On Conjugacy Classes of Congruence Subgroups of Psl(2, R)

London Mathematical Society ISSN 1461–1570

ON CONJUGACY CLASSES OF CONGRUENCE SUBGROUPS OF PSL(2, R).

C. J. CUMMINS

Abstract

Let G be a subgroup of PSL(2, R) which is commensurable with PSL(2, Z). We say that G is a congruence subgroup of PSL(2, R) if G contains a principal congruence subgroup Γ(N) for some N. An algorithm is given for determining whether two congruence subgroups are conjugate in PSL(2, R). This algorithm is used to determine the PSL(2, R) conjugacy classes of congruence subgroups of genus-zero and genus-one. The results are given in a table.

1. Introduction.

The principal congruence subgroups of Γ = SL(2, Z) are deﬁned as follows: a b a b 1 0 Γ(N) = { ∈ SL(2, ) | ≡ (mod N)}, c d Z c d 0 1

for N = 1, 2, 3,... . The image of Γ(N) in Γ = PSL(2, Z) = SL(2, Z)/{±12} is denoted by Γ(N). A subgroup of Γ is called a congruence subgroup if it contains some Γ(N). The level of a congruence subgroup G of Γ is the smallest N such that Γ(N) is contained in G. It is natural to extend this definition to subgroups of PSL(2, R). A subgroup of PSL(2, R) is commensurable with Γ if G ∩ Γ has finite index in both G and Γ. In this case, we say that G is a congruence subgroup if it contains some Γ(N). It was originally conjectured by Rademacher that there are only finitely many congruence subgroups of given genus in Γ. This problem was studied by several authors. Cox and Parry [3,4 ] gave effective bounds and computed a list of genus- zero congruence subgroups of Γ. Independently, Thompson [9] showed that, up to conjugation, there are only finitely many congruence subgroups of PSL(2, R) of fixed genus. Motivated by Thompson’s result, and using bounds due to Zograf [10], a list of congruence subgroups of PSL(2, R) of genus-zero and genus-one was found in [5] and fundamental domains for these groups were found in [6]. The strategy of [5] was, following Thompson, to use a result of Helling which states that every group commensurable with Γ is conjugate to a subgroup of a cer- tain class of maximal discrete subgroups of PSL(2, R). Thus to find all congruence subgroups of genus-zero and genus-one, it suffices to compute all the conjugacy classes of such subgroups inside these “Helling Groups”. Further work, however, is required to find the resulting PSL(2, R) conjugacy classes. In this paper we supply the necessary algorithm and the resulting conjugacy classes are given in Table1.

Work partly supported by NSERC Received 12 September 2007, revised 10 November 2008; published 21 November 2009. 2000 Mathematics Subject Classiﬁcation 11F06, 11F22, 20H10, 30F35 c 2009, C. J. Cummins

Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.126, on 02 Oct 2021 at 04:58:39, subject to the Cambridge Core terms of use, availableLMS at https://www.cambridge.org/core/terms J. Comput. Math. 12 (2009). https://doi.org/10.1112/S1461157000001510264–274 On Conjugacy Classes of Congruence Subgroups of PSL(2, R).

In Section 2 background deﬁnitions are given. The algorithm for the computation is described in Section 3 and some concluding comments are given in Section 4.

2. Background

If G is a discrete subgroup of PSL(2, R) which is commensurable with Γ, then G acts on the extended upper half plane H∗ = H ∪ Q ∪ {∞} by fractional linear transformations and the genus of G is deﬁned to be the genus of the corresponding Riemann surface H∗/G. From a computational point of view, it is easier to work with subgroups of Γ and SL(2, R), rather than Γ and PSL(2, R). There is a 1-1 correspondence between the subgroups of PSL(2, R) and the subgroups of SL(2, R) which contain −12, where 12 is the identity of SL(2, R). Thus in [5] and in this paper we consider subgroups of SL(2, R) which contain −12. If G is subgroup of SL(2, R) and G is its image in PSL(2, R), then when we refer to geometric invariants such as the genus or cusp number of G, we mean the corresponding invariants of G. We recall the following deﬁnition.

Definition 2.1. a b Γ (f)+ = {e−1/2 ∈ SL(2, ) a, b, c, d, e ∈ , e || f, e | a, e | d, 0 c d R Z f | c, ad − bc = e}, where e || f means e | f and gcd(e, f/e) = 1.

