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The geometry of secondary terms in arithmetic statistics

Michael Kural

TL;DR

This work proves the existence of a secondary term for counting cubic extensions of the function field \\mathbb{F}_q(t) by the discriminant norm, establishing the asymptotic \\mathrm{Cov}_3(2N) = c_1 q^{2N} - c_2^{i} q^{5N/3} + O(N^4 q^{3N/2}) with i \\equiv N \\pmod{3} and explicit constants. The authors implement a geometric parametrization of cubic covers via Casnati–Ekedahl and Miranda, embedding covers into Hirzebruch surfaces and counting sections of twisted line bundles, then sieve for smoothness. A key innovation is an elementary-transform-based bijection that converts singularities into vanishing conditions on a transformed surface, enabling a generating-function approach that yields both a main and a secondary term. The results extend secondary-term phenomena from cubic extensions over \\mathbb{Q} to function fields, using a robust geometric framework that handles inseparable and Galois-type cases, with potential for further applications to other degrees and global fields.

Abstract

In this thesis, we prove the existence of a secondary term for the count of cubic extensions of the function field $\mathbb{F}_q(t)$ of fixed absolute norm of discriminant. We show that the number of cubic extensions with absolute norm of discriminant equal to $q^{2N}$ is $c_1 q^{2N} - c_2^{i} q^{5N/3} + O_{\varepsilon}\left(q^{(3/2+\varepsilon)N}\right)$, where $c_1$ and $c_2^{i}$ are explicit constants and $c_2^{i}$ only depends on $N\pmod{3}$. This builds on the work of Bhargava-Shankar-Tsimerman and Taniguchi-Thorne, who proved the existence of a secondary term for the count of cubic extensions of $\mathbb{Q}$ with bounded discriminant. Our approach uses a parametrization of Miranda and Casnati-Ekedahl, which can be seen as a geometric version of the classical parametrization by binary cubic forms used by Davenport-Heilbronn. This allows us to count and sieve for smooth curves embedded in Hirzebruch surfaces, in the same spirit as Zhao and Gunther.

The geometry of secondary terms in arithmetic statistics

TL;DR

This work proves the existence of a secondary term for counting cubic extensions of the function field \\mathbb{F}_q(t) by the discriminant norm, establishing the asymptotic \\mathrm{Cov}_3(2N) = c_1 q^{2N} - c_2^{i} q^{5N/3} + O(N^4 q^{3N/2}) with i \\equiv N \\pmod{3} and explicit constants. The authors implement a geometric parametrization of cubic covers via Casnati–Ekedahl and Miranda, embedding covers into Hirzebruch surfaces and counting sections of twisted line bundles, then sieve for smoothness. A key innovation is an elementary-transform-based bijection that converts singularities into vanishing conditions on a transformed surface, enabling a generating-function approach that yields both a main and a secondary term. The results extend secondary-term phenomena from cubic extensions over \\mathbb{Q} to function fields, using a robust geometric framework that handles inseparable and Galois-type cases, with potential for further applications to other degrees and global fields.

Abstract

In this thesis, we prove the existence of a secondary term for the count of cubic extensions of the function field of fixed absolute norm of discriminant. We show that the number of cubic extensions with absolute norm of discriminant equal to is , where and are explicit constants and only depends on . This builds on the work of Bhargava-Shankar-Tsimerman and Taniguchi-Thorne, who proved the existence of a secondary term for the count of cubic extensions of with bounded discriminant. Our approach uses a parametrization of Miranda and Casnati-Ekedahl, which can be seen as a geometric version of the classical parametrization by binary cubic forms used by Davenport-Heilbronn. This allows us to count and sieve for smooth curves embedded in Hirzebruch surfaces, in the same spirit as Zhao and Gunther.

Paper Structure

This paper contains 47 sections, 59 theorems, 375 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main Result
  4. Outline of Approach
  5. Relationship to Zhao's Thesis
  6. Sections of the Paper
  7. Notation and Conventions
  8. Acknowledgments
  9. Parametrization of Cubic Covers
  10. Triple Covers
  11. Degree $3$ Sections and Binary Cubic Forms
  12. The correspondence
  13. Curves and Smooth Irreducible Covers
  14. Sieving for Smoothness
  15. Sections Corresponding to Covers
  16. ...and 32 more sections

Key Result

Theorem 1.1

Let $N$ be a nonnegative integer, and let $i \in \{0,1,2\}$ such that $N\equiv i\pmod{3}$. The number of degree $3$ extensions $K/\mathbb{F}_q(t)$ with $\mathop{\mathrm{Nm}}\nolimits(\mathop{\mathrm{Disc}}\nolimits_{K/\mathbb{F}_q(t)}) =q^{2N}$ is where $c_1, c_{2}^{0}, c_2^{1},$ and $c_2^{2}$ are explicit constants not depending on $N$.

Theorems & Definitions (167)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.1: Har77
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 157 more