Fermat descent

2025/08/18 by Arango-Piñeros, Santiago · 1 citation
#11D41 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.13059

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.

Citations

• Number fields generated by points in linear systems on curves
• The Brauer-Manin obstruction for stacky curves
• Smith Normal Form in Combinatorics
• Algebraic curves uniformized by congruence subgroups of triangle groups
• A "bottom up" characterization of smooth Deligne-Mumford stacks
• The canonical ring of a stacky curve
• Uniformization of Deligne-Mumford curves

Cited by

• Counting primitive integral solutions to spherical generalized Fermat equations

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