Let SLn(Q) be the set of matrices of order n over the rational numbers with determinant equal to 1. We study in this paper a subset Λ of SLn(Q), where a matrix B belongs to Λ if and only if the conjugate subgroup BΓq(n)B−1 of principal congruence subgroup Γq(n) of lever q is contained in modular group SLn(Z). The notion of least common denominator (LCD for convenience) of a rational matrix plays a key role in determining whether B belongs to Λ. We show that LCD can be described by the prime decomposition of q. Generally Λ is not a group, and not even a subsemigroup of SLn(Q). Nevertheless, for the case n=2, we present two families of subgroups that are maximal in Λ in this paper.
Citation: Guangren Sun, Zhengjun Zhao. SLn(Z)-normalizer of a principal congruence subgroup[J]. AIMS Mathematics, 2022, 7(4): 5305-5313. doi: 10.3934/math.2022295
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Let SLn(Q) be the set of matrices of order n over the rational numbers with determinant equal to 1. We study in this paper a subset Λ of SLn(Q), where a matrix B belongs to Λ if and only if the conjugate subgroup BΓq(n)B−1 of principal congruence subgroup Γq(n) of lever q is contained in modular group SLn(Z). The notion of least common denominator (LCD for convenience) of a rational matrix plays a key role in determining whether B belongs to Λ. We show that LCD can be described by the prime decomposition of q. Generally Λ is not a group, and not even a subsemigroup of SLn(Q). Nevertheless, for the case n=2, we present two families of subgroups that are maximal in Λ in this paper.
Denote by SLn(Q) the set
{A=(aij)n×n: aij∈Q,i,j=1,2,⋯,n,anddet(A)=1}, |
and SLn(Z) the set
{(aij)n×n∈SLn(Q): aij∈Z,i,j=1,2,⋯,n}. |
This set SLn(Z) is usually referred to as "modular group". It is well known that SL2(Z) is closely related to modular forms, see [3]. Many important subgroups of SLn(Z) have been widely studied, such as
Definition 1.1 Let n,q≥2 be positive integers. The principal congruence subgroup of level q is defined as
Γq(n)={(aij)n×n∈SLn(Z): aii≡1(mod q);aij≡0(mod q),i≠j}. |
Remark 1.2 A different but equivalent definition can refer to [4]. We always denote by In the n×n identity matrix throughout the paper, and use notation Eij to represent the matrix with all entries are 0 but the (i,j) entry is 1. For a matrix A, denoted by Aij its (i,j) entry.
Naturally, Γq(n) could be addressed in the larger group SLn(Q). In this paper, we concentrate our attention on the the following subset Λ of SLn(Z).
Definition 1.3 Let n,q≥2 be positive integers. The SLn(Z)-normalizer of Γq(n) is defined as
Λ={B∈SLn(Q): BΓq(n)B−1⊂SLn(Z)}, |
and matrix in Λ is called an SLn(Z)-normalization element of Γq(n).
This notion, which ties in nicely with "The congruence subgroup problem" (see [7], or [4]), is inspired by observing subgroup topologies in SLn(Q) (see [4]). As well known the normalizer of a congruence subgroup has acquired significance because it is related to some simple group in [2]. It has also played an important role in work on Weierstrass points on the Riemann surfaces in [5].
In order to figure out Λ, it is a natural approach to transform a matrix in SLn(Q) into the set of integral matrices by multiplying a suitable positive integer. The following concept plays a key role in our discussion on the structure of Λ.
Definition 1.4 Let n,r be positive integers, and B∈SLn(Q). We call r the least common denominator (LCD for convention) of B, denoted by cB, if r is the least positive integer such that rB is an integral matrix.
With above preparation, the remainder of this paper is organized as follows. In Section 2, the fundamental relationship between SLn(Q) and Λ is investigated. It is obvious that Λ contains SLn(Z), and we will show that this inclusion is proper when q is not square-free. With the aid of Smith normal form of an integral matrix, we first establish a necessary and sufficient condition for determining whether a matrix in SLn(Q) belongs to Λ. This statement does not give an explicit relationship between cB and q in the case n>2 yet. Fortunately one necessary condition is obtained, which enables us to bound the multiplicity of arbitrary prime factor of the LCD of a matrix in Λ by factoring q. Other than that, we also show that Λ is not a group, and not even a semigroup. In spite of this, Λ admits many subgroups distinguished from SLn(Q). In section 3, two families of subgroups that are maximal in Λ are presented in the case n=2.
