Convergence of Poincaré series on Hecke groups of large width. (English) Zbl 1457.11035

Pasles, Paul C.

Rocky Mt. J. Math. 49, No. 8, 2739-2746 (2019).

Given \(\lambda >1\), the Hecke group \(G_\lambda\) is given by \(\langle S_\lambda, T \rangle\) with \(S_\lambda = \left( \begin{smallmatrix} 1&\lambda \\ 0&1 \end{smallmatrix} \right)\) and \(T = \left( \begin{smallmatrix} 0&-1\\ 1&0 \end{smallmatrix} \right)\). If \(\mathcal H\) is the complex upper half plane, a multiplier system of weight \(k \in \mathbb C\) on \(G_\lambda\) is a function \(v:G_\lambda \to \mathbb C \setminus \{ 0 \}\) satisfying \[ v(M_3) (c_3z +d_3)^k = v(M_1) (c_1 M_2 z+d_1)^k v(M_2) (c_2 z+d_2)^k \] for all \(z \in \mathcal H\) and \(M_1, M_2, M_3 \in G_\lambda\) with \(M_1 M_2 = M_3\) and \(M_j = \left( \begin{smallmatrix} a_j&b_j\\ c_j&d_j \end{smallmatrix} \right)\) for \(j=1,2,3\). In his earlier work the author described a relation between the parameters associated to multiplier systems on \(G_\lambda\) for \(1 \leq \lambda <2\) and showed that parabolic Poincaré series of nonreal weight on the modular group are not absolutely convergent anywhere. In this paper he obtains a similar divergence result for all Hecke groups \(G_\lambda\) with \(\lambda >2\).

Reviewer: Min Ho Lee (Cedar Falls)

MSC:

11F12	Automorphic forms, one variable
40A05	Convergence and divergence of series and sequences

Keywords:

modular forms; automorphic forms; Poincaré series

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