Counting primes, groups, and manifolds

Statement of Results

Arithmetic Groups. Let n be a large integer, Γ a finitely generated group, and M a Riemannian manifold. Denote by π (n) the number of primes ≤ n, s_n(Γ) is the number of subgroups of Γ of index at most n and b_n(M) is the number of covers of M of degree at most n. The aim of this article is to announce results that show that, in some circumstances, these three seemingly unrelated functions are very much connected. This connection emerges, for example, when Γ is an arithmetic group, in which case it is also the fundamental group of a suitable locally symmetric finite volume manifold M. The studies of s_n(Γ) and b_n(M) are then almost the same. Moreover, if Γ has the congruence subgroup property, then estimating s_n(Γ) boils down to counting congruence subgroups of Γ. The latter is intimately related to the classical problem of counting primes. To present our results we need more notation.

Let G be an absolutely simple, connected, simply connected algebraic group defined over a number field k. For a finite subset S of valuations of k, including all the archimedean ones, let O_S denote the ring of S-integers of k and set Γ= G(O_S). A subgroup H ≤ Γ is called a congruence subgroup if there is some ideal I ⊲ O_S such that H contains the kernel of the homomorphism Γ → G(O_S/I).

Let c_n(Γ) denote the number of congruence subgroups of index at most n in Γ. The counting of congruence subgroups in arithmetic groups has already played a role in the proof of one of the main results of the theory of subgroup growth: A finitely generated, residually finite group Γ has polynomial subgroup growth [i.e., s_n(Γ) = n^{O (1)}] if and only if Γ is virtually solvable of finite rank (ref. 1 and references therein). That theorem required only a weak lower bound on the number congruence subgroups. In ref. 2, Lubotzky proved a more precise result: there exist numbers a, b depending on G, k, and S, such that∥ and, moreover, the sequence s_n(Γ) has much faster growth (at least n ^{log n}) if the congruence subgroup property fails for G. Below we determine the precise rate of growth of c_n(Γ). (All logarithms are in base e.)

Let X be the Dynkin diagram of the split form of G (e.g., X = A_{n -1} if G = SU_n). Let h be the Coxeter number of the root system Φ corresponding to X (it is the order of the Coxeter element of the Weyl group of X). Then , where , and for later use define R:= h/2. Let

Let GRH denote the Generalized Riemann Hypothesis (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 such that the cyclotomic field of q-th roots of unity is disjoint from k. Denote by π_k(x, q) the number of primes p with p ≤ x, p ≡ 1(mod q) and p splits completely at k. Then for some constant C = C(k) > 0 depending only on k (a more precise bound is given in ref. 5).

The lower bound for the limit in the Theorem 1 below was proved in ref. 3 and the upper bound in ref. 6.

Theorem 1. Let G, Γ, and γ(G) be as defined above. Assuming GRH we have and, moreover, this result is unconditional if G is of inner type (e.g., G splits) and k is either an abelian extension of or a Galois extension of degree <42.

An interesting aspect of Theorem 1 is not only that the limit exists but that it is completely independent of k and S and depends only on G. Although the independence on S is a minor point and can be proved directly, the only way we know to prove the independence on k is by applying the whole machinery of the proof.

In ref. 3 the crucial special case of Γ = SL ₂(O_S) is proved in full. There, we have . The lower bound follows using the Bombieri–Vinogradov Theorem (7) and the upper bound by a massive new combinatorial analysis.

Lattices. Let H be a connected characteristic 0 semisimple group. By this we mean that , where for each i, K_i is a local field of characteristic 0, and G_i is a connected simple algebraic group over K_i. We assume throughout that none of the factors G_i(K_i) is compact (so that rank_Ki(G_i) ≥ 1). Let Γ be an irreducible lattice of H; i.e., for every infinite normal subgroup N of H the image of Γ in H/N is dense there.

Assume now that By Margulis' Arithmeticity Theorem (8) every irreducible lattice Γ in H is arithmetic. Also the split forms of the factors G_i of H are necessarily of the same type and we set γ(H):= γ(G_i).

Moreover, a famous conjecture of Serre (9) asserts that such a group Γ has the (modified) congruence subgroup property. It has been proved in many cases. This enables us to prove Theorem 2.

Theorem 2. Assuming GRH and Serre's conjecture, then for every noncompact higher rank characteristic 0 semisimple group H and every irreducible lattice Γ in H the limit exists and equals γ(H); i.e., it is independent of the lattice Γ.

Moreover the above result holds unconditionally if H is a simple connected Lie group not locally isomorphic to and Γ is a nonuniform lattice in H (i.e., H/Γ is noncompact).

Theorem 2 shows, in particular, some algebraic similarity between different lattices Γ in the same Lie group G. This outcome is an addition to other results in the theory, e.g., Furstenberg's theorem showing that the boundaries of all such lattices Γ are the same or Margulis superrigidity showing that the finite dimensional representation theory of the different lattices Γ in the same G is similar (see ref. 8 and references therein).

