Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Maximum number of rational points on hypersurfaces in weighted projective spaces over finite fields

Yves Aubry, Marc Perret

TL;DR

The paper extends Serre-type point-count bounds from projective spaces to weighted projective hypersurfaces over finite fields. It establishes a constructive lower bound via explicit factorized polynomials and an upper bound through a screwing/averaging technique that reduces to the classical Serre bound on P^n. The main result proves that, when the second weight a_1 equals 1, the maximum number of F_q-rational points on a weighted-hypersurface of weighted degree d equals min{p_n, d q^{n-1} + p_{n-2}} for all d, in any dimension. This confirms the UCLA conjecture in the a_1=1 case and broadens the understanding of rational-point distributions in weighted projective settings.

Abstract

An upper bound for the maximum number of rational points on an hypersurface in a projective space over a finite field has been conjectured by Tsfasman and proved by Serre in 1989. The analogue question for hypersurfaces on weighted projective spaces has been considered by Castryck, Ghorpade, Lachaud, O'Sullivan, Ram and the first author in 2017. A conjecture has been proposed there and proved in the particular case of the dimension 2. We prove here the conjecture in any dimension provided the second weight is also equal to one.

Maximum number of rational points on hypersurfaces in weighted projective spaces over finite fields

TL;DR

The paper extends Serre-type point-count bounds from projective spaces to weighted projective hypersurfaces over finite fields. It establishes a constructive lower bound via explicit factorized polynomials and an upper bound through a screwing/averaging technique that reduces to the classical Serre bound on P^n. The main result proves that, when the second weight a_1 equals 1, the maximum number of F_q-rational points on a weighted-hypersurface of weighted degree d equals min{p_n, d q^{n-1} + p_{n-2}} for all d, in any dimension. This confirms the UCLA conjecture in the a_1=1 case and broadens the understanding of rational-point distributions in weighted projective settings.

Abstract

An upper bound for the maximum number of rational points on an hypersurface in a projective space over a finite field has been conjectured by Tsfasman and proved by Serre in 1989. The analogue question for hypersurfaces on weighted projective spaces has been considered by Castryck, Ghorpade, Lachaud, O'Sullivan, Ram and the first author in 2017. A conjecture has been proposed there and proved in the particular case of the dimension 2. We prove here the conjecture in any dimension provided the second weight is also equal to one.
Paper Structure (9 sections, 9 theorems, 47 equations, 1 figure)

This paper contains 9 sections, 9 theorems, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A lower bound for the number of rational points
  3. Weighted projective spaces
  4. A lower bound
  5. Some morphisms between weighted projective spaces
  6. The morphisms $\pi_i$
  7. Number of zeros of the pullback
  8. An upper bound for the number of rational points
  9. The main result

Key Result

Proposition 2.1

Let $k$ be a field and $\underline{a}=(a_0, a_1, \cdots, a_n)$ be a sequence of $n+1$ nonzero integers. Then each $k$-rational point $x\in {\mathbb P}(a_0,\ldots,a_n)$ has a representative $x=[x_0:x_1: \cdots: x_n]$ with $x_i\in k$ for any $0\leq i\leq n$.

Figures (1)

  • Figure 1: Screwing of weighted projective spaces

Theorems & Definitions (20)

  • Conjecture 1.1
  • Proposition 2.1
  • proof : Communicated by Laurent Moret-Bailly
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 10 more