The Hauptmodul at elliptic points of certain arithmetic groups

Jorgenson, Jay; Smajlović, Lejla; Then, Holger

doi:10.1016/j.jnt.2019.03.021

Cited by 2 publications

(1 citation statement)

References 20 publications

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“…Furthermore, for g N,+ ≤ 3, it is shown in [13] that all coefficients in the q-expansion for x + N (z) and y + N (z) are integers. For all such N, the precise values of these coefficients out to large order were computed, and the results are available at [15]. §3.…”

Section: The Function Field Of Meromorphic Functions On Y +mentioning

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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Section: The Function Field Of Meromorphic Functions On Y +mentioning

confidence: 99%

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

Cogdell¹,

Jorgenson

Smajlović³

2023

Nagoya Math. J.

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An approach for computing generators of class fields of imaginary quadratic number fields using the Schwarzian derivative

Jorgenson

Smajlović

Then³

2021

Math. Comp.

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Let N N be one of the 38 38 distinct square-free integers such that the arithmetic group Γ 0 ( N ) + \Gamma _0(N)^+ has genus one. We constructed canonical generators x N x_N and y N y_N for the associated function field (see Jorgenson, L. Smajlović, and H. Then [Exp. Math. 25 (2016), pp. 295–319]). In this article we study the Schwarzian derivative of x N x_N , which we express as a polynomial in y N y_N with coefficients that are rational functions in x N x_N . As a corollary, we prove that for any point e e in the upper half-plane which is fixed by an element of Γ 0 ( N ) + \Gamma _0(N)^+ , one can explicitly evaluate x N ( e ) x_N(e) and y N ( e ) y_N(e) . As it turns out, each value x N ( e ) x_N(e) and y N ( e ) y_N(e) is an algebraic integer which we are able to understand in the context of explicit class field theory. When combined with our previous article (see Jorgenson, L. Smajlović, and H. Then [Exp. Math. 29 (2020), pp. 1–27]), we now have a complete investigation of x N ( τ ) x_N(\tau ) and y N ( τ ) y_N(\tau ) at any CM point τ \tau , including elliptic points, for any genus one group Γ 0 ( N ) + \Gamma _0(N)^+ . Furthermore, the present article when combined with the two aforementioned papers leads to a procedure which we expect to yield generators of class fields, and certain subfields, using the Schwarzian derivative and which does not use either modular polynomials or Shimura reciprocity.

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The Hauptmodul at elliptic points of certain arithmetic groups

Cited by 2 publications

References 20 publications

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

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

An approach for computing generators of class fields of imaginary quadratic number fields using the Schwarzian derivative

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