Counting primes, groups, and manifolds
Dorian Goldfeld*, Alexander Lubotzky†‡, Nikolay Nikolov§, and La´ szlo´ Pyber¶
*Department of Mathematics, Columbia University, New York, NY 10027; †Einstein Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel; §Tata Institute for Fundamental Research, Colaba, Mumbai 400005, India; and ¶A. Re´nyi Institute of Mathematics, Rea´ltanoda u. 13-15, H-1053, Budapest, Hungary
Communicated by Hyman Bass, University of Michigan, Ann Arbor, July 9, 2004 (received for review April 1, 2004) ⌫ ⌳ ޚ ؍⌳ Let SL2( ) be the modular group and let cn( ) be the number and, moreover, the sequence sn( ) has much faster growth (at of congruence subgroups of ⌳ of index at most n. We prove that least nlog n) if the congruence subgroup property fails for G. ⌫ ͞ ͌ ؊ ؍ 2͞ ͞ ⌳ limn3ؕ (log cn( ) ((log n) log log n)) (3 2 2) 4. The proof is Below we determine the precise rate of growth of cn( ). (All based on the Bombieri–Vinogradov ‘‘Riemann hypothesis on the logarithms are in base e.) average’’ and on the solution of a new type of extremal problem Let X be the Dynkin diagram of the split form of G (e.g., X ϭ ϭ in combinatorial number theory. Similar surprisingly sharp esti- An-1 if G SUn). Let h be the Coxeter number of the root system mates are obtained for the subgroup growth of lattices in higher ⌽ corresponding to X (it is the order of the Coxeter element of ͉⌽͉ rank semisimple Lie groups. If G is such a Lie group and ⌫ is an the Weyl group of X). Then h ϭ ⁄l, where l ϭ rankރ(G) ϭ irreducible lattice of G it turns out that the subgroup growth of ⌫ rank(X), and for later use define R :ϭ h͞2. Let is independent of the lattice and depends only on the Lie type of ͑ͱ ͑ ϩ ͒ Ϫ ͒2 the direct factors of G. It can be calculated easily from the root h h 2 h ␥͑G͒ ϭ . system. The most general case of this result relies on the Gener- 4h2 alized Riemann Hypothesis, but many special cases are uncondi- tional. The proofs use techniques from number theory, algebraic Let GRH denote the Generalized Riemann Hypothesis groups, finite group theory, and combinatorics. (GRH) for Artin–Hecke L functions of number fields as stated in ref. 4. The GRH implies, in particular: Let k be a Galois number field of degree d over the rationals and let q be a prime n this announcement we present several results about counting such that the cyclotomic field of q-th roots of unity is disjoint primes, counting groups, and counting Riemmanian mani- I from k. Denote by (x, q) the number of primes p with p Յ x, folds, and we explore the connections between them. k p ϵ 1(mod q) and p splits completely at k. Then Statement of Results x 1 ͯ ͑ ͒ Ϫ ͯ Ͻ 2 Arithmetic Groups. Let n be a large integer, ⌫ a finitely generated k x, q Cx log x log q d͑q͒ log x group, and M a Riemannian manifold. Denote by (n) the Յ ⌫ ⌫ number of primes n, sn( ) is the number of subgroups of of for some constant C ϭ C(k) Ͼ 0 depending only on k (a more index at most n and bn(M) is the number of covers of M of degree precise bound is given in ref. 5). at most n. The aim of this article is to announce results that show The lower bound for the limit in the Theorem 1 below was that, in some circumstances, these three seemingly unrelated proved in ref. 3 and the upper bound in ref. 6. functions are very much connected. This connection emerges, for example, when ⌫ is an arithmetic group, in which case it is also Theorem 1. Let G, ⌫, and ␥(G) be as defined above. Assuming GRH the fundamental group of a suitable locally symmetric finite we have ⌫ volume manifold M. The studies of sn( ) and bn(M) are then ⌫ log c ͑⌫͒ almost the same. Moreover, if has the congruence subgroup n ϭ ␥͑ ͒ ⌫ lim ͑ ͒2͞ G , property, then estimating sn( ) boils down to counting congru- n3ϱ log n log log n ence subgroups of ⌫. The