Fermat descent

TL;DR

This paper reinterprets descent theory using quotient stacks to analyze the arithmetic of primitive solutions in generalized Fermat equations, offering a modern geometric perspective on classical Diophantine problems.

Contribution

It introduces a new geometric framework for descent theory via quotient stacks, applied to the study of primitive solutions in generalized Fermat equations.

Findings

Reinterpretation of descent theory through quotient stacks

Application to primitive solutions of Fermat equations

Enhanced understanding of arithmetic properties of solution stacks

Abstract

Descent theory (a modern formulation of Fermat's classical method of infinite descent) is a powerful tool in arithmetic geometry. In this article, we reinterpret descent theory through the lens of quotient stacks and apply it in the setting where it first arose: the Diophantine study of generalized Fermat equations (1) \[ Ax^a + By^b + Cz^c = 0. \] We focus on understanding the arithmetic of the stacks that arise from the study of primitive integral solutions to general Fermat equations, rather than on solving any particular instance of the equation.