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Improved Lang--Weil bounds for a geometrically irreducible hypersurface over a finite field

Kaloyan Slavov

Abstract

We sharpen to nearly optimal the known asymptotic and explicit bounds for the number of $\mathbb{F}_q$-rational points on a geometrically irreducible hypersurface over a (large) finite field. The proof involves a Bertini-type probabilistic combinatorial technique. Namely, we study the number of $\mathbb{F}_q$-points on the intersection of the given hypersurface with a random plane.

Improved Lang--Weil bounds for a geometrically irreducible hypersurface over a finite field

Abstract

We sharpen to nearly optimal the known asymptotic and explicit bounds for the number of -rational points on a geometrically irreducible hypersurface over a (large) finite field. The proof involves a Bertini-type probabilistic combinatorial technique. Namely, we study the number of -points on the intersection of the given hypersurface with a random plane.

Paper Structure

This paper contains 4 sections, 10 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. A collection of small intervals
  3. Probability estimates
  4. Explicit versions

Key Result

Theorem 1

Let $X\subset\mathbb{A}^n_{\mathbb{F}_q}$ be a geometrically irreducible hypersurface of degree $d$. Then where the implied constant depends only on $d$ and can be computed effectively.

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Example 3: Cylinder over a maximal curve
  • Remark 4
  • Theorem 5
  • Remark 6
  • Theorem 7
  • Theorem 8
  • Example 9
  • Corollary 10
  • ...and 15 more