Numerical cohomology for arithmetic surfaces and applications

TL;DR

This paper develops a numerical cohomology framework for arithmetic surfaces, deriving an absolute arithmetic Riemann-Roch formula and providing bounds on self-intersection numbers using geometric and arithmetic invariants.

Contribution

Introduction of numerical cohomology for arithmetic surfaces and an absolute arithmetic Riemann-Roch formula with applications to bounding self-intersection numbers.

Findings

Derived an upper bound for the self-intersection number of the relative dualizing sheaf.

Established a geometric analogue relating slopes of the Harder-Narasimhan filtration to self-intersection bounds.

Provided refined bounds considering the existence of sections and genus conditions.

Abstract

In this paper, we introduce numerical cohomology for arithmetic surfaces, which leads to an absolute version of arithmetic Riemann-Roch formula. As an application, we derive an upper bound for the self-intersection number of relative dualizing sheaf in terms of successive minima with respect to $L^2$-norm. The result has the geometric analogue that the slopes of the Harder-Narasimhan filtration of relative dualizing sheaf provide an upper bound for self-intersection number. Suppose that the arithmetic surface admits a section and has generic fiber of genus at least two, we obtain a refined upper bound for the self-intersection number, which is governed by the topological and arithmetic information of the section.