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Suppose that we have a variety $X \subset \mathbb{P}^{n}$ defined over $\mathbb{Q}$. Suppose we have $S$ many rational points on $X$ inside the box defined by $|x_i| \leq B_i$ for $i = 0, \cdots, n$, and $B_i \in \mathbb{R}_{\geq 0}$ for $i = 0, \cdots, n$. We denote this set by $$\displaystyle X(\mathbb{Q}; B_0, \cdots, B_n).$$ For a prime $p$, let $X_p$ denote the reduction of $X$ modulo $p$.

Now suppose that we fix a point $P$ on $X_p$, and consider the pre-image of $P$ in $X(\mathbb{Q}; B_0, \cdots, B_n)$ which we denote by $U_P$. Now consider a prime $q < p$. If $q, p, S$ are all sufficiently large, so that we don't have problems with biases towards small primes, can we say that the points in $U_P$ are well-distributed among the congruence classes modulo $q$?

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There are examples where such results are true, established in the study of the conjecture of Batyrev-Manin. For example, if $X$ is a toric variety (whose underlying torus is split), or a flag variety, or a quadric hypersurface which has a $\mathbb Q$-point, or an equivariant compactification of a vector group,...

See a survey by Emmanuel Peyre published in Journal Th. Nb. Bordeaux.

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Without any restrictions on $X$, the answer is no. Consider the following setup. Suppose that $X=E$ is an elliptic curve over $\mathbb{Q}$ with $E(\mathbb{Q})\cong\mathbb{Z}$ generated by an element $P \in E(\mathbb{Q})$. Let $p,q$ be primes such that the orders of the reductions of $P$ in $ \widetilde{X}(\mathbb{F}_p)$ and $\# \widetilde{X}(\mathbb{F}_q)$ are $m$ and $n$ respectively, and assume that $m$ and $n$ have greatest common divisor $d>1$.

Then the cardinality of the image of $E(\mathbb{Q})$ in $ \widetilde{X}(\mathbb{F}_p)$ is $m$, and the cardinality of its image in $ \widetilde{X}(\mathbb{F}_q)$ is $n$. If we would have equidistribution in the sense of your question, the image of $E(\mathbb{Q})$ in $\widetilde{X}(\mathbb{F}_p) \times \widetilde{X}(\mathbb{F}_q)$ should have size $mn$, whereas in actuality it is $mn/d<mn$.

The conditions are satisfied e.g. for $$ E: y^2=x^3+2, $$ with $P=(-1,1)$, $p=19$, and $q=103$. With these values, we have $m=n=d=13$.

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  • $\begingroup$ Can the above happen for an infinite set $(p,q)$ of primes? $\endgroup$ Oct 17, 2014 at 18:05
  • $\begingroup$ Probably yes. Don't have time to think about a proof though. $\endgroup$
    – R.P.
    Oct 17, 2014 at 18:11
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    $\begingroup$ Never mind two primes, the reduction map is not surjective modulo $p$ for infinitely many $p$ for an elliptic curve of rank one. $\endgroup$ Oct 17, 2014 at 21:13
  • $\begingroup$ You're absolutely right. I just wanted to show the answer is "no" even if one restricts attention to those elements of $X(\mathbb{F}_q)$ that are in the image of the reduction map. $\endgroup$
    – R.P.
    Oct 17, 2014 at 22:21

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