ON THE ORDER OF UNIMODULAR MATRICES MODULO INTEGERS PRELIMINARY VERSION

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1 ON THE ORDER OF UNIMODULAR MATRICES MODULO INTEGERS PRELIMINARY VERSION. Introduction Given an integer b and a prime p such that p b, let ord p (b) be the multiplicative order of b modulo p. In other words, ord p (b) is the smallest non negative integer k such that b k mod p. Clearly ord p (b) p, and if the order is maimal, b is said to be a primitive root modulo p. Artin conjectured (see the preface in []) that if b Z is not a square, then b is a primitive root for a positive proportion of the primes. What about the typical behaviour of ord p (b)? For instance, are there good lower bounds on ord p (b) that hold for a full density subset of the primes? In [3], Erdös and Murty proved that if b 0, ±, then there eists a δ > 0 so that ord p (b) is at least p /2 ep((log p) δ ) for a full density subset of the primes. However, we epect the typical order to be much larger. In [5] Hooley proved that the Generalized Riemann Hypothesis (GRH) implies Artin s conjecture. Moreover, if f : R + R + is an increasing function tending to infinity, it is possible to modify Hooley s argument to show that GRH implies that the order of b modulo p is greater than p/f(p) for full density subset of the primes. In this paper we will investigate a related question, namely lower bounds on the order of unimodular matrices modulo N Z. That is, if A SL 2 (Z), what can be said about lower bounds for ord N (A), the order of A modulo N, that hold for most N? It is a natural generalization of the previous questions, but our main motivation comes from mathematical physics (quantum chaos): In [6] Rudnick and I proved that if A is hyperbolic 2, then quantum ergodicity for toral automorphisms follows from ord N (A) being slightly larger than N /2, and we then showed that this condition does hold for a full density subset of the integers. Author supported in part by the National Science Foundation (DMS ). The constant is given by an Euler product that depends on b. 2 A is hyperbolic if tr(a) > 2.

2 2 Again, we epect that the typical order is much larger. In order to give lower bounds on ord N (A), it is essential to have good lower bounds on ord p (A) for p prime, and this is our first goal. Theorem. Let A SL 2 (Z) be hyperbolic, and let f : R + R + be an increasing function tending to infinity slower than log. Assuming GRH, there are at most O( ) primes p such that ord log f() ɛ p (A) < p/f(p). In particular, the set of primes p such that ord p (A) p/f(p) has density one. Using this we obtain an improved lower bound on ord N (A) that is valid for most integers. Theorem 2. Let A SL 2 (Z) be hyperbolic. Assuming GRH, the number of N such that ord N (A) N ɛ is o(). That is, the set of integers N such that ord N (A) N ɛ has density one. Remarks: If A is elliptic ( tr(a) < 2) then A has finite order (in fact, at most 6). If A is parabolic ( tr(a) = 2), then ord p (A) = p unless A is congruent to the identity matri modulo p, and hence there eists a constant c A > 0 so that ord N (A) > c A N. Apart from the application in mind, it is thus natural to only treat the hyperbolic case. As far as unconditional results for primes go, we note that the proof in [3] relies entirely on analyzing the divisor structure of p, and we epect that their method should give a similar lower bound on the order of A modulo p. An unconditional lower bound of the form () ord p (b) p η for a full proportion of the primes and η > /2 would be quite interesting. In this direction, Goldfeld proved [4] that if η < 3/5, then () holds for a positive, but not full, proportion of the primes. Clearly ord p (A) is related to ord p (ɛ), where ɛ is one of the eigenvalues of A. Since A is assumed to be hyperbolic, ɛ is a power of a fundamental unit in a real quadratic field. The question of densities of primes p such that ord p (λ) is maimal, for λ a fundamental unit in a real quadratic field, does not seem to have received much attention until quite recently; in [8] Roskam proved that GRH implies that the set of primes p for which ord p (λ) is maimal has positive density. (The work of Weinberger [2], Cooke and Weinberger [0] and Lenstra [7] does treat the case ord p (λ) = p, but not the case ord p (λ) = p +.) 2. Preliminaries 2.. Notation. If O F is the ring of integers in a number field F, we let ζ F (s) = a O F N(a) s denote the zeta function of F. By GRH we

