Numerical cohomology for arithmetic surfaces and applications
Wei He
TL;DR
The paper defines numerical cohomology h^i_𝔛(ℒ) for Hermitian line bundles on arithmetic surfaces and proves an absolute arithmetic Riemann–Roch formula linking Euler characteristics with both arithmetic and analytic data. It then derives practical upper bounds for the self-intersection number (ω_{𝔛/𝔬},ω_{𝔛/𝔬}) in terms of the successive minima of f_*ω_{𝔛/𝔬} under L^2-norms, mirroring the Harder–Narasimhan framework in geometry. The results connect Noether’s formula, geometry of numbers (via Minkowski), and analytic torsion to obtain explicit bounds that depend on genus, field invariants, and topological features of sections. These bounds have potential implications for effective arithmetic geometry questions, such as connections to Bogomolov-type and Mordell-type conjectures through refined invariants.
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.