Recently a colleague and I needed to use the fact that the natural map $SL\_2(\mathbb{Z}) \rightarrow SL\_2(\mathbb{Z}/N\mathbb{Z})$ is surjective for each $N$. I happily chugged my way through an elementary proof, but my colleague pointed out to me that this is a consequence of strong approximation.


After browsing through some references (including Ch. 7 of Platonov and Rapinchuk, and one of Kneser's papers) I now cheerfully agree with my colleague. Indeed, as P+R describe, strong approximation is "the algebro-geometric version of the Chinese Remainder Theorem".


But what about the proofs? To this particular novice in this area, the proofs seemed rather difficult -- involving a variety of cohomology calculations, nontrivial theorems about Lie groups, and much else besides. I confess I did not attempt to read them closely.


But perhaps this is because the theorems are proved in a great deal of generality, or perhaps such arguments are regarded as routine by experts. My question is this:



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For example, can a reasonably simple proof be given for $SL\_n$ which does not involve too much machinery, but which illustrates the concepts behind the general case?



Is strong approximation difficult?
There are two very-different questions here: the best arguments for surjectivity of the natural maps $SL(n,R)\rightarrow SL(n,R/I)$, and about Strong Approximation. While it is true that Strong Approximation more-than implies these surjectivities in situations that are not quite elementary, it is serious overkill, I think.


The most direct proofs of these surjectivities may very well strike one as needlessly and unilluminatingly messy, but I think this is a result of underestimating the issue. It's not as serious as (edit: "the strong version of") Strong Approximation, by far, but is not an easy exercise, either. The arguments in Rosenberg cannot be simplified much, apart from removing $K$-theory terminology, if one insists. A possibly irreducibly-minimal argument for $SL(2,\hbox{PID})$ is on-line at <http://www.math.umn.edu/~garrett/m/mfms/notes/07b_surjectivity.pdf> In any case, once one reconciles oneself to the not-quite-triviality of the surjectivity issue, the proofs one reasonably finds may seem less unreasonable.


The question(s) about proofs of (edit: "the strong forms of") Strong Approximation are in a different league. (Indeed, as in one comment or answer, the surjectivity issue by itself does not need adeles or cohomology.) In the late 1930s, Eichler had results amounting to Strong Approximation for simple algebras over number fields. M. Kneser's work starting about 1960 (and with two articles in the Boulder conference) mostly addressed orthogonal groups and classical groups *other* than SL(n) or unit groups of simple algebras. For the latter, including SL(n), the argument is essentially linear algebra and elementary analysis, with some clevernesses. 


(The argument in my old Hilbert Modular Forms book was (if I recall correctly) a distillation of remarks in Eichler's old papers, and remarks of Kneser. The SL(n) case was not the main focus of any of that, since, despite its non-triviality, Strong Approximation for SL(n), even over number fields, is much easier than the orthogonal group case. Perhaps I'll put a version of that argument on-line sometime soon, since the book is out of print, and there is no longer any electronic version...)


In brief, the reason the Platonov-Rapinchuk iconicized version uses cohomology is to address the orthogonal-group or unitary-group cases, and others with more genuine arithmetic content than has the SL(n) case. That is, again, the SL(n) case of Strong Approximation is "trivial" by comparison to the orthogonal-group case, although it itself is vastly more serious than the surjectivity question.


[EDIT 2:] Following up @LYX's note, indeed, for the usual invocations of Strong Approximation for SL(n), the Bourbaki Comm. Alg. citation suffices. Unsurprisingly, it does argue on elementary matrices (thus implicitly a bit of K-theory), and is purely algebraic, in the sense that finiteness of residue fields is irrelevant. One "usual" invocation is to know that $SL(n,F\_\infty)SL(n,F)$ is dense in $SL(n,\mathbb A\_F)$ for a number field $F$, where $F\_\infty$ is the product of archimedean completions ($F\_\infty=F \otimes\_{\mathbb Q} \mathbb R$). The application to automorphic forms is that quotients of this adele group are "merely" quotients of the Lie group $SL(n,F\_\infty)$.


The *stronger* version replaces $F\_\infty$ with $F\_v$ for any place $v$. That is, the general case does not specially suppress reference to archimedean places. (The argument I gave in my Hilbert modular forms book *did* suppress archimedean places, exactly because of the specific motivations. Nevertheless, "my" argument there did more resemble the viewpoint that would extend to the stronger assertion. The date on Bourbaki's Comm. Alg. is 1985, which might explain why, from late 1970s to mid-1980's, I saw no discussion of any form of "Approximation" other than Kneser and Eichler. Kneser's paper does also mention Rosenlicht and Steinberg for discussions of more-interesting cases than SL(n).)


The Kneser article "Strong Approximation" in the Boulder Conference (Proc Symp Pure Math AMS IX, 1966, pp 187-196 cites Eichler 1938 J. Reine und Angew. Math "Allgemeine..." as treating the simple algebra case, excepting the definite quaternion algebra case (in which SA does not hold). Thus, Eichler's case was already more complicated than SL(n).


In summary: any form of the SL(n) case is vastly simpler than orthogonal-group or other cases. There is a bifurcation already for SL(n), namely, whether or not archimedean places are suppressed (thus, giving a "purely algebraic" result).


The more delicate result can be relevant for "Shimura curves", made from quaternion division algebras over totally real fields, split at only a single real place, to know that the adelic construction is a single classical quotient. 


