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This question is a follow-up on question 2, posed in: On the distribution of roots modulo primes of an integral polynomial

In appendix B of [1] by Pink, and in [2,3] by Serre, there are definitions of the density of a subset $S$ of the closed points of a scheme $X$ of finite type (and positive dimension) over $\mathrm{Spec}(\mathbb{Z})$.

Quick recap of the definition of density. One can define this density via the counting measure: $$ d(S) = \lim_{N \to \infty} \frac{\#\{x \in S | q_x \le N\}}{\#\{x \in X^{\text{cl}} | q_x \le N\}}, $$ where $q_x$ denotes the number of elements in the residue field of $x$. (The density is only defined for subsets $S$ for which the limit exists.)

This definition is not well-behaved if the image of $X \to \mathrm{Spec}(\mathbb{Z})$ is not dominant. Then one has to define it via $\zeta$-functions: $$ \zeta_{S}(s) = \sum_{x \in S} q_x^{-s}, $$ where $s$ is a complex parameter. Assume that $X$ is irreducible, of dimension $d$. The function $\zeta_{S}(s)$ is holomorphic on the region $\Re(s) > d$. Now we define the density as $$ d(S) = \lim_{s \searrow d}\frac{\zeta_{S}(s)}{\zeta_{X^{\text{cl}}}(s)} $$ (again, if the limit exists).

Question. Let $X \to Y$ be a morphism of irreduble schemes of finite type over $\mathrm{Spec}(\mathbb{Z})$. Assume that $\dim(Y) > 0$. Let $\eta$ be the generic point of $Y$. Let $S$ be a subset of $X^{\text{cl}}$. For every point $x \in X_{\eta}^{\text{cl}}$, let $S_{x}$ denote $S \cap \bar{x}$; where $\bar{x}$ is the closure of $x$ in $X$.
Assume that for all $x \in X_{\eta}^{\text{cl}}$ the set $S_{x}$ has density $0$ (or $1$) in $\bar{x}$. Does this imply that $S$ has density $0$ (or $1$) in $X$?

Motivation. There are several theorems in algebraic geometry that involve statements about densities of places of number fields. For example:
Let $K$ be a number field. Let $Z/K$ be an elliptic curve, or a K3 surface. Then there exists a finite extension $L/K$ such that $Z_{L}$ has good and ordinary reduction at a density $1$ set of places of $L$.
A positive answer to the question would imply that the statement is also true if one asks for $K$ to be a finitely generated field of characteristic $0$.


References

[1]: Richard Pink. “The Mumford–Tate conjecture for Drinfeld-modules”. In: Kyoto University. Research Institute for Mathematical Sciences. Publications 33.3 (1997), pp. 393–425.
[2]: Jean-Pierre Serre. “Zeta and L-functions”. In: Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963). Harp er & Row, New York, 1965, pp. 82–92.
[3]: Jean-Pierre Serre. Lectures on $N_X(p)$. Vol. 11. Chapman & Hall/CRC Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2012, pp. x+163.

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  • $\begingroup$ I found your question a bit confusing at first read. What you have called the "degree" $\mathrm{deg}(x)$ of a closed point is usually called the "norm" $\mathrm{N}(x)$ (Serre follows this convention in his books). The degree $\mathrm{deg}(x)$ is normally instead defined to be the degree of the residue field of $x$ over its prime subfield (again, this is the convention from Serre's books). If the prime subfield is the finite field $\mathbb{F}_p$, then we have the relation $\mathrm{N}(x) = p^{\mathrm{deg}(x)}$. $\endgroup$ – Daniel Loughran Sep 22 '16 at 9:45
  • $\begingroup$ @DanielLoughran — Sorry, of course. I was typing too fast. I'll fix it. $\endgroup$ – user98708 Sep 22 '16 at 9:47

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