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

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.