latter is intimately related to the classical problem of counting primes. To present our results we and, moreover, this result is unconditional if G is of inner type (e.g., need more notation. G splits) and k is either an abelian extension of ޑ or a Galois Let G be an absolutely simple, connected, simply connected extension of degree Ͻ42. algebraic group defined over a number field k. For a finite subset An interesting aspect of Theorem 1 is not only that the limit S of valuations of k, including all the archimedean ones, let OS exists but that it is completely independent of k and S and ⌫ϭ denote the ring of S-integers of k and set G(OS). A subgroup depends only on G. Although the independence on S is a minor H Յ ⌫ is called a congruence subgroup if there is some ideal point and can be proved directly, the only way we know to prove k I “ OS such that H contains the kernel of the homomorphism the independence on is by applying the whole machinery of the ⌫ 3 G(O ͞I). proof. S ⌫ϭ ⌫ In ref. 3 the crucial special case of SL2(OS) is proved in Let cn( ) denote the number of congruence subgroups of full. There, we have ␥(SL ) ϭ 1 (3 Ϫ 2͌2). The lower bound index at most n in ⌫. The counting of congruence subgroups 2 4 in arithmetic groups has already played a role in the proof of follows using the Bombieri–Vinogradov Theorem (7) and the one of the main results of the theory of subgroup growth: A upper bound by a massive new combinatorial analysis. finitely generated, residually finite group ⌫ has polynomial ⌫ ϭ O(1) ⌫ Lattices. Let H be a connected characteristic 0 semisimple group. subgroup growth [i.e., sn( ) n ] if and only if is virtually r By this we mean that H ϭ⌸ ϭ G (K ), where for each i, K is solvable of finite rank (ref. 1 and references therein). That i 1 i i i a local field of characteristic 0, and G is a connected simple theorem required only a weak lower bound on the number i congruence subgroups. In ref. 2, Lubotzky proved a more precise result: there exist numbers a, b depending on G, k, and Abbreviation: GRH, Generalized Riemann Hypothesis. S, such that ‡To whom correspondence should be addressed. E-mail: [email protected]. a log n b log n The lower bound depended on GRH at the time but was made unconditional in ref. 3. log log n Յ ͑⌫͒ Յ log log n n cn n , © 2004 by The National Academy of Sciences of the USA
13428–13430 ͉ PNAS ͉ September 14, 2004 ͉ vol. 101 ͉ no. 37 www.pnas.org͞cgi͞doi͞10.1073͞pnas.0404571101 Downloaded by guest on September 28, 2021 ޒ algebraic group over Ki. We assume throughout that none of the Problem. Estimate mn(X) for the case of H having -rank equal to Ն ⌫ ϭ factors Gi(Ki) is compact (so that rankKi(Gi) 1). Let be an one. For H SO(n, 1) the results of ref. 12 suggest that irreducible lattice of H; i.e., for every infinite normal subgroup ͑ ͒ N of H the image of ⌫ in H͞N is dense there. log mn H lim Assume now that n3ϱ log n!
r may exist, but we do not have any clue what it could be. ͑ ͒ ϭ ͑ ͒ Ն rank H : rankKi Gi 2. iϭ1 Proofs The Lower Bound. We shall illustrate the main idea of the proof By Margulis’ Arithmeticity Theorem (8) every irreducible lattice with ⌫ϭSL (ޚ) and refer to ref. 3 for the full details. ⌫ d in H is arithmetic. Also the split forms of the factors Gi of H Choose any ⑀(0, 1). For x ϾϾ 0 and a prime q Ͻ x, let ␥ ϭ ␥ 2 are necessarily of the same type and we set (H): (Gi). P(x, q) be the set of primes p Յ x, such that p ϵ 1 mod q. Let Moreover, a famous conjecture of Serre (9) asserts that such a ϭ ͉ ͉ ϭ ¥ ⌫ L(x, q) P(x, q) and M(x, q) pʦP(x, q) log p. Then the group has the (modified) congruence subgroup property. It has Bombieri–Vinogradov Theorem (7) ensures the existence of a been proved in many cases. This enables us to prove Theorem 2. prime q ⑀ (xp͞log x, x) such that
