Reduction mod $n$ of symplectic group

According to Demazure, when he was a graduate student he asked Serre about the surjectivity of $G(\mathbf{Z}) \to G(\mathbf{Z}/(n))$ for all $n>0$ with $G = {\rm{Sp}}_{2g}$ or maybe ${\rm{SL}}_N$ (I can't remember which he said it was). Serre told him that he should ask Grothendieck for the "right" answer. Grothendieck told him that this was the "wrong question", and to approach such matters from the "right" point of view was the original goal of the SGA3 seminar on reductive group schemes over rings, etc. (killing a fly with a sledgehammer?).

That being said, a slick perspective which transforms the problem into something more robust over fields goes as follows. Let $\mathbf{A}_f$ be the ring of finite adeles for $\mathbf{Q}$ (this is a $\mathbf{Q}$-algebra, namely $\mathbf{Q} \otimes_{\mathbf{Z}} \widehat{\mathbf{Z}}$ in which $\widehat{\mathbf{Z}}$ is a compact open subring), so $K = G(\widehat{\mathbf{Z}})$ is a compact open subgroup of $G(\mathbf{A}_f)$ which meets $G(\mathbf{Q})$ in $G(\mathbf{Z})$. Since $G$ is $\mathbf{Z}$-smooth, the reduction map $K = G(\widehat{\mathbf{Z}}) \to G(\mathbf{Z}/(n))$ is surjective for Hensel's Lemma reasons and has open kernel. Thus, the original surjectivity question has an affirmative answer if $G(\mathbf{Q})$ is dense in $G(\mathbf{A}_f)$ (as this implies $G(\mathbf{Z})$ is dense in $G(\widehat{\mathbf{Z}})=K$, so $G(\mathbf{Z})$ maps onto $K/K'$ for any open normal subgroup $K'$ of $K$).

In other words, we can now forget about $\mathbf{Z}$-structures and instead attack a question entirely in terms of the $\mathbf{Q}$-group $H = G_{\mathbf{Q}}$: when is $H(\mathbf{Q})$ dense in $H(\mathbf{A}_f)$? We'd like an affirmative answer for at least $H$ equal to ${\rm{SL}}_N$ and ${\rm{Sp}}_{2g}$. In the case of $H=\mathbf{G}_{\rm{a}}$, such density holds and is just a special case of the classical "strong approximation" property for adele rings of global fields. This makes contact with Paul Broussous' comment about unipotent groups when $H$ is a split simply connected $\mathbf{Q}$-group due to the way in which such groups have an "open cell" expressed in terms of certain ${\rm{SL}}_2$-subgroups (especially the direct product structure for a split maximal torus in terms of a basis of simple coroots for the cocharacter lattice, one way of characterizing the simply connected case).

For an actual argument along these lines (i.e., bootstrapping from strong approximation for adele rings using the structure theory of split simply connected groups over fields), see http://math.stanford.edu/~conrad/248BPage/handouts/strongapprox.pdf (and note how it makes essential use of working over a field rather than over a ring of integers). This has a vast generalization to arbitrary simply connected groups over global fields relative to a suitable finite non-empty set of places (such as the place $\{\infty\}$ for $\mathbf{Q}$ when the group is split) via the "strong approximation theorem" for such groups, but that lies far deeper than the split case discussed above.

This was originally proved in

M. Newman, J. R. Smart, Symplectic modulary groups, Acta Arith 9 (1964), 83-89.

The canonical reference is Morris Newman's Integral Matrices, Theorem VII.21 (reference taken from Andy Putman's comment below), while gross overkill would be Platonov-Rapinchuk's book.