An arithmetic Riemann--Roch theorem for weighted pointed curves

Abstract. In this talk, we report on work in progress with Gerard Freixas generalizing

the arithmetic Riemann--Roch theorem for pointed stable curves to the case

where the metric is allowed to have conical singularities at the marked points.

One main analytical ingredient in the proof is the computation of special values

of certain spectral zeta functions associated to these points.

As particular arithmetic application we derive an analytic class number formula

for the Selberg zeta function of the group $\mathrm{PSL}_2(\mathbb{Z})$.