[Paper Review] Eisenstein series with non-unitary twists

Anton Deitmar, Frank Monheim|arXiv (Cornell University)|Dec 21, 2016

Advanced Algebra and Geometry2 references3 citations

TL;DR

This paper establishes the convergence and meromorphic continuation of Eisenstein series twisted by non-unitary but cusp-unitary representations of Fuchsian groups in PSL(2,R). By leveraging invariant integral operators and spectral theory on cusp forms, it proves that such Eisenstein series admit meromorphic continuation to the complex plane and satisfy a functional equation, extending trace formula techniques to non-compact quotients.

ABSTRACT

It is shown that for a non-unitary twist of a Fuchsian group, which is unitary at the cusps, Eisenstein series converge in some half-plane. It is shown that invariant integral operators provide a spectral decomposition of the space of cusp forms and that Eisenstein series admit a meromorphic continuation.

Motivation & Objective

• To extend trace formula techniques to non-compact quotients Γ\G by studying Eisenstein series with non-unitary twists.
• To establish convergence of Eisenstein series under the condition that the representation is unitary on parabolic elements (unitary at cusps).
• To show that invariant integral operators provide a spectral decomposition of the space of cusp forms.
• To prove meromorphic continuation of Eisenstein series and derive their functional equation.

Proposed method

• Uses geometric estimates on word representations in Fuchsian groups to bound the growth of twisted matrix norms.
• Defines a canonical Hilbert space structure on the space of automorphic forms with twisted action.
• Applies invariant integral operators to decompose the space of cusp forms via spectral theory.
• Employs Fredholm theory of integral operators and smoothing operator techniques from pseudo-differential operator theory to analyze resolvents.
• Replaces pointwise kernel estimates with operator norm bounds to handle vector-valued kernels in L2(Γ\H, χ).
• Uses the functional equation of the Eisenstein series by analyzing the difference F(s,z) = E(z,s,χ) − Φ(s)E(z,1−s,χ) and showing it must vanish via spectral contradiction.

Experimental results

Research questions

• RQ1 Under what conditions do Eisenstein series with non-unitary twists converge for non-compact Fuchsian quotients?
• RQ2 Can the space of cusp forms be spectrally decomposed via invariant integral operators under non-unitary twists?
• RQ3 Does the Eisenstein series admit meromorphic continuation to the entire complex plane when the twist is unitary at cusps?
• RQ4 What is the functional equation structure of the twisted Eisenstein series, and how does it relate to the constant term Fourier expansion?
• RQ5 How does the spectral theory of integral operators extend to vector bundle-valued kernels in the context of non-unitary representations?

Key findings

• Eisenstein series with non-unitary but cusp-unitary twists converge in some half-plane, provided the representation is unitary on parabolic elements.
• The space of cusp forms is stable under all invariant integral operators, implying a spectral decomposition via such operators.
• The Eisenstein series admit a meromorphic continuation to the entire complex plane, with poles only at isolated points.
• The vector-valued Eisenstein series satisfy a functional equation E(z,s,χ) = Φ(s)E(z,1−s,χ), where Φ(s) is a meromorphic matrix-valued function.
• The determinant of Φ(s) is non-vanishing, ensuring the functional equation is well-defined and non-degenerate.
• The singularity of the integral kernel vanishes in differences of resolvents, enabling the use of compact operator theory in the proof of analytic continuation.