Theorem 2. Assuming GRH and Serre’s conjecture, then for every x x noncompact higher rank characteristic 0 semisimple group H and L͑x, q͒ ϭ ϩ Oͩ ͪ ; ͑q͒ log x ͑q͒͑log x͒2 every irreducible lattice ⌫ in H the limit ͑⌫͒ x x log sn ͑ ͒ ϭ ϩ ͩ ͪ M x, q ͑ ͒ O ͑ ͒͑ ͒2 . lim ͑ ͒2͞ q q log x n3ϱ log n log log n Put L :ϭ L(x, q) and M :ϭ M(x, q). exists and equals ␥(H); i.e., it is independent of the lattice ⌫. By strong approximation (compare ref. 1, window 9), ⌫ maps Moreover the above result holds unconditionally if H is a simple ϭ⌸ ކ ރ ⌫ onto GP : pP(x,q) SLd( p). Let B(p) be the subgroup of upper connected Lie group not locally isomorphic to D4( ) and is a ކ nonuniform lattice in H (i.e., H͞⌫ is noncompact). triangular matrices of SLd( p) and set Theorem 2 shows, in particular, some algebraic similarity B :ϭ B͑p͒. between different lattices ⌫ in the same Lie group G. This P ͑ ͒ outcome is an addition to other results in the theory, e.g., p P x,q Furstenberg’s theorem showing that the boundaries of all such ⌸ ކ* dϪ1 The group BP maps onto the diagonal p( p) , which in turn lattices ⌫ are the same or Margulis superrigidity showing that the ͑ Ϫ ͒ maps onto ކ d 1 L. For fixed ʦ ͑0,1͒ പ 1 ގ, the latter finite dimensional representation theory of the different lattices q L͑d Ϫ 1͒ Ϫ Ϫ 2 2 ⌫ in the same G is similar (see ref. 8 and references therein). vector space has approximately q (1 )(d 1) L subgroups of Ϫ We point out the following geometric reformulation of the index q (d 1)L (see proposition 1.5.2 in ref. 1), each giving rise ϭ (dϪ1)L ⌫ special case. to a subgroup of index n [GP : BP]q in . Now, log[GP ϳ Ϫ ͞ 3 ϱ : BP] d(d 1)M 2asx and after some algebraic Theorem 3. Let H be a simple connected Lie group of ޒ-rank Ն2 manipulations, we obtain that for this chosen value of n, ރ ϭ ͞ that is not locally isomorphic to D4( ). Put X H K, where K is log c ͑⌫͒ ͑1 Ϫ ͒͑1 Ϫ ͒ a maximal compact subgroup of H. Let M be a finite volume n Ն Ϫ ͑ ͒ ͑ 3 ϱ͒ 2 2 o 1 , x , noncompact manifold covered by X and let bn(M) be the number ͑log n͒ ͞log log n ͑ϩ R͒ of covers of M of degree at most n. Then where in our case R ϭ d͞2. As shown in ref. 3, section 3, the ͑ ͒ log bn M maximum value of the expression above for , ʦ (0, 1) is lim ͑ ͒2͞ n3ϱ log n log log n precisely ␥ 2 exists, equals (H), and is independent of M. ͩͱ ͑ ϩ ͒ Ϫ ͪ It is interesting to compare Theorems 2 and 3 with the results R R 1 R ␥͑G͒ ϭ of Liebeck and Shalev (10) and T. W. Mu¨ller and J.-C. Puchta 4 R2 ϭ ޒ ⌫ MATHEMATICS (unpublished data): If H SL2( ) and is a lattice in H, then ϭ ϭ ͌ ϩ Ϫ and is achieved for 0 0 R(R 1) R. By taking x log s ͑⌫͒ n ϭ Ϫ͑⌫͒ sufficiently large we can choose ʦ (0, 1) പ 1͞[L(d Ϫ 1)] ގ lim , ϭ n3ϱ log n! to be arbitrarily close to 0, and take 0. This proves the lower bound. where is the Euler characteristic. The reason for invoking the GRH in Theorem 1 is that, in the We finally mention a conjecture and a question: Let X be the general case, we need an equivalent of the Bombieri–Vinogradov symmetric space associated with a simple Lie group H as in Theorem for k in place of ޑ. The work of Murty and Murty (13) Theorem 3. Denote by m (X) the number of manifolds covered n gives an analogue of it for number fields, but their result is by X of volume at most n. By a well known result of Wang (11), ޒ weaker in general. It suffices for our needs when, for example, this number is finite unless H is locally isomorphic to SL2( )or ͞ޑ ރ k is an abelian extension. SL2( ). The Upper Bound. The proof of the upper bound in ref. 6 is inspired Conjecture. If ޒ-rank(H) Ն 2 then by the special case solved in ref. 3 and has two parts: log m ͑X͒ n ϭ ␥͑ ͒ I. A reduction to an extremal problem for abelian groups, and lim ͑ ͒2͞ H . n3ϱ log n log log n II. Solving this extremal problem (see Theorem 6 below).