By the following theorem, the study of groups commensurable with Γ is essen- + tially the study of subgroups of the groups Γ0(f) , where f is a square-free integer.

Theorem 2.2. (Helling [7]. See also Conway [2]). If G is a subgroup of SL(2, R) + which is commensurable with Γ, then G is conjugate to a subgroup of Γ0(f) for some square-free f.

As noted in the introduction, we many deﬁne the notion of a congruence subgroup + for subgroups of Γ0(f) using the same deﬁnition as for subgroups of Γ. However, it turns out to be equivalent, and more convenient, to introduce, following Thompson, the appropriate generalization of Γ(N). Recall that a b Γ (N) = { ∈ SL(2, ) | c ≡ 0 (mod N)}. 0 c d Z

Definition 2.3. G(n, f) = Γ0(nf) ∩ Γ(n).

+ Note that G(n, f) is a normal subgroup of Γ0(f) and that G(n, 1) = Γ(n).

+ Definition 2.4. Call a subgroup G of Γ0(f) a congruence subgroup if G(n, f) ⊆ G for some n.

The following proposition shows that Deﬁnition 2.4 is equivalent, for subgroups + of Γ0(f) , to the standard deﬁnition of a congruence subgroup given in the introduction. Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.126, on 02 Oct 2021 at 04:58:39, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms265 . https://doi.org/10.1112/S1461157000001510 On Conjugacy Classes of Congruence Subgroups of PSL(2, R).

Proposition 2.5. A subgroup G of SL(2, R) contains a subgroup G(n, f) for some n and some f if and only if G contains Γ(m) for some m.

Proof. Suppose G contains Γ(m). Then Γ(m) = G(m, 1) and so G contains G(m, 1). Conversely, if G contains G(n, f), then we have Γ(nf) ⊆ Γ0(nf)∩Γ(n) and so Γ(nf) is contained in G. We may also extended the notion of the level of a congruence subgroup.

+ Definition 2.6. If G is a congruence subgroup of Γ0(f) , then let n = n(G, f) be the smallest positive integer such that G(n, f) ⊆ G. We call n(G, f) the level of G.

+ If f = 1, then Γ0(f) = Γ and this deﬁnition of level coincides with the usual + deﬁnition, since G(n, 1) = Γ(n). However, if G is a subgroup of Γ0(f) and f 6= 1, then it is not necessarily the case that n = n(G, f) is the smallest n such that Γ(n) + is contained in G. Moreover, if G is a congruence subgroup of both Γ0(f1) and + Γ0(f2) , with f1 6= f2, then n(G, f1) and n(G, f2) are not necessarily equal.

3. The Algorithm

By Helling’s Theorem, every subgroup of SL(2, R) which is commensurable with + SL(2, Z) is conjugate to a subgroup of Γ0(f) for some square-free, positive integer f. Thus to tabulate all conjugacy classes of congruence subgroups of genus-zero and + genus-one, it is sufficient to list the genus-zero and genus-one subgroups of Γ0(f) + for the (finite) set of values of f such that Γ0(f) is genus-zero or genus-one. This was done in [5]. However, in [5] the groups were found up to conjugacy in each + Γ0(f) . If a class occurred for two different values of f, then this was recorded in the tables, but the full SL(2, R) conjugacy classes were not computed. In this section we give an algorithm to find these classes. Table1 records the results. + + Suppose that K1 and K2 are subgroups of Γ0(f1) and Γ0(f2) respectively, and that K1 and K2 represent two of the classes in Table 2 of [5]. We want to test K1 and K2 for conjugacy in SL(2, R). It would initially seem that we have to consider the case f1 6= f2. It turns out, however, that we do not need to consider such cases, since we can find a square-free integer fmin such that K1 and K2 are + both subgroups of Γ0(fmin) .

+ Proposition 3.1. Suppose K1 and K2 are congruence subgroups of Γ0(f1) and + Γ0(f2) respectively. If K1 and K2 are conjugate in SL(2, R), then there is some + positive, square-free integer fmin, such that K1 and K2 are subgroups of Γ0(fmin) .

