A Kronecker limit type formula for elliptic Eisenstein series

Let $Γ\subset\mathrm{PSL}_{2}(\mathbb{R})$ be a Fuchsian subgroup of the first kind acting by fractional linear transformations on the upper half-plane $\mathbb{H}$, and let $M=Γ\backslash\mathbb{H}$ 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 $Γ\backslash\mathbb{H}$. 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 $Γ$. 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.

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