Eisenstein series twisted with non-expanding cusp monodromies

Abstract

Let Γ be a geometrically finite Fuchsian group and suppose that χ : Γ → GL(V ) is a finite-dimensional representation with non-expanding cusp monodromy. We show that the parabolic Eisenstein series for Γ with twist χ converges on some half-plane. Further, we develop Fourier-type expansions for these Eisenstein series.

Meromorphic continuation of Selberg zeta functions with twists having non-expanding cusp monodromy

“…Even though we do not need this result in this article, we record it here for purposes of reference. This result is crucial for the study of χ-twisted Eisenstein series [15]. Proof.…”

“…Answering this question we leave for future research. The first step in this direction is already done in [15], where the convergence of χ-twisted Eisenstein series is addressed. Let X := Γ\H denote the (two-dimensional real hyperbolic good) orbifold with fundamental group Γ.…”

“…Then (14) becomes d H (i, h.i) = ℓ(c h ). We continue to proceed as in the proof of Proposition 3.2 until (15), which now reads j∈Jp m d0−1 j ≪ e 4(d0−1)d H (i,h.i) .…”

Modular forms of virtually real-arithmetic type I: Mixed mock modular forms yield vector-valued modular forms

“…We also give an explicit bound on the weight of convergence. Both [38] and [29] contain a proof of convergence for much more general types. A bound on the weight of convergence in terms of geodesics is given in [29].…”

“…Müller established a trace formula for non-unitary twists [47], which includes modular forms for our notion of vra types except that Müller restricts to co-compact subgroups of SL 2 (R). Extensions of this trace formula and the associated continuous spectrum have since then been investigated in several papers [20,21,29]. Theorem 2 can be viewed as a partial description of the holomorphic discrete spectrum for vra types.…”