3 ON THE ORDER OF UNIMODULAR MATRICES MODULO INTEGERS 3 mean that all nontrivial zeroes of ζ F (s) lie on the line Re(s) = /2 for all number fields F. Let ɛ be an eigenvalue of A, satisfying the equation (2) ɛ 2 tr(a)ɛ + det(a). Since A is hyperbolic, K = Q(ɛ) is a real quadratic field. Let O K be the integers in K, and let D K be the discriminant of K. Since A has determinant one, ɛ is a unit in O K. For n Z + we let ζ n = e 2πi/n be a primitive n-th root of unity, and α n = ɛ /n be an n-th root of ɛ. Further, with Z n = K(ζ n ), K n = K(ζ n, α n ), and L n = K(α n ), we let σ p denote the Frobenius element in Gal(K n /Q) associated with p. We let F p k denote the finite field with p k elements, and we let F p F 2 p 2 be the norm one elements in F p 2, i.e., the kernel of the norm map from F p to F 2 p. Let A p be the group generated by A in SL 2 (F p ). A p is contained in a maimal torus (of order p or p + ), and we let i p be the inde of A p in this torus. Finally, let π() = {p : p is prime} be the number of primes up to Kummer etensions and Frobenius elements. We want to characterize primes p such that n i p, and we can relate this to primes splitting in certain Galois etensions as follows: Reduce equation (2) modulo p and let ɛ denote a solution to equation (2) in F p or F p 2. (Note that if p does not ramify in K then the order of A modulo p equals the order of ɛ modulo p.) If p splits in K then ɛ F p, and if p is inert, then ɛ F p 2 \ F p. In the latter case, ɛ F p since 2 the norm one property is preserved when reducing modulo p. Now, F p and F p are cyclic groups of order p and p + respectively. 2 Thus, if p splits in K then ord p (ɛ) p, whereas if p is inert in K then ord p (ɛ) p +. Lemma 3. Let p be unramified in K n, and let C n = {, γ} Gal(K n /Q), where γ is given by γ(ζ n ) = ζn and γ(α n ) = αn. Then the condition that n i p is equivalent to σ p C n. Moreover, C n is invariant under conjugation. Proof. The split case: Since n i p and i p p we have ζ n F p, i.e. F p contains all n-th roots of unity. Moreover, ɛ is an n-th power of some element in F p, and thus the equation n ɛ splits completely in F p. In other words, p splits completely in K n and σ p is trivial. The inert case: Since n divides i p, ɛ is an n-th power of some element in F p 2 and hence α n F p 2. Moreover, n p 2 implies that ζ n F p 2.

4 4 Now, N F p 2 F p (α n ) = and N F p 2 F p (ζ n ) = ζn p+ = implies that σ p (ζ n ) ζ n mod p, σ p (α n ) α n mod p. For p that does not ramify in K n we thus have (3) σ p (ζ n ) = ζn, σ p (α n ) = αn Now, an element τ Gal(K n /Q) is of the form { ζ n ζn t t Z τ : α n αnζ u n s s Z, u {, } Composing γ and τ then gives τ γ : { ζ n ζ n α n α n ζ t n α u n ζ s n and { ζ n ζn t ζn t γ τ : α n αnζ u n s αn u ζn s which shows that γ is invariant under conjugation The Chebotarev density Theorem. In [9] Serre proved that the Generalized Riemann Hypothesis (GRH) implies the following version of the Chebotarev density Theorem: Theorem 4. Let E/Q be a finite Galois etension of degree [E : Q] and discriminant D E. For p a prime let σ p G = Gal(E/Q) denote the Frobenius conjugacy class, and let C G be a union of conjugacy classes. If the nontrivial zeroes of ζ E (s) lie on the line Re(s) = /2, then for 2, {p : σ p C} = C G π() + O ( ) C G /2 (log D E + [E : Q] log ) Now, primes that ramify in K n divides nd K (see Lemma 9), so as far as densities are concerned, ramified primes can be ignored. The bounds on the size of D Kn (see Lemma 9) and Lemma 3 then gives the following: Corollary 5. If GRH is true then 2 (4) {p : n i p } = [K n : Q] π() + O ( /2 (log(n) ) Remark: For theorems and 2 to be true, it is enough to assume that the Riemann hypothesis holds for all ζ Kn, n >.

