Normalizers of intermediate congruence subgroups of the Hecke subgroups

Abstract For a square-free positive integer N, we study the normalizer of ΓΔ(N) in PSL2(ℝ) and investigate the group structure of its quotient by ΓΔ(N) under certain conditions.


Introduction
For each positive integer N , we let 0 .N / be the Hecke subgroup of the full modular group SL 2 .Z/ defined by We denote by N 0 .N / the normalizer of 0 .N / in PSL 2 .R/. Newman [14,17,18] obtained a result about N 0 .N /. This normalizer has acquired its importance in several areas of mathematics. For instance, the genus zero subgroups of N 0 .N / have a mysterious correspondence to the conjugacy classes of the monster simple group [6,7]. Moreover, the normalizer N 0 .N / played an important role in the work on Weierstrass points on the modular curve X 0 .N / associated to 0 .N / [14] and on ternary quadratic forms [15]. The automorphism group of the modular curve X 0 .N / is closely related to the quotient group N 0 .N /= 0 .N /. Kenku and Momose [12] determined the full automorphism group for X 0 .N / with N ¤ 63 and Elkies [8] completed the problem by treating the case N D 63. And recently Harrison [9] corrected the statement in [12] for the case N D 108. According to their results, there are exceptional automorphisms (not coming from the elements in the quotient group N 0 .N /= 0 .N /) only for the case N D 37; 63; 108. Meanwhile, as for the quotient group N 0 .N /= 0 .N /, Atkin and Lehner [2] stated its structure without proof. But the list in [2] turned out to contain several errors and later was corrected by Akbas and Singerman [1] and Bars [4].
Let be a congruence subgroup of SL 2 .Z/ and X./ the modular curve associated to . Motivated by the importance of the normalizer of 0 .N / and the automorphism group of X 0 .N /, there have been several works on the normalizer of and the automorphism group of X./. When D 1 .N /, the group of elements of SL 2 .Z/ that are congruent to 1 0 1 modulo N , the third author and Koo [11], and Lang [13] independently determined its normalizer in PSL 2 .R/. Furthermore for the modular curve X 1 .N / WD X. 1 .N // with N square-free, Momose [16] proved that there are no exceptional automorphisms. Let .N / be the principal congruence subgroup which consists of the elements of SL 2 .Z/ that are congruent to 1 0 0 1 modulo N , and let X.N / WD X..N //. Recently Bars, Knotogeorgis, and Xarles [5] considered the automorphism group of X.N / and proved that it is equal to the group PSL 2 .Z=N Z/, which is isomorphic to the normalizer of .N / in PSL 2 .R/ modulo˙.N /.
Let .N / be the congruence subgroup of SL 2 .Z/ defined by where is a subgroup of .Z=N Z/ and we always assume that 1 2 : We note that .N / is an intermediate subgroup between 0 .N / and 1 .N /. In particular, if D .Z=N Z/ (respectively D f˙1g), then we have .N / D 0 .N / (respectively .N / D˙ 1 .N /). In this article, we are concerned with the normalizer of .N / in PSL 2 .R/ and its underlying group structures. After the preprint was ready, we recognized the results in the paper [19], which independently obtained a criterion of normalizers (compare Corollary 2.6 of that reference with our Theorem 2.1). The reference aims only for determining the normalizers, while we also investigate the structure of quotient groups in case N is square-free.
This paper is organized as follows. In Section 2 we investigate the normalizer N .N / of .N / in PSL 2 .R/. In Section 3 we find the group structures of the quotient group N .N /= .N / for square-free N when the exact sequence splits. In fact, the sequence (1) is not well-defined in general, since 0 .N / will not always be a normal subgroup of N .N /. However, 0 .N / is a normal subgroup of N .N / for square-free N . We prove that in this case, where r is the number of distinct prime divisors of N , and we give some examples of such quotient groups for nontrivial . Finally, in Section 4 we study the case of composite N , which is a product of two distinct primes and find out what happens in the cases when the exact sequence (1)  We use the following notations through this paper.

Notations.
1. For integers a; b 2 Z such that a ¤ 0, we use a k b to mean that ajb and gcd.a; b a / D 1. 2. For a prime p and an integer a such that gcd.a; p/ D 1, we let a p Á denote the Legendre symbol if p ¤ 2, and we define a 2 Á D 1 conventionally. 3. By abuse of notation, for an integer a, we use a 2 to mean that the congruence class of a belongs to . 4. For a positive integer n and an integer a prime to n, we let ord n .a/ denote the order of a modulo n, i.e. the smallest positive integer k such that a k Á 1 .mod n/.

