A Kronecker limit type formula for elliptic Eisenstein series

TL;DR

This paper extends the theory of Eisenstein series by establishing a Kronecker limit formula for elliptic Eisenstein series on hyperbolic surfaces, providing explicit residue computations and formulas for the modular group.

Contribution

It proves the meromorphic continuation of elliptic Eisenstein series, computes their poles and residues, and derives a Kronecker limit formula applicable to general Fuchsian groups.

Findings

Meromorphic continuation of elliptic Eisenstein series established

Explicit poles and residues computed for these series

Kronecker limit functions expressed in terms of modular forms for PSL(2,Z)

Abstract

Let $\Gamma\subset\mathrm{PSL}_{2}(\mathbb{R})$ be a Fuchsian subgroup of the first kind acting by fractional linear transformations on the upper half-plane $\mathbb{H}$, and let $M=\Gamma\backslash\mathbb{H}$ be the associated finite volume hyperbolic Riemann surface. Associated to any cusp of $M$, there is the classically studied non-holomorphic (parabolic) Eisenstein series. In 1979, Kudla and Millson studied non-holomorphic (hyperbolic) Eisenstein series associated to any closed geodesic on $\Gamma\backslash\mathbb{H}$. In 2004, Jorgenson and Kramer introduced so-called elliptic Eisenstein series associated to any elliptic fixed point of $M$. In the present article, we prove the meromorphic continuation of the elliptic Eisenstein series and we explicitly compute its poles and residues. Further, we derive a Kronecker limit type formula for elliptic Eisenstein series for general $\Gamma$. Finally, for the full modular group $\mathrm{PSL}_{2}(\mathbb{Z})$, we give an explicit formula for the Kronecker's limit functions in terms of holomorphic modular forms.