients of certain weight 1/2 mock modular formsare given in terms of traces of cycle integrals of the j-function. Their result applies to those coefficients a(d, D) for which dD is not a square. Recently Bruinier, Funke, and Imamoḡlu employed a regularized theta lift to show that the coefficients a(d, D) for square dD are traces of regularized integrals of the j-function. In the present paper we provide an alternate approach to this problem. We introduce functions jm,Q (for Q a quadratic form) which are related to the j-function and show, by modifying the method of Duke, Imamoḡlu, and Tóth, that the coefficients for which dD is a square are traces of cycle integrals of the functions jm,Q.

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confidence: 71%

“…This is the obstruction to a geometric interpretation of the modular trace for square discriminants. In a recent paper, Bruinier, Funke, and Imamoḡlu [2] address this issue by regularizing the integral (1.5) and showing that the corresponding modular traces…”

Section: Introductionmentioning

confidence: 99%

Periods of the j -function along infinite geodesics and mock modular forms

Andersen

2015

Bulletin of the London Mathematical Society

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“…is a mock modular form of weight 1 2 and multiplier system χ θ on Γ 0 (4) such that its shadow is the weakly holomorphic modular form −2g 1 , where g 1 (z) := −q −1 +2+ d<0 Tr d (j 1 )q |d| . The following theorem is the extension of [14], by Bruinier, Funke, and Imamoḡlu [11], to weakly holomorphic modular functions on modular curves of arbitrary genus.…”

Section: Modular Tracesmentioning

confidence: 99%

“…The remainder of this paper is organized as follows. In Section 2, we review the basic notions of modular traces and their modularity which were proved by Duke, Imamoḡlu, Tóth [14] and Bruinier, Funke, Imamoḡlu [11]. In Section 3, we define regularized Eichler integrals for weakly holomorphic modular forms, and then we prove that the regularized Eichler integrals are well defined and that, for a harmonic weak Maass form f , the nonholomorphic part of f is the same as the regularized Eichler integral of its shadow.…”

Section: Introductionmentioning

confidence: 99%

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Limits of traces of singular moduli

Choi

Lim

2019

Trans. Amer. Math. Soc.

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Let f and g be weakly holomorphic modular functions on Γ 0 (N ) with the trivial character. For an integer d, let Tr d (f ) denote the modular trace of f of index d. Let r be a rational number equivalent to i∞ under the action of Γ 0 (4N ). In this paper, we prove that, when z goes radially to r, the limit QĤ (f ) (r) of the sum H(f )(z) = d>0 Tr d (f )e 2πidz is a special value of a regularized twisted L-function defined by Tr d (f ) for d ≤ 0. It is proved that the regularized L-function is meromorphic on C and satisfies a certain functional equation. Finally, under the assumption that N is square free, we prove that if QĤ (f ) (r) = QĤ (g) (r) for all r equivalent to i∞ under the action of Γ 0 (4N ), then Tr d (f ) = Tr d (g) for all integers d.

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Harmonic Eisenstein Series of Weight One

2017

Contributions in Mathematical and Computational Sciences

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In this short note, we will construct a harmonic Eisenstein series of weight one, whose image under the ξ-operator is a weight one Eisenstein series studied by Hecke [6]. 1 2 YINGKUN LĨ ϑ(τ ) and related to the holomorphic Eisenstein series ϑ(τ ) constructed by Hecke via ξθ(τ ) = ϑ(τ ),where ξ = ξ 1 is the differentiable operator introduced by Bruinier and Funke [4]. In the notion loc. cit.,θ(τ ) is a harmonic Maass form of weight one. For any k ∈ 1 2 Z, a harmonic Maass form of weight k is a real analytic functions on the upper half-plane H := {τ = u+iv : v > 0} that transforms with weight k with respect to a discrete subgroup of SL 2 (R), and is annihilated by the weight k hyperbolic LaplacianHarmonic Maass forms can be written as the sum of a holomorphic part and a non-holomorphic part. The Fourier coefficients of their holomorphic parts are expected to contain interesting arithmetic information concerning the ξ k -images of the non-holomorphic parts (see e.g.[2, 5]).In [11], Kudla, Rapoport and Yang considered an Eisenstein series, which is harmonic. The Fourier coefficients of its holomorphic part are logarithms of rational numbers, and can be interpreted as the arithmetic degree of special divisors on an arithmetic curve. In view of their work and the Kudla program [8], we expect the Fourier coefficients of the harmonic Eisenstein series we construct to have a similar interpretation as well.The idea to constructθ(τ ) is rather straightforward. If we can construct a functionΘ(τ, t) such that it is modular in τ and satisfies ξΘ(τ, t) = Θ(τ, t) for each t, then simply integrating it in t will produce the desirableθ(τ ). This idea has already been used in [3], where ξ 1/2 connected the theta kernels constructed from the Gaussian and the Kudla-Millson Schwartz form. In our setting, we will introduce an L ∞ functionφ τ , which is a ξ-preimage of the Schwartz function used in constructing Θ(τ, t) under ξ (see Prop. 3.4). We will then use this function to form a theta kernelΘ(τ, t) and integrate it to obtain the harmonic Eisenstein seriesθ(τ ) in Theorem 4.3 in the last section.

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Regularized theta liftings and periods of modular functions

Cited by 35 publications

References 28 publications

Periods of the j -function along infinite geodesics and mock modular forms

Periods of the j -function along infinite geodesics and mock modular forms

Limits of traces of singular moduli

Harmonic Eisenstein Series of Weight One

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