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Strong Approximation for Algebraic Groups

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Strong Approximation for Algebraic Groups Thin groups and superstrong approximation MSRI Publications Volume 61, 2013 Strong approximation for algebraic groups ANDREI S. RAPINCHUK We survey results on strong approximation in algebraic groups, considering in detail the classical form of strong approximation as well as more recent results on strong approximation for arbitrary Zariski-dense subgroups. Some other topics, ranging from strong approximation in homogeneous spaces of algebraic groups to various applications of strong approximation, are also discussed. 1. Introduction This article is a survey of known results related to strong approximation in algebraic groups. We focus primarily on two aspects: the classical form of strong approximation, which is really strong approximation for S-arithmetic groups (Section 2), and its more modern version for arbitrary Zariski-dense subgroups (Section 3). Along the way we will also mention results dealing with strong approximation in arbitrary varieties and particularly homogeneous spaces (which are probably not so well known to the general audience as some other results in the article) and some applications. The reader will find more applications of strong approximation for Zariski-dense subgroups in other articles in this volume. 1A. Strong approximation and congruences. The most elementary way to start thinking about strong approximation is in terms of lifting solutions of integer polynomial equations mod m for all m > 1, to integer solutions. So, suppose we have a family of polynomials f˛.x1;:::; x / ZŒx1;:::; x ; ˛ I; d 2 d 2 and we let X Ad denote the closed affine subscheme defined by these poly- Z nomials. Thus, for any Z-algebra R, the scheme X has the following set of R-points: ˚ d « X.R/ .a1;:::; a / R f˛.a1;:::; a / 0 for all ˛ I : D d 2 j d D 2 Then for any integer m > 1, we have a natural reduction modulo m map m X.Z/ X.Z=mZ/; W ! 269 270 ANDREI S. RAPINCHUK and the question is whether these maps are surjective for all m. (Of course, this question is meaningful only if we assume that X.Z=mZ/ ? for all m.) ¤ Observe that for m n, there is a canonical homomorphism Z=nZ Z=mZ, j ! hence a natural map n X.Z=nZ/ X.Z=mZ/. Clearly, X.Z=mZ/; n is m W ! f mg an inverse system, so we can assemble all the X.Z=mZ/’s together by taking the inverse limit: lim X.Z=mZ/ X.Z/; where Z lim Z=mZ: D y y D Q Recall that the Chinese remainder theorem furnishes an isomorphism Z Zp, y ' p where Zp is the ring of p-adic integers and the product is taken over all primes, which allows us to identify X.Z/ with Q X.Z /. y p p Just as above, for any integer m > 1, there is a natural map X.Z/ X.Z=mZ/ X.Z=mZ/: O W y ! y y D The preimages of points under the m form a basis of a natural topology on O X.Z/, which coincides with either of the following topologies: y the topology of the inverse limit on limX.Z=mZ/ — cf.[Klopsch et al. 2011, Chapter I, 5.3]; the topology induced by the embedding X.Z/, Zd , where Z is endowed y ! y y with the inverse limit topology on lim Z=mZ; Q the direct product topology on X.Zp/, where X.Zp/ gets its topology from p d the embedding X.Zp/, Z , and Zp is endowed with the natural p-adic ! p topology. As an immediate consequence, we have: Lemma 1.1. The following conditions are equivalent: (1) m X.Z/ X.Z=mZ/ is surjective for all integers m > 1; W ! (2) the natural embedding X.Z/, X.Z/ has a dense image in the above W ! y topology. In this situation, we say X has strong approximation if it satisfies the equiv- alent conditions of Lemma 1.1 (of course, this is only a first approximation to the precise definition(s) of strong approximation that will be given later; see Section 2A). Intuitively, strong approximation should not be very common as there are plentiful examples where X.Z/ ? but X.Z/ ? (i.e., the Hasse y ¤ D principle fails — note that here we actually omit the archimedean place of Q), and also examples where X.Z/ is nonempty but so “small” that it cannot possibly be STRONGAPPROXIMATIONFORALGEBRAICGROUPS 271 dense in X.Z/. A classical example of the second situation is a cubic hypersurface y X A3 given by the equation 3x3 4y3 5z3 0 C C D I it is known that X.Z/ .0; 0; 0/ but X.Zp/ .0; 0; 0/ (hence infinite as any D f g ¤ f g point on X other than the origin is smooth) for all prime p. In fact, very little appears to be known about strong approximation for schemes (varieties) lying outside some special classes such as homogeneous spaces — one can only give some necessary conditions (see Proposition 2.2 and subsequent remarks). So, in this article we will deal almost exclusively with algebraic groups. 