Generalized Fermat equations, stacks, and arithmetic statistics Open Access

Abstract

Let (a, b, c) be a triple of positive integers. The Belyi stack P1(a, b, c) is the algebraic stack obtained by rooting the projective line at 0, 1, and infinity with multiplicities a, b, and c, respectively.

In this thesis, we study the relationship between primitive integral solutions to generalized Fermat equations

F: A x^a + B y^b + C z^c = 0 (1)

and the S-integral points on P1(a, b, c).

We find that, after inverting a suitable finite set S of rational primes, the stack P1(a, b, c) is isomorphic to the quotient [U/H], where U is the punctured cone defined by F, and H is the stabilizer group scheme of U in G_m^3. By descent theory, the S-integral points of P1(a, b, c) are partitioned into H(Z_S)-orbits of U_tau(Z_S) for an explicit set of twists F_tau of Equation (1). From this perspective, we reformulate the proofs of the landmark results of Darmon–Granville [13, Theorem 2] and Beukers [7, Theorem 1.2].

Finally, we obtain a winsome application in arithmetic statistics. Suppose that the Euler characteristic

chi(F) := 1/a + 1/b + 1/c - 1

is positive, and that there exists at least one primitive integral solution to Equation (1). Then, we prove that there is an explicitly computable constant kappa(F) > 0 such that the number of primitive integral solutions (x, y, z) to Equation (1) of height max{|A x^a|, |C z^c|} not exceeding h is asymptotic to

kappa(F) * h^chi.

Table of Contents

1 Introduction

1.1 Parametrizing Pythagorean triples via the method of Fermat descent — p. 3

1.2 Counting integral points on the projective line with three fractional points — p. 9

2 Background — p. 16

2.1 The groupoid of points on a stack — p. 16

2.2 H¹, torsors, and quotient stacks — p. 18

2.2.1 Nonabelian Čech cohomology — p. 18

2.2.2 Torsor sheaves — p. 19

2.2.3 Torsor schemes — p. 21

2.2.4 Quotient stacks — p. 23

2.2.5 The method of descent — p. 24

2.3 The root stack construction — p. 28

2.3.1 Generalized effective Cartier divisors — p. 28

2.3.2 Definition of a root stack — p. 29

2.3.3 The projective line rooted at a point — p. 31

3 Stacks associated to generalized Fermat equations — p. 35

3.1 The projective line with three fractional points — p. 35

3.1.1 A brief discussion of triangle groups — p. 35

3.1.2 Existence of Galois Belyi maps — p. 37

3.1.3 The Belyi stack — p. 39

3.2 The Fermat stack — p. 42

3.2.1 The graded ring — p. 43

3.2.2 The stacky proj — p. 45

3.3 The group scheme H — p. 47

3.4 The Belyi stack as a quotient — p. 50

3.5 The method of descent on the Belyi stack — p. 54

3.5.1 The theorem of Darmon–Granville — p. 55

3.5.2 The theorem of Beukers — p. 56

4 Counting primitive integral solutions — p. 58

4.1 Rational points of bounded height in the image of a rational function — p. 58

4.1.1 The primitivity defect set — p. 60

4.1.2 Proofs — p. 62

4.2 Counting integral points on the Belyi stack — p. 65

4.3 Counting primitive integral solutions to generalized Fermat equations — p. 66

Bibliography — p. 69

About this Dissertation

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School	Laney Graduate School
Department	Mathematics
Degree	Ph.D.
Submission	Dissertation
Language	English
Research Field	Mathematics
Keyword	Generalized Fermat equations
Committee Chair / Thesis Advisor	John Voight, University of Sydney David Zureick-Brown, Emory University
Committee Members	Bjorn Poonen, MIT David Borthwick, Emory University Suresh Venapally, Emory University

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