# $X(\mathbb{Z}/p\mathbb{Z})$ versus $\{X(\mathbb{Z})\pmod{p}\}$

Let $$P_1$$,...,$$P_m$$ be polynomials in $$n$$ variables with coefficients in $$\mathbb{Z}$$ and consider the set $$X(\mathbb{Z})=\{(x_1,...,x_n)\in \mathbb{Z}^n \ |\ P_i(x_1,...,x_n)=0 ~ ,\ \forall i \in\{1,...,m\}\}.$$

In algebraic geometry, we are often interested in the number of elements in $$X(\mathbb{F}_{p})=\{(x_1,...,x_n)\in \mathbb{F}_p^n \ |\ P_i(x_1,...,x_n)=0 \pmod{p},\ \forall i \in\{1,...,m\}\}.$$

Are there known results about $$\{X(\mathbb{Z})\pmod{p}\}:=\{(x_1,...,x_n) \pmod{p}\ |\ (x_1,...,x_m)\in X(\mathbb{Z})\}~ ?$$

Certainly, $$|\{X(\mathbb{Z})\pmod{p}\}|\leq |X(\mathbb{F}_{p})|$$. Can we say more?

I'm interested in any notes/papers on the subject.

• I would recommend you to take a look at "A Course in Arithmetic" of Jean-Pierre Serre for the case of quadratic forms. – Wille Liou May 3 '19 at 14:44
• You can have everything in between. Namely there are many examples where $X(\mathbb{Z}) = \emptyset$. On the other hand, your inequality can also be an equality, and this property is closely related to strong approximation, – Daniel Loughran May 3 '19 at 19:35

I found the MSRI survey Strong approximation for algebraic groups by Andrei Rapinchuk very well-written and informative (and it has a lengthy bibliography for further exploration).

To answer your question: the cubic hypersurface $$X\subset \mathbb{A}^3$$ defined by $$3x^3 + 4y^3 + 5z^3 =0$$ has only one $$\mathbb{Z}$$-point $$(0,0,0)$$ but has more than one $$\mathbb{Z}/p\mathbb{Z}$$-points for prime $$p$$.

For example, when $$p=2$$ the locus is $$\{(0,0,0),(0,1,0),(1,0,1),(1,1,1)\}$$.

So your inequality is generally strict.

For a variety $$X$$ defined over $$\mathbb{Z}$$, say that $$X$$ has strong approximation if the reduction map $$X(\mathbb{Z})\to X(\mathbb{Z}/m\mathbb{Z})$$ is surjective for all $$m\geq 1$$.

Some necessary conditions for this property are that $$X$$ is absolutely irreducible, and also that the $$\mathbb{Z}$$-points are Zariski dense.

An example where strong approximation does hold is $$\mathrm{SL}_2$$; and more generally simply-connected algebraic groups and homogeneous spaces whose $$\mathbb{Z}$$-locus is Zariski dense (see the above reference for a precise formulation).

• Thanks a lot, this is very helping! I will look at your reference – Stabilo May 4 '19 at 19:02