A Finitary Hasse Principle for Diagonal Curves

TL;DR

This paper establishes a Hasse principle for certain Diophantine equations and diagonal curves over fields with finitely generated Galois groups, advancing understanding of rational solutions in algebraic number theory.

Contribution

It proves a new Hasse principle for equations involving finite index subgroups of rational multiplicative groups and diagonal curves over specific algebraic fields.

Findings

Hasse principle holds for equations ax+by+cz=0 with subgroup constraints

Hasse principle extends to diagonal curves over fields with finitely generated Galois groups

Provides new criteria for the existence of rational solutions in algebraic number fields

Abstract

We prove a Hasse principle for solving equations of the form ax+by+cz=0 where x, y, z belong to a given finite index subgroup of the multiplicative group of rational numbers. From this we deduce a Hasse principle for diagonal curves over fields of algebraic numbers with finitely generated Galois group.