In this section, we will show that it can be determined whether a matrix in SLn(Q) belongs to Λ when its LCD is specified. In order to go on our following discussion, we need a widely understood theorem (see [1,6,8,9]), which arises from basic module theory over a principal ideal domain. The Smith normal form of integral matrices, which will be presented in the following paragraph, is our main tool in this section.
Let n be a positive integer, and A an integral matrix of order n. Then there exists unimodular matrices P,Q such that PAQ is a diagonal matrix
PAQ=diag(d1,d2,⋯,dl,0,⋯,0), | (2.1) |
where di, unique up to the sign, are nonzero integers for all 1≤i≤l, and di∣di+1. Moreover, di is called the i-th invariant factor of A, and (2.1) is the Smith normal form of A.
With the aid of this result, we give a characterization for the LCD of matrices in Λ as following theorem states.
Theorem 2.1 Let n,q≥2 be positive integers, and B∈SLn(Q). Then B is an SLn(Z)-normalization element of Γq(n) if and only if the n-th invariant factor dn of cBB divides q.
Proof. Suppose that B is an SLn(Z)-normalization element of Γq(n) and A=cBB. Let P,Q be unimodular matrices such that PAQ is the Smith normal form in (2.1). It follows from the fact B is invertible that the number l in (2.1) is equal to the order n of B. It is obvious that di∣di+1, and a positive integer r is the LCD of A if and only if the greatest common divisor of all entries of rA is 1. Thus d1=±1. Set C=PBQ, then C is also an SLn(Z)-normalization element of Γq(n). Hence, for any X=(xij)n×n∈Γq(n), we have
CXC−1=(x11d1d−12x12…d1d−1nx1nd2d−11x21x22…d2d−1nx2n⋮⋮⋮dnd−11xn1dnd−12xn2…xnn)∈SLn(Z) |
Take X=In+qE1n. It follows from above statements that (1,n) entry of CXC−1, which is equal to ±qd−1n, is an integer, and thus this means that dn∣q. This completes the proof of necessity.
Conversely, if dn∣q, then di∣q for all 1≤i≤n. We can assert from this fact that all entries in CXC−1 are integers, and thus the sufficiency follows.
It is worth pointing out that in the case n=2, we have d1d2=±d2=c2B, and thus the fact d2 of cBB divides q is equivalent to that c2B divides q.
Unfortunately, above theorem does not give an explicit relationship between cB and q in the case n>2. However, the following proposition can help us to understand the problem stated to some extent.
Theorem 2.2 Let n,q≥2 be positive integers, pe11⋯pekk the prime decomposition of q, and B=(bij)n×n∈SLn(Q). If B is an SLn(Z)-normalization element of Γq(n), then cB=pf11⋯pfkk, where 0≤fi≤ei−1 for all 1≤i≤k. In particular, if q is square-free, then Λ=SLn(Q).
Proof. Let A=cBB, and PAQ be the Smith normal form of A. We claim here that the following conclusion is true. Claim: Suppose that p is a prime number and f a positive integer. Then pf∣cB implies pf+1∣dn.
If above assertion was established, then the result follows by Theorem 2.1.
Assume the claim is false, then dn is divisible by at most f-th power of p, so does di, i=1,⋯,n. Note here that d1=±1, hence d1⋯dn is divisible by at most (n−1)f-th power of p. On the other hand, the determinants of PBQ and B are both equal to 1. Therefore, d1⋯dn=cnB, and this means that pnf∣d1⋯dn=cnB. So, above statements are contrary to the fact (n−1)f<nf, and thus the assumption is not correct.
Corollary 2.3 Let q≥2 be a square-free positive integer, then the normalizer of Γq(n) in SLn(Q) is SLn(Z).
The previous conclusions present some description for the elements of Λ. It has to be pointed out that as the subset of the group SLn(Q), Λ does not possess a fine structure. In fact, we can show that Λ is not a semigroup of the group SLn(Q).
Proposition 2.4 Let n≥2 be a positive integer, q=m2sr, where m>1,r≥1,s≥1 are integers with m2∤r. Then SLn(Z) is properly contained in Λ, and Λ is also a proper subset of SLn(Q), but not a subsemigroup. In addition, any subgroup of SLn(Q) which contains SLn(Z) must have an element, which does not belong to Λ.
Proof. It is clear that B=diag(1m,1,⋯,1,m)∉SLn(Z). Moreover, m2 is the n-th invariant factor of cBB and divides q, hence B∈Λ, and thus we can get the first proper inclusion. On the other hand, we know that LCD of matrices in SLn(Q) is unbounded. However, the fact B∈Λ implies cB<q by Theorem 2.2, then the second inclusion follows.
