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The Circle Method for Quadrics over Function Fields

Johanna Mettasch

Abstract

We use the circle method to count $\mathbb{F}_q(t)$-rational points of bounded naive height on a quadric hypersurface $X\subseteq \mathbb{P}^{n-1}$ defined over $\mathbb{F}_q$, provided that $\mathrm{char}(\mathbb{F}_q)>2$ and $n\ge 3$. Viewing these points as morphisms $\mathbb{P}^1 \to X$ of fixed degree, we obtain exact formulas for their number depending on the parity of $n$ and on the determinant of the quadratic form defining $X$, including secondary terms in some cases.

The Circle Method for Quadrics over Function Fields

Abstract

We use the circle method to count -rational points of bounded naive height on a quadric hypersurface defined over , provided that and . Viewing these points as morphisms of fixed degree, we obtain exact formulas for their number depending on the parity of and on the determinant of the quadratic form defining , including secondary terms in some cases.

Paper Structure

This paper contains 8 sections, 19 theorems, 232 equations.

Table of Contents

  1. Introduction
  2. Background on Function Fields
  3. The Circle Method
  4. Quadratic Gauß Sums
  5. Exact Formulas for N(P)
  6. Computation of N(P) if 2|n and -1 to the n/2 a1...an is a square in Fq
  7. Computation of N(P) if 2|n and -1 to the n/2 a1...an is not a square in Fq
  8. Computation of N(P) if 2!|n

Key Result

Theorem 1.1

Let $n$ be even, and let $(-1)^{\frac{n}{2}}\det(f)$ be a square in $\mathbb{F}_q^\times$. For $n=4$ we have and for $n\ge 6$ we have $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • Lemma 4.2
  • ...and 24 more