Normalizers of intermediate congruences subgroups
From Q 2 xw Nyz D Q, we have that Qxw N Q yz D 1 and hence the following holds: Note that N a is the multiplicative inverse of a modulo Q. Now we define an isomorphism t Q W .Z=N Z/ ! .Z=N Z/ by t Q .a/ Á ( a .mod N Q /; N a .mod Q/: Since .Z=N Z/ is isomorphic to the direct product .Z=QZ/ .Z= N Q Z/ , one can show that the condition (2) holds if and only if t Q .a/ 2 . Therefore we have the following result: Taking the trace, we see that 2 D a C d . Since d is a multiplicative inverse of a in .Z=N Z/ , .a 1/ 2 Á 0 .mod N /; and hence a Á 1 .mod q/: Now consider the natural homomorphism Then ker. / D f1; q C 1; 2 q C 1; :::; . 1/ q C 1g is the cyclic group of order generated by q C 1. Thus equation (4) is equivalent to that a 2 ker. /.
In [11], the third author and Koo prove that N .N / is generated by the elements of 0 .N / and W Q for all QjjN when N ¤ 4 and D f˙1g, and its proof mainly depends on the following two conditions: If .=f˙1g/ \ ker. / D f1g holds, then Eq. (3) is the same as Eq. (6). Similarly Eq. (7) is the same as the following condition: By exactly the same arguments as those in [11], we have the following result: From Theorem 2.3, one can easily obtain the following result: Proof. If N is square-free, then defined in (5) is an isomorphism, and hence ker. / is trivial.
3 The group structures of the quotient group N .N /= .N / for square-free N : the split case In this section, we assume that N is square-free and for simplicity we assume that t Q ./ D for all QjjN . As the main result of this section, we find a condition for so that the exact sequence (1) splits. For that, we state a well-known result as follows: Put N D p 1 p 2 p r with distinct primes p 1 ; p 2 ; :::; p r . Then the exact sequence (1) splits if and only if one can find W p i for all i so that the following two conditions hold: We give necessary and sufficient conditions for the splitting property of the sequence .1/ in turn when r D 1; 2 and r 3.

The case when N D p
First, we consider the case when N is a prime p.
In this case W p is always contained in N .N /, and hence we have the following result: Theorem 3.2. Let Ä .Z=pZ/ then the sequence .1/ splits and where m D p 1 jj and D m is a dihedral group of order 2m.
Proof. One can easily check that 1 p p W p Á 2 D 1, and hence the conditions (8) and (9) hold. Since ..Z=pZ/ =/ is a cyclic group of order m, N .p/= .p/ Š Z=mZ Ì Z=2Z. Also one can easily prove that the following holds: Our result comes from this relation.

The case when N D pq
Next, we consider the case when N D pq for two distinct primes p and q.
Note that gcd.x; q/ D 1 and gcd.x 0 ; p/ D 1. Hence there exist y; z; y 0 ; z 0 2 Z such that .px/z qy D 1; .qx 0 /z 0 py 0 D 1: Then by the uniqueness of a and b modulo pq, a Á qx 02 C py 0 .mod pq/; b Á px 2 C qy .mod pq/: Then det.W p / D p and det.W q / D q and the first component of . 1 hence it is a 0 b .mod pq/, which is in since a 0 ; b; 1 2 . So the condition (9) holds. Hence the sequence (1) splits.
Conversely, suppose the sequence (1) splits. Then there exist W p D px y pqz pw (8) and (9). By a similar computations of the first components of Proof. By using the quadratic reciprocity law, we can prove that the conditions (1) and (2) are equivalent to that It is based on a having to be 1, and the same value must be attained by b if p > 2.

3.3
The case when N is a square-free integer with more than 2 prime divisors  (1) and (2). Then by the condition (1), for each i D 1; : : : ; r, there exist x i 2 Z such that For each i D 1; : : : ; r, let Then det.W p i / D p i , and the first component of . 1 which is in by condition (1). Hence the condition (8) holds. Note that if we let a 0 i 2 Z such that which is in by condition (2). Thus the condition (9) holds, and hence the sequence (1) splits. Suppose the sequence (1) splits. As explained in the proof of Theorem 3.3, we can show that the conditions (1) and (2) (1) does not split since otherwise the condition (1) in Theorem 3.6 implies that a i Á 1 .mod N / for all i , which shows that there is no b ij 2 f˙1g satisfying the condition (2) in Theorem 3.6. This is a different phenomenon from the case when r D 1 or 2 referring to Theorem 3.2 and Corollary 3.4.