5 ON THE ORDER OF UNIMODULAR MATRICES MODULO INTEGERS Bounds on degrees. In order to apply the Chebotarev density Theorem we need bounds on the degree [K n : Q]. We will first assume that ɛ is a fundamental unit. Lemma 6. If ɛ is a fundamental unit in K and if n = 4 or n = q, for q an odd prime, then Gal(K n /K) is nonabelian. Proof. We start by showing that [K n : Z n ] = n. Consider first the case n = q. If α q Z q then β = N Zq K (α q) = α q [Zq:K] ζq t K R for some integer t. Since q is odd we may assume that α q R, and this forces ζq t =, which in turn implies that α q [Zq:K] K. Because ɛ is a fundamental unit this means that q [Z q : K]. On the other hand, [Z q : K] φ(q), a contradiction. Thus α q Z q, and hence K q /Z q is a Kummer etension of degree q. For n = 4 we note that i Z 4 = K(i). Thus α 2 = ɛ Z 4 implies that ɛ Z 4. However, either ɛ or ɛ is real and generates a real degree two etension of K, whereas K(i) is a non-real quadratic etension of K, and hence α 2 Z 4. Now, if α 4 Z 4 (α 2 ) then N Z 4(α 2 ) Z 4 (α 4 ) = α4i 2 t Z 4 for some t Z, and thus α4 2 = α 2 Z 4 which contradicts α 2 Z 4. Therefore, [Z 4 (α 4 ) : Z 4 ] = [Z 4 (α 4 ) : Z 4 (α 2 )][Z 4 (α 2 ) : Z 4 ] = 4. Finally we note that the commutator of any nontrivial element σ Gal(K n /Z n ) with any nontrivial element σ 2 Gal(K n /L n ) is nontrivial (we may regard Gal(K n /Z n ) and Gal(K n /L n ) as subgroups of Gal(K n /K)). Hence Gal(K n /K) is nonabelian. Lemma 7. If ɛ is a fundamental unit then [K n : Z n ] n/2. Proof. Clearly Z n (α q k) K n, and since field etensions of relative prime degrees are disjoint, it is enough to show that if q k n is a prime power then q k [Z n (α q k) : Z n ] if q is odd, and q k [Z n (α q k) : Z n ] if q = 2. If q is odd then Lemma 6 implies that α q Z n since Gal(Z n /K) is abelian. Hence, if m Z and α m Z q k n, we must have q k m. Now, if σ Gal(Z n (α q k)/z n ) then σ(α q k) = α q kζ tσ for some integer t q k σ. Thus there eists an integer t such that β = N Zn(α q k ) Z n (α q k) = α [Zn(α q k ):Z n] ζq t Z n Multiplying β by ζq t Z n we find that α [Zn(α q k ):Z n] q k q k [Z n (α q k) : Z n ]. q k Z n, and hence

