Topics in arithmetic statistics Open Access
Keyes, Christopher (Summer 2023)
Published
Abstract
Arithmetic statistics encompasses a broad class of questions in number theory and arithmetic geometry of a distinctly quantitative flavor. In this thesis the author addresses three such questions, the first two of which are related to superelliptic curves, which are given by an equation of the form Cf : y^m = f(x,z). For a fixed such curve defined over the rational numbers Q and an appropriately chosen degree n, we give an asymptotic lower bound on the number of finite extensions K/Q of degree n arising as the minimal field of definition for an algebraic point on Cf , counted by absolute discriminant. Rather than fixing the curve, we could instead ask how often a family of superelliptic curves has certain arithmetic properties. In particular, we study how often such curves are everywhere locally soluble, computing exactly the density of f such that Cf has points everywhere locally. Finally, we interpret the Mertens’ classical product theorem as a statement about the density of integers lacking small prime factors. We then prove a generalization to Chebotarev sets of prime ideals in Galois extensions of number fields.
Table of Contents
1 Introduction 1
1.1 Counting number fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Solubility for families of curves . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Mertens’ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Preliminaries 9
2.1 Local fields and Newton polygons . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Generating symmetric groups . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Chebotarev’s density theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Fields generated by points on curves 38
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 The parametrization strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Polynomial families from hyperelliptic curves . . . . . . . . . . . . . . . . . 47
3.4 Polynomial families from superelliptic curves . . . . . . . . . . . . . . . . . 58
3.5 Accounting for multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.6 Geometric sources of higher degree points . . . . . . . . . . . . . . . . . . . 74
4 Solubility densities in families of superelliptic curves 84
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.2 The proportion is positive . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.3 Lower bounds for the proportion . . . . . . . . . . . . . . . . . . . . . . . . 98
4.4 Upper bounds for the proportion . . . . . . . . . . . . . . . . . . . . . . . . 119
4.5 An exact formula for the m = 3 and d = 6 case . . . . . . . . . . . . . . . . 121
4.6 Bounds for ρ m,d (p) via computer search . . . . . . . . . . . . . . . . . . . . 176
4.7 Counting binary forms by factorization type . . . . . . . . . . . . . . . . . . 179
4.8 Explicit formulas for rational functions . . . . . . . . . . . . . . . . . . . . . 183
5 Mertens’ theorem for Chebotarev sets 190
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
5.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
5.3 Proof of Theorem 5.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
5.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Bibliography 213
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