Eisenstein Metrics

TL;DR

This paper investigates Eisenstein series-derived metrics on automorphic vector bundles, revealing conditions under which residues form harmonic metrics and highlighting open problems related to matrix-valued Kloosterman sums.

Contribution

It introduces a new class of Eisenstein metrics on automorphic bundles and analyzes their residues, connecting harmonicity to the pole structure of Eisenstein series.

Findings

Residues at rightmost poles can be harmonic metrics.

Analysis of indecomposable but non-irreducible representations.

Open questions relate to matrix-valued Kloosterman sums.

Abstract

We study families of metrics on automorphic vector bundles associated to representations of the modular group. These metrics are defined using an Eisenstein series construction. We show that in certain cases, the residue of these Eisenstein metrics at their rightmost pole is a harmonic metric for the underlying representation of the modular group. The last section of the paper considers the case of a family of representations that are indecomposable but not irreducible. The analysis of the corresponding Eisenstein metrics, and the location of their rightmost pole, is an open question whose resolution depends on the asymptotics of matrix-valued Kloosterman sums.