On the congruence class modulo prime numbers of the number of rational points of a variety

TL;DR

This paper investigates the distribution of the number of rational points on varieties modulo primes, showing that certain sets of primes are either empty or have positive density, and provides bounds on the smallest such primes.

Contribution

It establishes conditions under which the set of primes with specific point-count congruences is either empty or has positive density, and applies sieve methods to bound the smallest primes in related sets.

Findings

The set of primes with certain point-count congruences is either empty or has positive lower density for varieties of dimension ≤ 3.

An upper bound on the smallest prime in the set where the point count is not divisible by the prime is obtained.

Sieve methods are used to estimate the size of the least prime in specific families of hyperelliptic curves.

Abstract

Let $X$ be a scheme of finite type over $\mathbf{Z}$. For $p \in \mathcal{P}$ the set of prime numbers, let $N_{X}(p)$ be the number of $\mathbf{F}_{p}$-points of $X/\mathbf{F}_{p}$. For fixed $n\geq 1$ and $a_{1}, \ldots, a_{n} \in \mathbf{Z}$, we study the set $\bigcap_{i=1}^{n}\lbrace p\in\mathcal{P}-\Sigma_{X}, N_{X}(p)\neq a_{i}\ [\bmod\ p]\rbrace$ where $\Sigma_{X}$ is the finite set of primes of bad reduction for $X$. In case $\dim X\leq 3$, we show the set is either empty or has positive lower-density. We also address the question of the size of the smallest prime in that set. Using sieve methods, we obtain for example an upper bound for the size of the least prime of $\lbrace p\in\mathcal{P}, p\nmid N_{X}(p)\rbrace$ on average in particular families of hyperelliptic curves.