Now we give some examples in the split case.
Example 3.8. Let N D 21 D 3 7 and D f˙1;˙8g Ä .Z=N Z/ . In this case, the maps t 3 and t 7 are the identity map, and so they preserve . Indeed, consists precisely of those residues that are congruent to˙1 modulo 7; hence it will make it immediately evident that is a subgroup and that it is preserved under the involutions t 3 and t 7 . If we let a D 8 and b D 1, then a and b satisfy the conditions of Theorem 3.3 when we take p D 3 and q D 7, and hence N .21/= .21/ Š .Z=3Z/ Ì .Z=2Z/ 2 : More precisely, we take OE2 D 2 1 21 11 , W 3 D 9 4 21 9 and W 7 D 7 2 21 7 . Then hOE2i D Z=3Z and hW 3 ; W 7 i D .Z=2Z/ 2 , and we can check that i.e. exactly one involution of .Z=2Z/ 2 operates tirivally on Z=3Z, and the other two operate nontrivially on Z=3Z. Thus OE2W 3 has order 6 and W 7 .OE2W 3 / D .OE2W 3 / 1 W 7 in N .21/= .21/, and hence N .21/= .21/ is isomorphic to the dihedral group D 6 of order 12.  (1) does not split. In this section we find the group structure of N .N /= .N / when N D pq with distinct primes p; q for which the exact sequence (1) does not split, and D f˙1g Ä .Z=N Z/ . If we take then one can easily check that Put w 1 D px 2 1 C qy 1 and w 2 D qx 2 2 C py 2 . Then from the fact that det.W p / D p and det.W q / D q the following holds: 1 .mod q/; (13) and hence 1 N W q W p W q W p OE1; 1 Á 1 .mod N /, which shows its triviality in the quotient group N .N /= .N /. Now as the complement of Corollary 3.4, consider the non-split cases for N D pq with distinct two primes p; q which can be divided into the following five sub-cases depending on the congruences of p and q: (i) p D 2 and q Á 5 .mod 8/.
(ii) p Á q Á 3 .mod 4/, in which case we choose p and q such that p q Á D 1.
For the non-split case for N D pq, we have the following group presentations of the quotient group N .N /= .N / where D f˙1g.
In this case, N .N /= .N / is isomorphic to the Dihedral group D q 1 of order 2.q 1/.
Proof. From (10) and Euler's criterion, we have Let d D ord q .2/. Then q 1 d should be odd. Suppose that q 1 d is even, then d j q 1 2 which is a contradiction to (14). Take a primitive root r 2 .Z=N Z/ of q so that 2 Á r q 1 d .mod q/, and put x 1 to be an integer satisfying x 1 Á r m .mod q/ where m is the integer with q 1 d C 2m D 1. Then 2x 2 1 Á r .mod q/, and ord q .2x 2 1 / D q 1. If we take for some y 1 ; z 1 , and let w 1 D 2x 2 1 C qy 1 , then W 2 2 D OEw 1 in N .N /= .N / by (11). Since ord N .w 1 / D ord q .2x 2 1 / D q 1 and w q 1 2 1 Á 1 .mod N /, the order of W 2 in N .N /= .N / is equal to q 1. We recall that we work modulo D f˙1g, so that the order q 1 of w 1 means an order of q 1 2 of W 2 2 , whence an order of q 1 of W 2 itself.
Since w 1 generate .Z=N Z/ , N .N /= .N / can be generated by W 2 and W q . From (13) we know that Then the map a 7 ! W 2 and b 7 ! W q can be extended to a unique homomorphism from G to N .N /= .N / because W 2 and W q satisfy all the relations in G if we replace a and b by W 2 and W q . Clearly, the order jGj of G is equal to 2.q 1/ which is the same as jN .N /= .N /j. Thus G is isomorphic to N .N /= .N /.  Proof. Let us consider the case when p Á q Á 3 .mod 4/ by choosing the notation for p and q such that p q Á D 1. By the same reason as in the proof of Theorem 4.1, q 1 d 1 is odd where d 1 WD ord q .p/. Thus we can take a primitive root r 2 .Z=N Z/ of q and an integer x 1 so that px 2 1 Á r .mod q/, and hence ord q .px 2 1 / D q 1. Take for some y 1 ; z 1 , and let w 1 D px 2 1 C qy 1 . By the Chinese Remainder Theorem, we know that and hence ord N .w 1 / D lcm.ord q .px 2 1 /; ord p .qy 1 // D lcmfq 1; 2g D q 1 by (12). From (12), we also know that w .mod p/, and put x 2 to be an integer satisfying x 2 Á s m 2 .mod p/ where m 2 is the integer with q 1 d 1 C2m 2 D 2. Then px 2 2 Á s 2 .mod p/, and ord p .qx 2 2 / D p 1 2 . In fact, we cannot take x 2 so that ord p .qx 2 2 / D p 1. Now we take W q D qx 2 y 2 N qz 2 ! for some y 2 ; z 2 , and let w 2 D qx 2 2 C py 2 .
