Certain aspects of holomorphic function theory on some genus-zero arithmetic groups

Jorgenson, Jay; Smajlović, Lejla; Then, Holger

doi:10.1112/s1461157016000425

Cited by 11 publications

(5 citation statements)

References 11 publications

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“…(2) 4 (z) would be a weight 2 holomorphic modular form which vanishes only at e 2 , which is not possible since there is no weight two modular form on X N for any squarefree N such that the surface X N has genus zero; see [JST15]. Therefore, E…”

Section: Moonshine Groups Of Square-free Levelmentioning

confidence: 99%

Applications of Kronecker’s limit formula for elliptic Eisenstein series

2017

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We develop two applications of the Kronecker's limit formula associated to elliptic Eisenstein series: A factorization theorem for holomorphic modular forms, and a proof of Weil's reciprocity law. Several examples of the general factorization results are computed, specifically for certain moonshine groups, congruence subgroups, and, more generally, non-compact subgroups with one cusp. In particular, we explicitly compute the Kronecker limit function associated to certain elliptic points for a few small level moonshine groups.

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Section: Moonshine Groups Of Square-free Levelmentioning

confidence: 99%

Applications of Kronecker’s limit formula for elliptic Eisenstein series

2017

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“…Let be a positive square-free integer which is one of the possible values for which the quotient space has genus zero (see [5] for a list of such N as well as [14]). Let be the Kronecker limit function on associated with the parabolic Eisenstein series; it is given by formula (30) above.…”

Section: Examplesmentioning

confidence: 99%

Kronecker Limit Functions and an Extension of the Rohrlich–jensen Formula

Cogdell¹,

Jorgenson

Smajlović³

2023

Nagoya Math. J.

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In [20], Rohrlich proved a modular analog of Jensen’s formula. Under certain conditions, the Rohrlich–Jensen formula expresses an integral of the log-norm $\log \Vert f \Vert $ of a ${\mathrm {PSL}}(2,{\mathbb {Z}})$ modular form f in terms of the Dedekind Delta function evaluated at the divisor of f. In [2], the authors re-interpreted the Rohrlich–Jensen formula as evaluating a regularized inner product of $\log \Vert f \Vert $ and extended the result to compute a regularized inner product of $\log \Vert f \Vert $ with what amounts to powers of the Hauptmodul of $\mathrm {PSL}(2,{\mathbb {Z}})$ . In the present article, we revisit the Rohrlich–Jensen formula and prove that in the case of any Fuchsian group of the first kind with one cusp it can be viewed as a regularized inner product of special values of two Poincaré series, one of which is the Niebur–Poincaré series and the other is the resolvent kernel of the Laplacian. The regularized inner product can be seen as a type of Maass–Selberg relation. In this form, we develop a Rohrlich–Jensen formula associated with any Fuchsian group $\Gamma $ of the first kind with one cusp by employing a type of Kronecker limit formula associated with the resolvent kernel. We present two examples of our main result: First, when $\Gamma $ is the full modular group ${\mathrm {PSL}}(2,{\mathbb {Z}})$ , thus reproving the theorems from [2]; and second when $\Gamma $ is an Atkin–Lehner group $\Gamma _{0}(N)^+$ , where explicit computations of inner products are given for certain levels N when the quotient space $\overline {\Gamma _{0}(N)^+}\backslash \mathbb {H}$ has genus zero, one, and two.

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“…Let N = r ν=1 p ν be a positive square-free integer which is one of the 44 possible values for which the quotient space Y + N = Γ + 0 (N )\H has genus zero; see [Cum04] for a list of such N as well as [JST16b]. Let ∆ N (z) be the Kronecker limit function on Y + N associated to the parabolic Eisenstein series; it is given by formula (26) above.…”

Section: Genus Zero Atkin-lehner Groupsmentioning

confidence: 99%

Kronecker limit functions and an extension of the Rohrlich-Jensen formula

Cogdell¹,

Jorgenson²,

Smajlović³

2021

Preprint

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In [Ro84] Rohrlich proved a modular analogue of Jensen's formula. Under certain conditions, the Rohrlich-Jensen formula expresses an integral of the log-norm log f of a PSL(2, Z) modular form f in terms of the Dedekind Delta function evaluated at the divisor of f . In [BK20] the authors re-interpreted the Rohrlich-Jensen formula as evaluating a regularized inner product of log f and extended the result to compute a regularized inner product of log f with what amounts to powers of the Hauptmoduli of PSL(2, Z).In the present article, we revisit the Rohrlich-Jensen formula and prove that it can be viewed as a regularized inner product of special values of two Poincaré series, one of which is the Niebur-Poincaré series and the other is the resolvent kernel of the Laplacian. The regularized inner product can be seen as a type of Maass-Selberg relation. In this form, we develop a Rohrlich-Jensen formula associated to any Fuchsian group Γ of the first kind with one cusp by employing a type of Kronecker limit formula associated to the resolvent kernel. We present two examples of our main result: First, when Γ is the full modular group PSL(2, Z), thus reproving the theorems from [BK20]; and second when Γ is an Atkin-Lehner group Γ 0 (N ) + , where explicit computations are given for certain genus zero, one and two levels.

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Certain aspects of holomorphic function theory on some genus-zero arithmetic groups

Cited by 11 publications

References 11 publications

Applications of Kronecker’s limit formula for elliptic Eisenstein series

Applications of Kronecker’s limit formula for elliptic Eisenstein series

Kronecker Limit Functions and an Extension of the Rohrlich–jensen Formula

Kronecker limit functions and an extension of the Rohrlich-Jensen formula

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