On the divisibility by $p$ of the number of $F_{p}$-points of a variety

Lucile Devin

Published 2016-01-11Version 1

Let $X$ be an affine scheme of finite type over $\mathbf{Z}$, we study the set $\lbrace p\in\mathcal{P}, p\nmid N_{X}(p)\rbrace$ where $N_{X}(p)$ is the number of $\mathbf{F}_{p}$-points of $X/\mathbf{F}_{p}$. We prove a simple condition for the set to have positive lower-density in case $\dim X\leq 3$. Then we study the size of the smallest element of the set. We use sieve methods to bound 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.