On the rationality of cycle integrals of meromorphic modular forms

Alfes-Neumann, Claudia; Bringmann, Kathrin; Schwagenscheidt, Markus

doi:10.1007/s00208-019-01914-4

Cited by 8 publications

(18 citation statements)

References 23 publications

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“…Remark Similar theta lifts of meromorphic modular forms were studied by Bruinier, Imamoglu, Funke and Li in [8] and by Bringmann and the authors of the present work in [1]. It was shown there that the generating series of traces of cycle integrals of meromorphic modular forms of positive even weight can be completed to real‐analytic modular forms of half‐integral weight whose images under the lowering operator are given by certain indefinite theta functions.…”

Section: Introduction and Statement Of Resultssupporting

confidence: 68%

“…Let

g : H \to C

be a real‐analytic and

normalΓ

‐invariant function, and assume that it is of moderate growth at

\infty

. We define the regularized Petersson inner product of

f

and

g

\begin{matrix} true \\ right {〈f, g〉}^{reg} = \lim_{ε_{1}, \dots, ε_{r} \to 0} \int_{F^{*} ∖ ⋃_{ℓ = 1}^{r} B_{ε_{ℓ}} (ϱ_{ℓ})} f (z) \bar{g false(z false)} \frac{d x d y}{y^{2}} . \end{matrix}

It was shown in Proposition 3.2 of [1] that this regularized inner product exists under the present assumptions on

f

and

g

. In particular, the theta lift defined in (1.1) converges due to the rapid decay of the Kudla–Millson theta function at

\infty

.…”

Section: Preliminariesmentioning

confidence: 99%

“…Let

f \in {double-struckS}_{0}

be a meromorphic modular form of weight 0 which decays like a cusp form toward

\infty

and let

false[ϱ_{1} false], \dots, false[ϱ_{r} false] \in normalΓ ∖ double-struckH

be the classes of poles of

f

mod

normalΓ

. We define the Kudla–Millson theta lift of

f

\begin{matrix} true \\ right Φ_{KM} (f, τ) = {〈f, \bar{Θ_{KM} (\cdot, τ)}〉}^{reg} = \lim_{ε_{1}, \dots, ε_{r} \to 0} \int_{F^{*} ∖ ⋃_{ℓ = 1}^{r} B_{ε_{ℓ}} (ϱ_{ℓ})} f (z) Θ_{KM} (z, τ) \frac{d x d y}{y^{2}} . \end{matrix}

Since

{normalΘ}_{normalKM} false(z, τ false)

is real‐analytic in

z

and decays square exponentially as

y

goes to

\infty

, it follows from [1, Proposition 3.2] that the theta lift converges for every

τ \in H

. In particular, it transforms like a modular form of weight

3 / 2

for

{normalΓ}_{0} false(4 false)

.…”

Section: The Kudla–millson Theta Liftmentioning

confidence: 99%

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Traces of reciprocal singular moduli

Alfes-Neumann

Schwagenscheidt

2020

Bull. London Math. Soc.

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Section: Introduction and Statement Of Resultssupporting

confidence: 68%

“…Let

g : H \to C

be a real‐analytic and

normalΓ

‐invariant function, and assume that it is of moderate growth at

\infty

. We define the regularized Petersson inner product of

f

and

g

\begin{matrix} true \\ right {〈f, g〉}^{reg} = \lim_{ε_{1}, \dots, ε_{r} \to 0} \int_{F^{*} ∖ ⋃_{ℓ = 1}^{r} B_{ε_{ℓ}} (ϱ_{ℓ})} f (z) \bar{g false(z false)} \frac{d x d y}{y^{2}} . \end{matrix}

It was shown in Proposition 3.2 of [1] that this regularized inner product exists under the present assumptions on

f

and

g

. In particular, the theta lift defined in (1.1) converges due to the rapid decay of the Kudla–Millson theta function at

\infty

.…”

Section: Preliminariesmentioning

confidence: 99%

“…Let

f \in {double-struckS}_{0}

be a meromorphic modular form of weight 0 which decays like a cusp form toward

\infty

and let

false[ϱ_{1} false], \dots, false[ϱ_{r} false] \in normalΓ ∖ double-struckH

be the classes of poles of

f

mod

normalΓ

. We define the Kudla–Millson theta lift of

f

\begin{matrix} true \\ right Φ_{KM} (f, τ) = {〈f, \bar{Θ_{KM} (\cdot, τ)}〉}^{reg} = \lim_{ε_{1}, \dots, ε_{r} \to 0} \int_{F^{*} ∖ ⋃_{ℓ = 1}^{r} B_{ε_{ℓ}} (ϱ_{ℓ})} f (z) Θ_{KM} (z, τ) \frac{d x d y}{y^{2}} . \end{matrix}

Since

{normalΘ}_{normalKM} false(z, τ false)

is real‐analytic in

z

and decays square exponentially as

y

goes to

\infty

, it follows from [1, Proposition 3.2] that the theta lift converges for every

τ \in H

. In particular, it transforms like a modular form of weight

3 / 2

for

{normalΓ}_{0} false(4 false)

.…”

Section: The Kudla–millson Theta Liftmentioning

confidence: 99%

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Traces of reciprocal singular moduli

Alfes-Neumann

Schwagenscheidt

2020

Bull. London Math. Soc.

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“…A further example lies in [2], where Alfes-Neumann, Bringmann, Schwagenscheidt, and the author used the higher Siegel theta lift 1 on lattices of signature (1,2) to investigate traces of cycle integrals of a certain cusp form. By relating the lift to the Fourier coefficients of certain modular objects called harmonic Maass forms, the results also gave an elegant proof of the rationality of such traces of cycle integrals, which had previously been shown in [3].…”

Section: Introductionmentioning

confidence: 75%

“…Since Θ P is a unary theta function it has weight 1 2 , and its preimage G P has weight 3 2 . In fact, via [13] we can recover examples involving the Hurwitz class numbers H (each of whose generating function has weight 3 2 ). For example, combining Theorems 1.1 and 1.2 for the input function f t , and evaluating at the special point described in Section 4 yields the classical formula…”

Section: Introductionmentioning

confidence: 99%

Higher Siegel theta lifts on Lorentzian lattices, harmonic Maass forms, and Eichler-Selberg type relations

Males¹

2021

Preprint

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We investigate so-called "higher" Siegel theta lifts on Lorentzian lattices in the spirit of Bruinier-Ehlen-Yang and Bruinier-Schwagenscheidt. We give an series representation of the lift in terms of Gauss hypergeometric functions, and evaluate the lift as the constant term of a Fourier series involving the Rankin-Cohen bracket of harmonic Maass forms and theta functions. We also note how one could obtain the Fourier expansion. We discuss one explicit example, and show how one could obtain infinite families of relationships between Hurwitz class numbers and divisor power sums.

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Harmonic weak Maass forms and periods II

Alfes,

Bruinier,

Schwagenscheidt

2024

Math. Ann.

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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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On the rationality of cycle integrals of meromorphic modular forms

Cited by 8 publications

References 23 publications

Traces of reciprocal singular moduli

Traces of reciprocal singular moduli

Higher Siegel theta lifts on Lorentzian lattices, harmonic Maass forms, and Eichler-Selberg type relations

Harmonic weak Maass forms and periods II

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