In what follows, we show that Λ is not a subsemigroup. As a matter of fact, it suffices to verify that there exist two matrices whose product does not belong to Λ. Let K=diag(1ms,1,⋯,1,ms). Analogously, m2s is the n-th invariant factor of cKK and divides q, hence K∈Λ. With B defined as previous paragraph, set J=BK. Then the n-th invariant factor of cJJ is m2s+2 which does not divide q, the conclusion follows.
The last assertion can be restated as: Let D∈Λ∖SLn(Z), then the subgroup H generated by SLn(Z)⋃{D} must have an element which does not belong to Λ.
Let A=cDD, and PAQ be the Smith normal form of A. We note that PDQ∈H, and thus s-th power of PDQ
(PDQ)s=(1cDPAQ)s=diag(ds1csD,⋯,dsncsD) |
is also in the group H. It is easily deduced from d1=±1 that c(PDQ)s=csD. If (PDQ)s∈Λ, then csD divides q by Theorem 2.2. However cD>1 for D∉SLn(Z), and s can be chose arbitrarily, it is absurd.
In view of sparsity of set Λ as Proposition 2.4 pointed out, we are led to focus our attention on subgroups of SLn(Q) those are maximal in Λ, which is strictly defined as follows.
Definition 3.1 With n,q defined as Proposition 2.4, we call a subgroup M of SLn(Q) contained in Λ a maximal SLn(Z)-normalizer or simply maximal normalizer, if the following conditions hold
1. M is a subgroup of SLn(Q);
2. If H is a subgroup of SLn(Q) and properly contains M, then there is a matrix A∈H∖Λ.
It follows immediately from Proposition 2.4 that SLn(Z) satisfies above two conditions, and we call it the trivial maximal normalizer which is not our focus. In fact, we are interested in the following problem.
Problem 3.2 Find all nontrivial maximal normalizers.
To our knowledge, the problem we mentioned above is difficult to solve completely. In the rest of this section, two families of nontrivial maximal normalizers will be presented in the case n=2. In what follows, we focus attention on the case n=2. In other words, we will address related problems in SL2(Q).
Proposition 3.3 Let q=μ2ν>1 be a positive integer, where ν is a square-free integer. If τ1 and τ2 are two coprime positive integers with product τ1τ2 dividing μ, then
H(τ1,τ2)={1τ1τ2(aτ1τ2bτ21cτ22dτ1τ2): a,b,c,d∈Z;ad−bc=1} | (3.1) |
is a maximal normalizer. And if either τ1 or τ2 is greater than 1, then H(τ1,τ2) is nontrivial.
Proof. By Theorem 2.1, we know that the SL2(Z)-normalizer of Γq(2) is Λ={B∈SL2(Q): cB∣μ}.
It is easy to verify that H(τ1,τ2) is a subset of Λ, and also a subgroup of SL2(Q). Therefore, we only need to show H(τ1,τ2) is maximal in Λ. Namely, we have to prove that for any K∈Λ∖H(τ1,τ2), the subgroup L generated by H(τ1,τ2)⋃{K} must have an element, which does not belong to Λ.
To this end, let A=cKK. First, we show there exists J∈H(τ1,τ2) such that JA is a upper triangular matrix. If A21=0, it is trivial. And when A11=0, we need only take J as τ1τ2E12−τ1τ2E21. As a consequence, we can assume that neither of A11,A21 is equal to 0. Let α=gcd(A11τ2,A21τ1), and c′=A21τ1α, d′=−A11τ2α. As well known that there exists two integers a′,b′ such that a′d′−b′c′=1, so we can take
J=1τ1τ2(a′τ1τ2b′τ21c′τ22d′τ1τ2). |
Set D=JK, then D11D22=1 as the determinant of JK is 1. We show that the subgroup generated by D is not contained in Λ. The scenarios will be considered for the following two cases:
Case 1. D11,D22∉{±1}. Let D11=ω1θ1 be the reduced fraction with θ1>1. Since D∈L implies Dm∈L, we deduce cDm∣μ, i.e., θm1∣μ for any positive integer m, but this is in contradiction with θ1>1.
