Quantitative Riemann existence theorem over a number field

TL;DR

This paper constructs explicit algebraic models for coverings of the projective line over number fields, providing bounds on their defining equations and fields, thus making the Riemann existence theorem effective in arithmetic settings.

Contribution

It introduces a method to explicitly realize algebraic curves corresponding to ramified coverings over number fields, with bounds on equations and fields, extending classical Riemann existence results.

Findings

Explicit models of algebraic curves over number fields

Bounds on defining equations of curves

Effective realization of Riemann existence theorem

Abstract

Given a covering of the projective line with ramifications defined over a number field, we define a plain model of the algebraic curve realizing the Riemann existence theorem for this covering, and bound explicitly the defining equation of this curve and its definition field.