6 6 For q = 2 the proof is similar, ecept that a factor of two is lost if α 2 Z n. Remark: K 2 /Q is a Galois etension of degree four, hence abelian and therefore contained in some cyclotomic etension by the Kronecker- Weber Theorem, and it is thus possible that α 2 Z n for some values of n. Lemma 8. We have nφ(n) K [K n : Q] 2nφ(n) Proof. We first note that [Z n : K] equals φ(n) or φ(n)/2 depending on whether K Q(ζ n ) or not. We also have the trivial upper bound [K n : Z n ] n. For a lower bound of [K n : Z n ] we argue as follows: Let γ K be a fundamental unit. Since the norm of ɛ is one we may write ɛ = γ k for some k Z. (Note that k does not depend on n.) As [Z n (γ /n ) : Z n (ɛ /n )] k, Lemma 7 gives that [Z n (ɛ /n ) : Z n ] n/k. The upper and lower bounds now follows from Bounds on discriminants. [K n : Q] = [K n : Z n ][Z n : K][K : Q] Lemma 9. If p ramifies in K n then p nd K. Moreover, Proof. First note that log(disc(k n /Q)) K [K n : K] log(n) disc(k n /Q) = N K Q (disc(k n /K)) disc(k/q) [Kn:K]. From the multiplicativity of the different we get disc(k n /K) = disc(z n /K) [Kn:Zn] N Zn K (disc(k n/z n )), Since ɛ is a unit, so is ɛ /n. Thus, if we let f() = n ɛ then f () = n n, and therefore the principal ideal f (ɛ /n )O Kn equals no Kn. In terms of discriminants this means that and similarly it can be shown that disc(k n /Z n ) N Kn Z n (no Kn ) disc(z n /K) N Zn K (no Z n ). Thus disc(k n /Q) divides ( NQ K N K n K (no K n ) N Zn K (no Z n ) [Kn:Zn]) disc(k/q) [Kn:K].

7 ON THE ORDER OF UNIMODULAR MATRICES MODULO INTEGERS 7 which proves the two assertions. = n 4[Kn:K] disc(k/q) [Kn:K] 3. Proof of Theorem In order to bound the number of primes p < for which i p > /2 we will need the following Lemma: Lemma 0. The number of primes p such that ord p (A) y is O(y 2 ). Proof. Given A there eists a constant C A such that det(a n I) = O(CA n ). Now, if the order of A mod p is n, then certainly p divides det(a n I) 0. Putting M = y n= det(an I) we see that any prime p for which A has order n y must divide M. Finally, the number of prime divisors of M is bounded by y log(m) n log(c A ) y 2. n= First step: We consider primes p such that i p ( /2 log, ). By Lemma 0 the number of such primes is ( ( ) ) 2 ( ) (5) O = O /2 log log 2. Second step: Consider p such that q i p for some prime q ( /2, log 3 /2 log ). We may bound this by considering primes p such that p ± mod q for q ( /2, log 3 /2 log ). Since q /2 log, Brun s sieve gives (up to an absolute constant) the bound φ(q) log() and the total contribution from these primes is at most (6) φ(q) log(/q) log q ( /2 log 3,/2 log ) q ( /2 log 3,/2 log ) Now, summing reciprocals of primes in a dyadic interval, we get q π(2m) M log M q [M,2M] q.

8 8 Hence q ( /2 log 3,/2 log ) q ( ) /2 log log log 2 /2 / log 3 log log log. log log and equation (6) is O( ). log 2 Third step: Now consider p such that q i p for some prime q (f() 2, /2 ). We are now in the range where GRH is applicable; by log 3 Corollary 5 and Lemma 8 we have {p : q i p } qφ(q) log + O(/2 log(q 2 )) Summing over q (f() 2, /2 ) we find that the number of such p log 3 is bounded by ( ) (7) q 2 log + O(/2 log(q 2 )) Now, q (f() 2, /2 log 3 ) and thus equation (7) is q (f() 2, /2 log 3 ) q 2 f() f() log + log 2. Fourth step: For the remaining primes p, any prime divisor q i p is smaller than f() 2. Hence i p must be divisible by some integer d (f(), f() 3 ). Again Lemmas 5 and 8 give {p : d i p } dφ(d) log + O(/2 log(d 2 )) Noting that φ(d) d ɛ and summing over d (f(), f() 3 ) we find that the number of such p is bounded by ( ) (8) d 2 ɛ log + O(/2 log(d 2 )) Now, d (f(),f() 3 ) d (f(),f() 3 ) d 2 ɛ f() ɛ

