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Dedekind eta function, Kronecker limit formula and Dedekind sum for the Hecke group
Abstract
Let $\Gamma$ be a discrete subgroup of $SL(2, \doubz$) with fundamental domain of finite invariant measure ${dydx\over y\sp2}$. $\Gamma$ acts discontinuously on the complex upper half-plane $\Pi$. This action can be extended to $\IR \cup \{\infty\}$. To each cusp $k\sb{i}$ of $\Gamma$ L. Goldstein in (1) attaches an eta function $\eta\sbsp{\Gamma}{k\sb{i}}$. Then $\eta\sbsp{SL(2,\doubz)}{\infty}$ coincides with the classical Dedekind eta function. It is well known that the Eisenstein series$$E\sbsp{\Gamma,s}{k\sb{i}}(z)=\sum\sb\sigma y(\sigma\sbsp{i}{-1}\sigma z)\sp{s}$$(where the sum is over a complete set of representatives $\sigma$ of the cosets $\Gamma\sb{k\sb{i}}\\\Gamma, \Gamma\sb{k\sb{i}}$ is the stabilizer of $k\sb{i}$ in $\Gamma, Re(s) >$ 1 and $z \in\ \Pi)$ converges absolutely and uniformly for s in any compact subset of $Re(s) >$ 1. This series can be analytically continued to a meromorphic function in the entire s-plane. The continllation has all its poles in the interval (0,1) and s = 1 is always a pole. $E\sbsp{\Gamma,s}{k\sb{i}}$ is an automorphic function for $\Gamma$ and an eigenfunction of the Laplace-Beltrami operator. Therefore it can be expanded in a Fourier series at the cusp $k\sb{i}$. For the Eisenstein series for the modular group there exist explicit formulae for the coefficients of this expansion. The generalized function $\eta\sbsp{\Gamma}{k\sb{i}}$ satisfies a transformation formula similar to the one satisfied by the Dedekind eta function $\eta\sbsp{SL(2,\doubz)}{\infty}$ for the modular group. This formula involves a generalization $S\sbsp{\Gamma}{k\sb{i}}(\gamma)$, where $\gamma \in\ \Gamma$, of the Dedeklnd sum. We consider the structure of $\eta\sbsp{\Gamma\sb0(N)}{\infty}$ and $S\sbsp{\Gamma\sb0(N)}{\infty}$ for the Hecke subgroup $\Gamma\sb0(N)$. A key point for the purposes of studying this eta function is to obtain an explicit Fourier expansion for the Eisenstein series for $\Gamma\sb0(N)$ at the cusp $\infty$. We derive a Kronecker limit formula for this case and explicitly describe $\eta\sbsp{\Gamma\sb0(N)}{\infty}$ and $S\sbsp{\Gamma\sb0(N)}{\infty}$. Knowing the generalized Dedekind sum allowed the study of some of its arithmetic properties. In particular, we verify a conjecture of L. Goldstein on the rationality of $S\sbsp{\Gamma\sb0(N)}{\infty}$. In the last chapter we turn to a general cusp $k\sb{i}$ of $\Gamma$ and express the eta function $\eta\sbsp{\Gamma}{k\sb{i}}$ in terms of the eta function at the cusp at infinity. Our results are motivated by, but independent, of L. Goldstein's considerations in (?).
Subject Area
Mathematics
Recommended Citation
Vassileva, Irina Natcheva, "Dedekind eta function, Kronecker limit formula and Dedekind sum for the Hecke group" (1996). Doctoral Dissertations Available from Proquest. AAI9639046.
https://scholarworks.umass.edu/dissertations/AAI9639046