Elliptic Eisenstein series for $${PSL}_{2}(\mathbb{Z})$$

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 \(\Gamma \setminus \mathbb{H}\)be the associated finite volume hyperbolic Riemann surface. Associated to any cusp of \(\Gamma \setminus \mathbb{H}\), there is the classically studied non-holomorphic (parabolic) Eisenstein series. In [11], Kudla and Millson studied non-holomorphic (hyperbolic) Eisenstein series associated to any closed geodesic on \(\Gamma \setminus \mathbb{H}\). Finally, in [9], Jorgenson and the first named author introduced so-called elliptic Eisenstein series associated to any elliptic fixed point of \(\Gamma \setminus \mathbb{H}\). In this article, we study elliptic Eisenstein series for the full modular group \(\mathrm{{PSL}}_{2}(\mathbb{Z})\). We explicitly compute the Fourier expansion of the elliptic Eisenstein series and derive from this its meromorphic continuation.