1B. SL versus GL . Let us start with two elementary examples: G1 SL2 2 2 D and G2 GL2. One doesn’t see much of a difference between these examples D just by looking at the defining equations, which can be written as follows, with the obvious labeling of coordinates: G can be realized as a hypersurface in A4, given by 1 x11x22 x12x21 1: D G can be realized as a hypersurface in A5, given by 2 y.x11x22 x12x21/ 1: D However, G1 has strong approximation, and G2 does not. Lemma 1.2. For any m > 1, the reduction modulo m map m SL2.Z/ SL2.Z=mZ/ W ! is surjective. Proof. The argument does not use equations (in fact, it is not a completely trivial task to prove strong approximation using equations in this case — see the discussion after Proposition 2.4). The crucial observation is that any g N 2 SL2.Z=mZ/ can be written as a product of elementary matrices: Y g ei j .a / with .i ; j / .1; 2/; .2; 1/ and a Z=mZ: (1) N D k k Nk k k 2 f g Nk 2 k (As usual, for i j , we let eij .a/ denote the elementary matrix having a as its ¤ ˛1 ˛r ij -entry.) For this, one needs to observe that if m p pr then it follows D 1 from the Chinese remainder theorem that ˛1 ˛r SL2.Z=mZ/ SL2.Z=p Z/ SL2.Z=p Z/; D 1 r 272 ANDREI S. RAPINCHUK which reduces the problem to the case where m p˛. Now, given D Â Ã x11 x12 ˛ g SL2.Z=p Z/; N D x21 x22 2 ˛ we see that either x11 or x12 is a unit mod p , so using Gaussian elimination one can easily write g as a product of elementaries over Z=p˛Z. N Next, given an arbitrary g SL2.Z=mZ/, pick a factorization (1), and further- N 2 more, for each k pick an integer a in the class a modulo m. Set k Nk Y g ei j .a / SL2.Z/: D k k k 2 k Then m.g/ g, proving the surjectivity of m. DN Note that the proof of Lemma 1.2 relies on the consideration of unipotent elements, so it is worth pointing out that, as we will see in the course of this article, unipotent elements are involved in one way or another in most known results on strong approximation (even when the group at hand does not contain any nontrivial rational unipotent elements, i.e., is anisotropic). The fact that G2 GL2 does not have strong approximation is much easier: D in fact, already the map 5 GL2.Z/ GL2.Z=5Z/ W ! fails to be surjective. (Indeed, since all matrices in GL2.Z/ have determinant 1, ˙ the matrices in 5.GL2.Z// have determinant 1 .mod 5/, and therefore, for 1 0 ˙ N N example, 0 2 does not lie in the image of 5.) One can conceptually articulate N N the obstruction that prevents GL2 from having strong approximation in this case by saying that in order for an affine Q-variety X to have strong approximation, X.Z/ must be Zariski-dense in X: Indeed, let Y X.Z/ be the Zariski closure of X.Z/ in X , and assume that D Y X . Pick a point a X.Q/ Y .Q/, where Q is an algebraic closure of Q. ¤ 2 N n N N Then one can find a polynomial f ZŒx1;:::; x that vanishes on Y and such 2 d that f .a/ 0. It follows from Chebotarev’s density theorem that for infinitely ¤ many primes p, we have a X.Zp/ and f .a/ 0 .mod p/. Let a X.Fp/ be 2 6Á N 2 the reduction of a modulo p, where Fp Z=pZ Zp=pZp. (Note that it would .p/ D D .p/ be more appropriate to write X .Fp/ instead of X.Fp/, where X denotes the reduction of X modulo p, but we will slightly abuse the notations in this introductory section in order to keep them simple.) Then clearly a X.Fp/ Y .Fp/; N 2 n STRONGAPPROXIMATIONFORALGEBRAICGROUPS 273 and in particular, X.Fp/ Y .Fp/. On the other hand, the image of the reduction ¤ map p X.Z/ X.Fp/ is obviously contained in Y .Fp/. Thus, if X.Z/ is W ! not Zariski-dense in X then p fails to be surjective for infinitely many p, which certainly prevents X from having strong approximation.
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