Arithmetic over trivially valued field and its applications

TL;DR

This paper explores arithmetic over trivially valued fields and applies these findings to Arakelov geometry, demonstrating partial continuity of arithmetic χ-volume and providing bounds for the arithmetic Hilbert-Samuel function.

Contribution

It introduces new applications of trivially valued field arithmetic to Arakelov geometry, including partial continuity results and bounds for key functions.

Findings

Partial continuity of arithmetic χ-volume along semiample divisors

Upper bound estimate of arithmetic Hilbert-Samuel function

Applications to Arakelov geometry over adelic curves

Abstract

By some result on the study of arithemtic over trivially valued field, we find its applications to Arakelov geometry over adelic curves. We prove a partial result of the continuity of arithmetic $\chi$-volume along semiample divisors. Moreover, we give a upper bound estimate of arithmetic Hilbert-Samuel function.