From (12), we know ord N .w 2 / D lcm. p 1 2 ; 2/; hence it is equal to p 1 because p 1 2 is odd by our assumption about p. From (12) again, From (13) we know that .W q W p / 2 D 1. For u 2 .Z=N Z/ , by the action of the Atkin-Lehner involution W q on .Z=N Z/ via the t q operator which is in correspondence with conjugation by the W q on 0 .N / modulo 1 .N /, we have the following: W q OEuW 1 q OEuOE1; 1 Á 1 .mod q/: Thus W q OEw 1 W 1 q OEw 1 OE1; 1 Á 1 .mod q/, and clearly W q OEw 1 W 1 q OEw 1 OE1; 1 Á 1 .mod p/ because w 1 Á 1 .mod p/. Therefore, W q W 2 p W 1 q W 2 p D 1 holds. Let G D ha; b j a q 1 D b 2.p 1/ D .ba/ 2 D ba 2 b 1 a 2 D 1i. Then there is a unique homomorphism from G to N .N /= .N / determined by the map a 7 ! W p and b 7 ! W q . From the relations in G, we know that and hence every element of G can be expressed as a i b j with 0 Ä i < q 1 and 0 Ä j < 2.p 1/. Thus the order jGj of G is less than or equal to 2.p 1/.q 1/. Since jN .N /= .N /j Ä jGj and N .N /= .N / is of order 2.p 1/.q 1/, G is isomorphic to N .N /= .N /.
Finally consider the isomorphism. Let G D ha; b j a 2.q 1/ D b 2.p 1/ D .ba/ 2 D ba 2 b 1 a 2 D a q 1 b p 1 D 1i. From the first four relations, we know that any element of G can be expressed as a i b j with 0 Ä i < 2.q 1/ and 0 Ä j < 2.p 1/. However due to the relation a q 1 b p 1 D 1, it can be boiled down to be a i b j with 0 Ä i < q 1 and 0 Ä j < 2.p 1/. Thus jGj Ä 2.p 1/.q 1/, and hence the result follows by the same argument as in the proof of Theorem 4.2.   Suppose it can be generated by two elements, say˛andˇ. In the sequence (1), they map to generators of N .N /= 0 .N / under the map g. Since N .N /= 0 .N / is the Klein four-group and it is generated by W p and W q , we can assume˛andˇare same as W p and W q . In fact, if one of g.˛/ and g.ˇ/ is equal to W N , say g.ˇ/, we can take˛and˛ˇas generators of N .N /= .N /, and then g.˛/ and g.˛ˇ/ are equal to W p and W q . However, W p and W q cannot generate N .N /= .N / as shown in the proof of Theorem 4.4.
(2) Consider the non-split cases for N D pq and let Ä .Z=N Z/ . Suppose t Q ./ D for all QjjN . Then one can obtain a group presentation of N .N /= .N / by using the methods used in the proofs of Theorem 4.1, Theorem 4.2, Theorem 4.3, and Theorem 4.4. Since N .N / D N f˙1g .N /, there is a natural projection N f˙1g .N /= f˙1g .N / ! N .N /= .N /, and hence we can take the same generators of N .N /= .N / as of N f˙1g .N /= f˙1g .N /. Thus it suffices to change the order of generators and the relations between them for getting a group presentation.
We give an example in the non-split case which shows the orders that generators W p and W q can have and their relations depending on . then w 1 D 19 and w 2 D 27. Since w 3 1 Á 1 .mod 35/, the order of W 5 in N 1 .N /= 1 .35/ is 6, and since w 2 2 Á 6 .mod 35/, the order of W 7 in N 1 .N /= 1 .35/ is 4. Consider the group G 1 D ha; b j a 6 D b 4 D .ba/ 2 D ba 2 b 1 a 2 D 1i. Then W 5 and W 7 satisfy all the relations of G 1 in N 1 .N /= 1 .35/ if we replace a and b by W 5 and W 7 . Clearly jG 1 j D 24 which is the same as the order of N 1 .N /= 1 .35/, and hence N 1 .N /= 1 .35/ is isomorphic to G 1 .
Since w 1 2 2 , the order of W 5 in N 2 .N /= 2 .35/ is 2, and since w 4 2 Á 1 .mod 35/, the order of W 7 in N 2 .N /= 2 .35/ is 8. Consider the group G 2 D ha; b j a 2 D b 8 D .ba/ 2 D 1i. Since the order of a is 2, we can remove the relation ba 2 b 1 a 2 D 1. Then one can easily confirm that N 2 .N /= 2 .35/ is isomorphic to G 2 which is a dihedral group of order 16.