An Effective Version of Chevalley-Weil Theorem for Projective Plane Curves

TL;DR

This paper provides a quantitative version of the Chevalley-Weil theorem for plane projective curves, offering effective bounds on the discriminant of number fields generated by rational points and their preimages.

Contribution

It introduces an explicit, effective bound for the discriminant in the Chevalley-Weil theorem applied to non-singular plane projective curves over number fields.

Findings

Derived an explicit upper bound for the norm of the relative discriminant.

Extended classical Chevalley-Weil theorem to an effective, quantitative form.

Applicable to unramified morphisms of non-singular plane projective curves.

Abstract

We obtain a quantitative version of the classical Chevalley-Weil theorem for curves. Let $\phi : \tilde{C} \to C$ be an unramified morphism of non-singular plane projective curves defined over a number field $K$. We calculate an effective upper bound for the norm of the relative discriminant of the number field $K(Q)$ over $K$ for any point $P\in C(K)$ and $Q\in{\phi}^{-1}(P)$