Sur la th\'eorie spectrale des m\'etriques int\'egrables sur une surface de Riemann compacte

TL;DR

This paper extends the spectral theory of Laplacians to line bundles with 1-integrable metrics on compact Riemann surfaces, providing new identities and generalizations of previous results in the field.

Contribution

It introduces the concept of 1-integrable metrics on line bundles and extends spectral theory to these metrics on compact Riemann surfaces.

Findings

Extended spectral theory to 1-integrable metrics.

Derived identity relating zeta function derivative and torsion.

Reproduced known results through generalized framework.

Abstract

We continue the study of the spectral theory associated to integrable metrics, started in our previous paper arXiv:1301.1793 [math.SP]. We introduce the notion of 1-integrable metric on line-bundles on a compact Riemann surface. We extend the spectral theory of generalized Laplacians to line-bundles equipped with 1-integrable metrics. As an application, we recover the following identity: [\zeta'_{\Delta_{\bar{\mathcal{O}(m)}_\infty}}(0)=T_g\bigl((\p^1,\omega_\infty); \bar{\mathcal{O}(m)}_\infty \bigr),] obtained using direct computations in arXiv:1301.1792 [math.NT].