Case 2. D11,D22∈{±1}. We can assume D11=D22=1 because of −I2∈H(τ1,τ2). Let D12=βγ be the reduced fraction. Note that D∉H(τ1,τ2), then the ratio βτ2γτ1 of βγ and τ1τ2 is not an integer. For any positive integer m, let
Pm=(ambmcmdm)=[D(−τ1τ2E12+τ2τ1E21)]m. |
By this equality, we can obtain the following recurrence relation
a1=βτ2γτ1,a2=(βτ2γτ1)2+1,am+2=βτ2γτ1am+1−am. |
Hence am is a polynomial in βτ2γτ1 with integral coefficients. We note here that βτ2γτ1 is not an integer, so we can take the reduced fraction βτ2γτ1=ωθ with θ>1. Suppose now
am=ωm+sm−1ωm−1θ+⋯+s1ωθm−1+s0θmθm, | (3.2) |
where si is an integer for 1≤i≤m, and m is any positive integer. The right hand side of (3.2) is obviously reduced also, and this implies θm∣μ. Similar to case 1, it is absurd.
Remark 3.4 It is worth pointing out that the group H(τ1,τ2) in above proposition is a conjugate subgroup of SL2(Z) in SL2(Q). It is natural to raise a problem here: which conjugate subgroups of SL2(Z) in SL2(Q) are maximal normalizers.
It is not hard to see from above discussion that if the subgroup ⟨A⟩, generated by A∈Λ, is contained in Λ, then there must exist a maximal normalizer in Λ which contains A. In order to go on our story, the following definition is needed.
Definition 3.5 Let A∈SLn(Q). A is called σ−stable if there exists a positive integer σ such that cAm≤σ for any integer m. Otherwise, A is called unbounded.
When a matrix A∈SLn(Q) is σ−stable, we call A stable instead of σ−stable for convenience if we do not need to know σ exactly.
Example 3.6 Let q=p2, where p is a prime number. Then the matrix
B=1p(p1p⋱⋱1p) |
belongs to the SLn(Z)-normalizer of Γq(n), and is pn−1−stable. It is not hard to check that B is not p−stable when n>2, and thus Λ does not contain the cyclic group generated by B, i.e., no maximal normalizer contains B.
Having taken a short tour to the general case, we now concentrate our attention on the case n=2 again. The following result gives some description for stable matrix in SL2(Q).
Lemma 3.7 Let A∈SL2(Q). Then A is stable if and only if the trace tr(A) is an integer. In particular, under assumptions of Proposition 3.3, there exists a maximal normalizer which contains A whenever tr(A) is an integer and cA∣μ.
Proof. Set
A:=(abcd), Am:=(ambmcmdm) |
It is easy to show the following recurrence relation holds
{am+1=afm−fm−1bm+1=bfmcm+1=cfmdm+1=dfm−fm−1 | (3.3) |
where f0=1,f1=tr(A) and fm+1=tr(A)fm−fm−1 for any positive integer m. Hence fm is a polynomial in tr(A) with integral coefficients, and tr(A) is an integer implies in turn fm is also an integer. Then we can deduce cAm∣cA for m≥1 by (3.3). When m≤−1, a recurrence relation analogous to (3.3) holds clearly, and thus the sufficiency follows.
Conversely, suppose tr(A) is not an integer, we show that cAm+1 is unbounded for m≥1. Let tr(A)=ωθ be the reduced fraction with θ>1, we divide proof of the necessity into the following two cases as Proposition 3.3:
Case 1. b=0 and c=0. Proof is the same as Case 1 in Proposition 3.3.
Case 2. b≠0 or c≠0. It is sufficient to deal with b≠0. Let b=αβ be the reduced fraction. Set
f=max{fp: pfp∣θ,pfp+1∤θ}, |
and m≥f+1. Let α1θm be the reduced fraction of αθm. Then α1 and θ are coprime numbers, and pm−f∣θm as long as p is a prime divisor of θ. Hence bm+1 can be expressed as
α1(ωm+sm−1ωm−1θ+⋯+s1ωθm−1+s0θm)βθm, | (3.4) |
where si is an integer for 0≤i≤m−1. It is easy to check that θm and the integer in bracket of (3.4) are coprime, and the proof is the analogy of Case 2 in Proposition 3.3. Then the denominator in the reduced fraction of bm+1 is divisible by θm, i.e., θm∣cAm+1. This completes the proof of necessity.
With the help of above lemma, we can give another maximal normalizer in Λ except Proposition 3.3 presented.
Proposition 3.8 With q,μ defined as Proposition 3.3. Let
A=1μ(abcd) |
be a matrix in SL2(Z)-normalizer Λ of Γq(2), where a,b,c,d are integers, d,μ are coprime and a+d≡0(mod μ). Let λ=d−1b where 1≤d−1≤μ−1 and dd−1≡1(mod μ). Then the unique maximal normalizer H which contains A is composed of
1μ(lμ+λskμ+λ(mμ−λs)smμ−λs) |
where l,k,m,s are integers satisfy that l(mμ−λs)−ks=μ and la+kc≡lb+kd≡0(mod μ).
