On the Belyi degree(s) of a curve defined over a number field

TL;DR

This paper studies the minimal degree of Belyi covers of algebraic curves over number fields, providing bounds based on primes of bad reduction and establishing finiteness results for curves with bounded Belyi degree.

Contribution

It introduces the absolute and relative Belyi degrees of curves over number fields, establishes lower bounds depending on bad reduction primes, and proves finiteness of curves with bounded Belyi degree.

Findings

Lower bounds for Belyi degrees depending on bad reduction primes.

Finiteness of isomorphism classes of curves with bounded Belyi degree.

Sharpness of the established lower bounds.

Abstract

Belyi's theorem asserts that a smooth projective curve $X$ defined over a number field can be realized as a cover of the projective line unramified outside three points. In this short paper we investigate the bejaviour of the minimal degree of such a cover. More precisely, we start by defining the absolute Belyi degree of X, which only depends on the $\bar{\bold Q}$-isomorphism class of $X$. We then give a lower bound of this invariant, only depending on the stable primes of bad reduction (as defined in the paper) and we show that this bound is sharp. In the second part of the paper, we introduce the relative Belyi degree of a curve X defined over a fixed number field $K$. We first prove that there exist finitely many $K$-isomorphism classes of curves of bounded (relative) Belyi degree and we then obtain a lower bound, only depending on the primes of bad reduction of the minimal regular model of $X$ over (the ring of integers of) $K$.