Improved bounds on the number of holomorphic maps between compact Riemann surfaces

TL;DR

This paper establishes new upper bounds on the number of nonconstant holomorphic maps between compact Riemann surfaces, depending solely on their genus, improving previous estimates through advanced geometric techniques.

Contribution

It introduces improved bounds for holomorphic maps between Riemann surfaces based on genus, utilizing differential pullbacks and geometric number theory methods.

Findings

New upper bounds depend only on genus

Bounds are tighter than previous estimates

Method combines differential pullbacks with Jacobian variety techniques

Abstract

We give new upper bounds for the number of nonconstant holomorphic maps depending only on the genus. Our estimates improve previously known bounds. The proof is based on the study of pullbacks of holomorphic differentials, together with techniques from the geometry of numbers and the theory of Jacobian varieties.