Proof. It is not hard to verify that cA=μ and A∈H⊆Λ. The product of any two matrices in H can be expressed as
P=1μ(a1b1c1d1)=1μ2(l1μ+λs1k1μ+λt1s1t1)(l2μ+λs2k2μ+λt2s2t2), |
where ti=miμ−λsi for i=1,2, and
{a1=(l1l2+s2λl1+k1μ)μ+λ(s1l2+s2m1)b1=[l1k2+(m2μ−λs2)λl1+k1μ]μ+λ[(m1m2+s2λl2+k2μ)μ−λ(s1l2+s2m1)]c1=s1l2+s2m1d1=(m1m2+s2λl2+k2μ)μ−λ(s1l2+s2m1). |
Note here that d(λli+ki)=(dd−1)lib+kid≡0(mod μ) for i=1,2, so λli+ki≡0(mod μ). Let
{s=s1l2+s2m1l=l1l2+s2λl1+k1μk=l1k2+(m2μ−λs2)λl1+k1μm=m1m2+s2λl2+k2μ. |
It is easy to see l(mμ−λs)−ks=μ for detP=1. Since
la+kc=l1(l2a+k2c)+[m2cμ+s2(a−λc)]λl1+k1μ, |
we deduce la+kc≡0(mod μ) from the congruence l2a+k2c≡0(mod μ) and a−λc=a(1−dd1)+d1μ2≡0(mod μ). Similarly, we can get lb+kd≡0(mod μ), and this implies in turn P∈H. On the other hand, the inverse of arbitrary matrix in H is
1μ(mμ−λs−kμ−λ(mμ−λs)−slμ+λs)=1μ(mμ+λ(−s)(−k−λm−λl)μ+λ[lμ−λ(−s)]−slμ−λ(−s)), |
where
ma+(−k−λm−λl)c=m(a−λc)+c(λl+k)≡0(mod μ) |
and
mb+(−k−λm−λl)d≡0(mod μ), |
and thus H is a subgroup of SL2(Q).
In what follows, we show that H is the maximal and the unique normalizer. In fact, we will prove that any B∈Λ contained in the same maximal normalizer with A must be in H.
Set
B=1μ(uvst), |
and compute AB and BA respectively. It follows from the fact cAB and cBA divide μ that au+bs≡av+bt≡bs+dt≡0(mod μ). As d and μ are coprime, we can find integers l,k,m such that u=lμ+λs, v=kμ+λt,t=mμ−λs. On the other hand, we get by Lemma 3.7 that
au+bs+cv+dt≡0(mod μ2) | (3.5) |
Substitute u,v,t in (3.5), we have
(la+kc)μ+(1−dd1)(1+ad1)≡0(mod μ2) |
By Lemma 3.7 again, we obtain that a+d≡0(mod μ), so 1+ad1≡(a+d)d1≡0(mod μ), and thus la+kc≡0(mod μ).
Finally, let D=μ3ABA, then
D12=a(lb+kd)μ+b2s(1−dd1)(1+ad1)+dbm(1+ad1)μ |
The fact ABA∈Λ implies D12≡0(mod μ2), and then a(lb+kd)≡0(mod μ), that is lb+kd≡0(mod μ). Our proof of necessity is completed.
Remark 3.9 For coprime numbers d,μ, the maximal normalizer H in Proposition 3.8 is clearly different from H(τ1,τ2) in Proposition 3.3.
We give some remarks about our results obtained above to conclude our paper. In this paper, we study a subset Λ of SLn(Q), where a matrix B belongs to Λ if and only if the conjugate subgroup BΓq(n)B−1 of principal congruence subgroup Γq(n) is contained in modular group SL2(Z). We demonstrated that this subset Λ is fairly loose, and even not a semigroup. However, we presented in previous two sections two families of nontrivial maximal normalizers, i.e., two families of subgroups that are maximal in Λ. It is worth being pointed out that the maximal normalizers we provided are only one part of solutions to Problem 3.2 when n=2. In the light of Proposition 3.3 and 3.8, we need to consider matrices whose diagonals are not invertible or 0 modulo μ. As for the case n>2, Example 3.6 indicates that it is not enough to consider tr(A) only. However, it seems difficult to our knowledge to get a simple formula which is analogous to (3.3).
We thank the referees for careful reading of the manuscript and the suggestions on the writing of the paper. This work was supported by National Nature Science Foundation of China(No. 11971382).
All authors declare no conflicts of interest in this paper.
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