## Is strong approximation difficult?

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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:

Is the proof of strong approximation genuinely difficult?

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?

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The proof in Section 5 of Yoshida's notes (dpmms.cam.ac.uk/~ty245/2008_AGR_Fall/…), boils down to the elementary fact in your first sentence! I wonder if there really is a proof which is logically independent of this kind of elementary lemma, since otherwise your colleague's assertion seems a bit circular. (I am very far from an expert here.) – David Hansen Nov 22 2011 at 17:39

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Paul Garrett - Hilbert modular form pg.279-282 for $SL(n)$. – Marc Palm Nov 22 2011 at 17:42
@pm: Care to say a few words about the ingredients in Garrett's proof? – David Hansen Nov 22 2011 at 17:55

## 4 Answers

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.

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Unfortunately, the link seems to do not work.  – Ralph Nov 23 2011 at 0:29
Oop, sorry about the bad link. Should be fixed now... – paul garrett Nov 23 2011 at 1:03
A very illuminating answer. Thank you! – Frank Thorne Nov 24 2011 at 19:35

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The $SL_n$ case can be treated rather general in an elementary way:

Let $R \to S$ be a surjective homomorphism of commutative rings. If $SL_n(S)$ is generated by elementary matrices then the induced map $SL_n(R) \to SL_n(S)$ is surjective.

[This is, since an elementary matrix over $S$ obviously lifts to an elementary matrix over $R$].

The cases when $SL_n(S)$ is generated by elementary matrices include:

1. $S$ is Euclidean
2. $S$ is local
3. $S$ is a finite product of rings $S_i$ such that the $SL_n(S_i)$ are generated by elementary matrices.
4. $S$ is finite

[The proof of 1. and 2. is elementary and can be found in 2.3.2 / 2.2.2 of "Rosenberg: Algebraic K-Theory and its Applications" and 3. follows from $SL_n(S_1 \times S_2) \cong SL_n(S_1) \times SL_n(S_2)$. 4. holds since each finite comm. ring is a direct product of local rings.]

Example: Let $S = \mathbb{Z}/N\mathbb{Z}$ or more generally, let $S = D/I$ where $D$ is a Dedekind domain and $0 \neq I \trianglelefteq D$. By prime ideal factorization $I=\prod_{i=1}^m P_i^{k_i}$ and by Chinese Remainder Theorem $D/I \cong \prod_{i=1}^m D/P_i^{k_i}$ where $D/P_i^{k_i}$ is local with maximal ideal $P_i/P_i^{k_i}$. Thus $SL_n(D) \to SL_n(D/I)$ is surjective. Alternatively use that $D/I$ is finite.

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What a delightful proof! Thank you. – Frank Thorne Nov 24 2011 at 19:32

I am quite surprised that it seems to be so poorly known, but an elementary proof of Strong Approximation for $SL_n$ over a Dedekind domain can already be found in Bourbaki (Algebre Commutative, VII, $\S$2, n.4). Essentially, the idea is to deduce it from the Chinese Remainder Theorem, by means of elementary matrices, as hinted in Ralph's answer. Actually, I would be curious to know a bit more about the history of this case of Strong Approximation: who and when did prove it first?

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@LYX, is it meant to talk about the surjectivity of $SL(n,R)$ to $SL(n,R/I)$, or about the "strong" (genuine) version of Strong Approximation? I don't have Bourbaki's Algebre Comm here... but I would be mildly surprised if (what I call) "Strong Approximation" appears there, since archimedean places of a number field are very relevant. E.g., there is no "sense" of the assertion for abstract Dedekind domains. The surjectivity assertion is certainly much more elementary, and (I'd wager) would have been clear to Kronecker, at least, for example. But the "strong" version? Thanks for clarification. – paul garrett Nov 23 2011 at 1:56

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Bourbaki's Algebre Commutative, VII, §2, n.4 has the title: "The Approximation Theorem for Dedekind domains". They take a Dedekind domain $A$ with fraction field $K$ and define the ring of restricted adeles $\bf A$ as the restricted product of completions of $K$ over all (non-zero) primes of $A$. Proposition 4 states: $SL(n,K)$ is dense in $SL(n,{\bf A})$. – LYX Nov 23 2011 at 2:42
@LYX: Ah, thanks for the clarification! So this notion of "adeles" does not include archimedean factors, in the number field case. But if there is only one archimedean place, as for $\mathbb Q$, perhaps this gives (what I think of as) Strong Approximation. Thanks again for the reference: I'd not noticed it, and had not heard a reference to it, even while looking for such things some years ago! :) – paul garrett Nov 23 2011 at 17:04

The proof for $SL(n, \mathbb{Z})$ is not very difficult, and takes a couple of pages in M. Newman's "Integral Matrices". Newman uses no fancy language at all (no adeles, no cohomology), so this is highly recommended. The book should be in your university library, if not, I am quite sure it is on gigapedia. The book is a classic, and is highly recommended for pretty much everything it covers.

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Another reference of the same kind is Shimura, Introduction to the Arithmetic Theory of Automorphic Forms, lemma 1.38 and its proof. – Matthieu Romagny Nov 22 2011 at 20:35
@Matthieu: I have been told that Shimura's books/papers are generally really difficult to read, and that, for example, we should be glad that Deligne and Milne wrote very good books on Shimura varieties. Would you say this book is an exception? – M Turgeon Nov 22 2011 at 22:07

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@M Turgeon: I have heard the same complaint about this book, but I read the proof of this lemma at Matthieu's suggestion and it is quite nice! (I have not yet looked up Newman's book.) – Frank Thorne Nov 23 2011 at 0:51
@M Turgeon : I think that that particular proof is neat. – Matthieu Romagny Nov 23 2011 at 21:52
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