We point out the following geometric reformulation of the special case.

Theorem 3. Let H be a simple connected Lie group of that is not locally isomorphic to . Put X = H/K, where K is a maximal compact subgroup of H. Let M be a finite volume noncompact manifold covered by X and let b_n(M) be the number of covers of M of degree at most n. Then exists, equals γ(H), and is independent of M.

It is interesting to compare Theorems 2 and 3 with the results of Liebeck and Shalev (10) and T. W. Müller and J.-C. Puchta (unpublished data): If and Γ is a lattice in H, then where χ is the Euler characteristic.

We finally mention a conjecture and a question: Let X be the symmetric space associated with a simple Lie group H as in Theorem 3. Denote by m_n(X) the number of manifolds covered by X of volume at most n. By a well known result of Wang (11), this number is finite unless H is locally isomorphic to or .

Conjecture. If then

Problem. Estimate m_n(X) for the case of H having equal to one. For H = SO(n, 1) the results of ref. 12 suggest that may exist, but we do not have any clue what it could be.

Proofs

The Lower Bound. We shall illustrate the main idea of the proof with and refer to ref. 3 for the full details.

Choose any ρ ε (0, ½). For x » 0 and a prime q < x, let P(x, q) be the set of primes p ≤ x, such that p ≡ 1 mod q. Let L(x, q) = |P(x, q)| and M(x, q) = Σ_{p ∈ P ( x , q)} log p. Then the Bombieri–Vinogradov Theorem (7) ensures the existence of a prime q ε (x^p/log x, x ^ρ) such that Put L:= L(x, q) and M:= M(x, q).

By strong approximation (compare ref. 1, window 9), Γ maps onto . Let B(p) be the subgroup of upper triangular matrices of and set The group B_P maps onto the diagonal , which in turn maps onto . For fixed , the latter vector space has approximately subgroups of index q ^σ(d–1)L (see proposition 1.5.2 in ref. 1), each giving rise to a subgroup of index n = [G_P: B_P]q ^σ(d–1)L in Γ. Now, log[G_P: B_P] ∼ d(d–1)M/2 as x → ∞ and after some algebraic manipulations, we obtain that for this chosen value of n, where in our case R = d/2. As shown in ref. 3, section 3, the maximum value of the expression above for σ, ρ ∈ (0, 1) is precisely and is achieved for . By taking x sufficiently large we can choose σ ∈ (0, 1) ∩ 1/[L(d – 1)] to be arbitrarily close to σ₀, and take ρ = ρ₀. This proves the lower bound.

The reason for invoking the GRH in Theorem 1 is that, in the general case, we need an equivalent of the Bombieri–Vinogradov Theorem for k in place of . The work of Murty and Murty (13) gives an analogue of it for number fields, but their result is weaker in general. It suffices for our needs when, for example, is an abelian extension.

The Upper Bound. The proof of the upper bound in ref. 6 is inspired by the special case solved in ref. 3 and has two parts:

1. A reduction to an extremal problem for abelian groups, and

2. Solving this extremal problem (see Theorem 6 below).

Part I. The subgroup structure of the groups is completely known. By using this, it is shown in ref. 3 that Theorem 1 for is equivalent to the following extremal result on counting subgroups of abelian groups.

Let C_m denote the cyclic group of order m. For all pairs and of disjoint sets of primes, let where the maximum is taken over all sets , and , such that (here ).

Theorem 4. We have

By contrast, no such precise description of the subgroup structure exists even for . Still, surprisingly, the proof of the general upper bound reduces to a similar extremal problem for abelian groups by using some ideas of refs. 3 and 14 and Theorem 5 (see below), which is the main new ingredient in ref. 6.

Let be a finite quasisimple group of Lie type X over the finite field of characteristic p > 3. For a subgroup H of define where H ^⋄ denotes the maximal abelian quotient of H whose order is coprime to p. Set t(H) = ∞ if |H ^⋄| = 1.

Recall that R = R(X) = h/2, where h is the Coxeter number of the root system of the split Lie type corresponding to X.

Theorem 5. Given the Lie type X then

The proof of this theorem does not depend on the classification of the finite simple groups; we use instead the work of Larsen and Pink (15) [which is a classification-free version of a result of Weisfeiler (16)] and Liebeck et al. (17) (the latter for groups of exceptional type).

Part II. Once Part I is proved, the argument reduces to an extremal problem on abelian groups.

Theorem 6. Let d and R be fixed positive numbers. Suppose is an abelian group such that the orders x ₁ , x ₂ ,..., x_t of its cyclic factors do not repeat more than d times each. Suppose that r |A|^R ≤ n for some positive integers r and n. Then as n and r tend to infinity we have where .

The starting point of the proof of this theorem in ref. 3 is a well known formula for counting subgroups of finite abelian groups (see ref. 18). We refer the reader to ref. 3 for the details that are too complicated to be given here.