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Explicit formulas and exact values for the number of rational points on singular curves over finite fields

Lorenzo Beninati

Abstract

We provide new explicit formulas for bounding the number of rational points on singular curves over finite fields. This enables us to obtain exact values of N q (g, $π$) which is defined as the maximum number of rational points over F q on a curve of geometric genus g and arithmetic genus $π$. We also give special attention to the case g = 2 in order to extend the work of Aubry and Iezzi on N q (0, $π$) and N q (1, $π$).

Explicit formulas and exact values for the number of rational points on singular curves over finite fields

Abstract

We provide new explicit formulas for bounding the number of rational points on singular curves over finite fields. This enables us to obtain exact values of N q (g, ) which is defined as the maximum number of rational points over F q on a curve of geometric genus g and arithmetic genus . We also give special attention to the case g = 2 in order to extend the work of Aubry and Iezzi on N q (0, ) and N q (1, ).
Paper Structure (9 sections, 16 theorems, 47 equations)

This paper contains 9 sections, 16 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Explicit formulas for singular curves
  3. Preliminaries
  4. Explicit formulas
  5. Doubly positive trigonometric polynomials
  6. Some exact values of $N_q(g,\pi)$
  7. Exacts values deduced from explicit formulas
  8. The case $g=2$
  9. Results on smooth curves deduced from singular curves

Key Result

Theorem 1.1

Let $X$ be a smooth curve defined over $\mathbb{F}_q$ of genus $g$ and let $\omega_j$ be the reciprocal roots of $L_X(T)$ expressed in the form $\omega_j=\sqrt{q}e^{i\theta_j}$, with $0\leq \theta_j \leq \pi$. Then we have where $B_d(X)$ denotes the number of closed points of degree $d$ of $X$.

Theorems & Definitions (27)

  • Theorem 1.1: Serre
  • Corollary 1.2: Corollary 5.3.4 in serre
  • Proposition : Proposition \ref{['prop_fe_LT']}
  • Proposition : Proposition \ref{['N_q(g,pi)']}
  • Theorem : Theorem \ref{['g=2']}
  • Proposition 2.1
  • Remark 2.2
  • Example 2.3
  • Proposition 2.4
  • Lemma 2.5: Lemma 3.8 in lachaud
  • ...and 17 more