Goldfeld et al. PNAS ͉ September 14, 2004 ͉ vol. 101 ͉ no. 37 ͉ 13429 Downloaded by guest on September 28, 2021 ކ Part I. The subgroup structure of the groups SL2( p) is completely Theorem 5. Given the Lie type X then known. By using this, it is shown in ref. 3 that Theorem 1 for ޚ ͕ ͑ ͉͒ Յ ͑ކ ͖͒ Ն SL2( ) is equivalent to the following extremal result on counting lim inf min t H H X q R. subgroups of abelian groups. q3ϱ Let C denote the cyclic group of order m. For all pairs PϪ m The proof of this theorem does not depend on the classifica- and Pϩ of disjoint sets of primes, let tion of the finite simple groups; we use instead the work of Larsen and Pink (15) [which is a classification-free version of a ͑ ͒ ϭ ͑ ͉͒ ϭ ϫ f n : maxͭ sr X X CpϪ1 Cpϩ1ͮ , result of Weisfeiler (16)] and Liebeck et al. (17) (the latter for pʦPϪ pʦPϩ groups of exceptional type). Part II. Once Part I is proved, the argument reduces to an extremal where the maximum is taken over all sets PϪ, Pϩ, and r ʦ ގ, Ն ⌸ ϭ ഫ problem on abelian groups. such that n r pʦP p (here P PϪ Pϩ). ϭ Theorem 6. Let d and R be fixed positive numbers. Suppose A Cx Theorem 4. We have ϫ ϫ ϫ 1 Cx2 ... Cxt is an abelian group such that the orders x1, ͑ ͑ޚ͒͒ ͑ ͒ x2, ...,xt of its cyclic factors do not repeat more than d times each. log cn SL2 log f n ϭ ͉ ͉R Յ lim sup 2 lim sup 2 . Suppose that r A n for some positive integers r and n. Then 3ϱ ͑log n͒ ͞log log n 3ϱ ͑log n͒ ͞log log n n n as n and r tend to infinity we have By contrast, no such precise description of the subgroup ކ ͑␥ϩ ͑ ͒͒ log n structure exists even for SLd( p). Still, surprisingly, the proof of ͑ ͒ Յ o 1 sr A n , the general upper bound reduces to a similar extremal problem log log n for abelian groups by using some ideas of refs. 3 and 14 and ␥ ϭ ͌ ϩ Ϫ 2 ͞ 2 Theorem 5 (see below), which is the main new ingredient in ref. 6. where [( R(R 1) R) ] 4R . ކ Let X( q) be a finite quasisimple group of Lie type X over the The starting point of the proof of this theorem in ref. 3 is a well ކ Ͼ ކ finite field q of characteristic p 3. For a subgroup H of X( q) known formula for counting subgroups of finite abelian groups define (see ref. 18). We refer the reader to ref. 3 for the details that are too complicated to be given here. ͓ ͑ކ ͒ ͔ log X q : H t͑H͒ ϭ छ , log͉H ͉ We thank H. Bass for several helpful comments. This work was sup-
छ ported by grants from the National Science Foundation (to D.G. and where H denotes the maximal abelian quotient of H whose A.L.), the Israel Science Foundation, and the U.S.–Israel Binational छ order is coprime to p. Set t(H) ϭϱif ͉H ͉ ϭ 1. Science Foundation (to A.L.), and by Hungarian National Foundation Recall that R ϭ R(X) ϭ h͞2, where h is the Coxeter number for Scientific Research Grant T037846 (to L.P.). N.N. received a Golda of the root system of the split Lie type corresponding to X. Meir Postdoctoral Fellowship at the Hebrew University of Jerusalem.
1. Lubotzky, A. & Segal, D. (2003) Subgroup Growth, Progress in Mathematics 10. Liebeck, M. W. & Shalev, A. (2004) J. Algebra 276, 552–601. 212 (Birkhauser, Boston), pp. 91–97. 11. Wang, H. C. (1972) in Symmetric Spaces, eds. Boothby, W. & Weiss, G. 2. Lubotzky, A. (1995) Inv. Math. 119, 267–295. (Dekker, New York), pp. 460–487. 3. Goldfeld, D., Lubotzky, A. & Pyber, L. (2004) Acta Math., in press. 12. Burger, M., Gelander, T., Lubotzky, A. & Mozes, S. (2002) Geom. Funct. Anal. 4. Weil, A. (1980) in Oeuvres Scientifiques (Springer, New York), corrected 2nd 12, 1161–1173. printing, Vol. 2, pp. 48–61. 13. Murty, M. R. & Murty, V. K. (1987) Can. Math. Soc. Conf. Proc. 7, 243–272. 5. Murty, M. R., Murty, V. K. & Saradha, N. (1988) Am. J. Math. 110, 253–281. 14. Liebeck, M. & Pyber, L. (2001) Duke Math. J. 107, 159–171. 6. Lubotzky, A. & Nikolov, N. (2004) Acta Math., in press. 15. Larsen, M. & Pink, R. (2004) J. Am. Math. Soc., in press. 7. Bombieri, E. (1965) Mathematika 12, 201–225. 16. Weisfeiler, B. (1984) Proc. Natl. Acad. Sci. USA 81, 5278–5279. 8. Margulis, G. (1991) Discrete Subgroups of Semisimple Lie Groups, Ergebnisse 17. Liebeck, M., Saxl, J. & Seitz, G. (1992) Proc. London Math. Soc. 65, 297– der Math. 17 (Springer, Berlin). 325. 9. Serre, J.-P. (1970) Ann. Math. 92, 489–527. 18. Butler, L. M. (1987) Proc. Am. Math. Soc. 101, 771–775.
13430 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0404571101 Goldfeld et al. Downloaded by guest on September 28, 2021