9 and ON THE ORDER OF UNIMODULAR MATRICES MODULO INTEGERS 9 d (f(),f() 3 ) therefore equation (8) is /2 log(d 2 ) f() 3 /2 log( 2 ) f() ɛ log 4. Proof of Theorem 2 With the results from the previous section we may now deduce Theorem 2. If p is prime such that ord p (A) p/ log(p), or p ramifies in K, we say that p is bad. We let P B denote the set of all bad primes, and we let P B (z) be the set of primes p P B such that p z. Since only finitely many primes ramify in K, Theorem gives that the number of bad primes p is O( ). A key observation is the following: Lemma. We have (9) In particular, if we let log 2 ɛ p P B p < β(z) = p P B (z) /p, then β(z) tends to zero as z tends to infinity. Proof. Immediate from partial summation and the O( ) estimate log 2 ɛ in Theorem. Given N Z, write N = s 2 N G N B where N G N B is square free and N B is the product of bad primes dividing N. By the following Lemma, we find that few integers have a large square factor: Lemma 2. {N : s 2 N, s y} = O ( ) y Proof. The number of N such that s 2 N for s y is bounded by s 2 y s y Net we show that there are few N for which N B is divisible by p P B (z). In other words, for most N, N B is a product of small bad primes.

10 0 Lemma 3. The number of N such that p P B (z) divides N B is O(β(z)). Proof. Let p P B (z). The number of N such that p N is less than /p. Thus, the total number of N such that some p P B (z) divides N, is bounded by p P B (z) p = p P B (z) p = β(z). Combining the previous results we get that the number of N = s 2 N G N B such that N B is z-smooth and s y is ( + O (β(z) + /y)). For such N we have N B p z p ez. Letting z = log log and y = log we get that N G = N N s 2 N B log 3 for N with at most O ((β(log log ) + (log ) )) = o() eceptions. Now, the following Proposition gives that, for most N, ord N (A) is essentially given by p N ord p(a). Proposition ([6], Proposition ). Let D A = 4(tr(A) 2 4). For almost all 3 N, p d ord N (A) 0 ord p (A) ep(3(log log ) 4 ) where d 0 is given by writing N = ds 2, with d = d 0 gcd(d, D A ) squarefree. Finally, since ord p (A) we thus find that ord N (A) for most N. p log p p ɛ for p N G and p sufficiently large, p N G ord p (A) ep(3(log log ) 4 ) N ɛ G ep(3(log log ) 4 ) N 2ɛ 3 By for almost all N we mean that there are o() eceptional integers N that are smaller than.

11 ON THE ORDER OF UNIMODULAR MATRICES MODULO INTEGERS References [] E. Artin. The collected papers of Emil Artin. Addison Wesley Publishing Co., Inc., Reading, Mass.-London, 965. [2] G. Cooke and P. J. Weinberger. On the construction of division chains in algebraic number rings, with applications to sl 2. Comm. Algebra, 3:48 524, 975. [3] P. Erdős and M. R. Murty. On the order of a (mod p). In Number theory (Ottawa, ON, 996), pages Amer. Math. Soc., Providence, RI, 999. [4] M. Goldfeld. On the number of primes p for which p + a has a large prime factor. Mathematika, 6:23 27, 969. [5] C. Hooley. On Artin s conjecture. J. Reine Angew. Math., 225: , 967. [6] P. Kurlberg and Z. Rudnick. On quantum ergodicity for linear maps of the torus. Comm. Math. Phys., 222():20 227, 200. [7] H. W. Lenstra, Jr. On Artin s conjecture and Euclid s algorithm in global fields. Invent. Math., 42:20 224, 977. [8] H. Roskam. A quadratic analogue of Artin s conjecture on primitive roots. J. Number Theory, 8():93 09, [9] J.-P. Serre. Quelques applications du théorème de densité de Chebotarev. Inst. Hautes Études Sci. Publ. Math., (54):323 40, 98. [0] P. J. Weinberger. On Euclidean rings of algebraic integers. In Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 972), pages Amer. Math. Soc., Providence, R. I., 973. Department of Mathematics, Chalmers University of Techology, SE Gothenburg, Sweden URL: address: kurlberg@math.chalmers.se