A Height Inequality

TL;DR

This paper establishes a height inequality for arithmetic surfaces by developing a mathematical structure that estimates the canonical norm of relative differential forms, linking algebraic and arithmetic properties.

Contribution

It introduces a new mathematical framework on arithmetic surfaces that provides lower bounds for canonical norms, leading to the proof of a height inequality.

Findings

Lower bound for the canonical norm established

Height inequality proved for arithmetic surfaces

Framework links algebraic and arithmetic properties

Abstract

We give a mathematical structure on an arithmetic surface, that has algebraic meanings over finite places and can estimate the canonical norm for a relative differential form on the arithmetic surface. This will give a lower bound for the canonical norm for a relative differential form on an arithmetic surface, which proves a Height Inequality on the arithmetic surface.