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

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 $X\mathrm{/}{\mathbf{F}}_p$. For fixed $n\ge 1$ and $a_{1}, \ldots, a_{n} \in $\mathbf{Z}$$,westudytheset$\bigcap_{i=1}^{n}\lbrace p\in$\mathcal{P}$-\Sigma_{X}, N_{X}(p)\neq a_{i}\ [\bmod\ p]\rbrace$where$\Sigma_{X}$isthefinitesetofprimesofbadreductionfor$X$\mathrm{.}Incase$\dim X\leq 3$, 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 $\{p\in \mathcal{P},p\nmid N_X(p)\}$ on average in particular families of hyperelliptic curves.