−1 m Proof. First note that if m K1m = K1 = K2 for some m in SL(2, R), then m is a multiple of a primitive integer matrix, since K1 and K2 are commensurable with ∗ SL(2, Z). (This follows easily from the fact that K1 and K2 both have Q = Q∪{∞} as the set fixed by parabolic elements and so m must map Q∗ to Q∗. ) Next, for any element m of SL(2, R) which is equal to λA, where A is a primitive integer matrix, we define the normalized determinant hmi to be the determinant of A. It is not difficult to verify that hmi is well defined. If B ∈ SL(2, Z), then hmBi = hmi. So the set of normalized determinants of the elements of K1 is finite, since K1 contains some Γ(N) with finite index. Let Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.126, on 02 Oct 2021 at 04:58:39, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms266 . https://doi.org/10.1112/S1461157000001510 On Conjugacy Classes of Congruence Subgroups of PSL(2, R).

{e1, . . . , ek} be the set of normalized determinants of the elements of K1. These + must be square-free, since K1 is contained in Γ0(f1) , so the normalized determinants divide f1. Then fmin = lcm(e1, ..., ek) is the smallest square-free integer such + that K1 is contained in Γ0(fmin) . The matrices of K2 have the same normalized determinants as K1 up to squares, since they are obtained by conjugating by a multiple of a primitive integer matrix. But K2 is contained in some Helling group and so the normalized determinants of its elements must also be square-free. So the set of its normalized determinants is also {e1, . . . , ek}. Thus K2 is also contained in + Γ0(fmin) .

So to find all SL(2, R) conjugates of the congruence subgroups of interest, we can + first find all SL(2, R) conjugates within Γ0(f) for each f. Then the full SL(2, R) classes are obtained using the known intersections of classes for different values of f already computed in Table 2 of [5]. The area of a fundamental domain is invariant under conjugation in SL(2, R) + and so if K1 and K2 are subgroups of Γ0(f) which are conjugate in SL(2, R), they + have the same index in Γ0(f) . Also, the number of classes of elliptic fixed points of given order and the cusp number are invariant under SL(2, R) conjugation. Thus when testing K1 and K2 for SL(2, R) conjugacy these invariants are first tested for equality. Also, by [5] Corollary 4.8, the set of primes dividing the levels of K1 and K2 are equal if the groups are conjugate in SL(2, R). Given two groups which pass these initial tests, we now want to bound the + number of conjugating matrices m which have to be considered. As Γ0(f) acts + transitively on Q ∪ {∞}, by replacing K2 by a conjugate group in Γ0(f) we can p q arrange for m to have the form m = λ , with p, q, r integers such that 0 r gcd(p, q, r) = 1, pr > 0 and λ = (pr)−1/2. By [5] Proposition 4.7 we have the following constraints: p | `1, r | (`1/p) gcd(f, p), 0 6 q < p and also `2 | gcd(pr, `1)`1, where ì is the level of Ki, i = 1, 2. This assumes that we fix K1 and conjugate K2 + + in Γ0(f) . If we also allow conjugations of K1 in Γ0(f) , then we can also impose 1 t −1 0 q < gcd(p, r), since we can then conjugate K by . As Km = K we 6 1 0 1 2 1 can apply the same arguments to get the conditions: r | `2 and p | (`2/r) gcd(f, r), + but again this assumes we allow a conjugation of K1 in Γ0(f) . This produces a finite list of possible conjugating matrices m. + To summarize: given K1 and K2, subgroups of Γ0(f) , we first test the obvious invariants for equality. Then, given the levels of the two groups, we can find a finite list of possible conjugating matrices with the property that, if K1 and K2 are 0 conjugate in SL(2, R), then there is some m in the list and some conjugate K1 of K1 + 0 + 0 in Γ0(f) and some conjugate K2 of K2 in Γ0(f) such that m conjugates K1 to 0 0 0 K2. Since there are only finitely many conjugates K1 and K2, this leads to a finite number of cases to test and so we have an algorithm for finding all the SL(2, R) conjugacy classes of congruence subgroups of a given genus. Since the groups are infinite there is still the problem of giving an algorithm to perform the test for equality. This can be done as follows. It is not difficult to p q verify that if m = λ as above, then mG(pr` , f)m−1 ⊆ G(` , f) ⊆ K . Let 0 r 1 1 1 −1 `3 = lcm(pr`1, `2), then since G(`3, f) ⊆ G(pr`1), we have mG(`3, f)m ⊆ K1. Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.126, on 02 Oct 2021 at 04:58:39, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms267 . https://doi.org/10.1112/S1461157000001510 On Conjugacy Classes of Congruence Subgroups of PSL(2, R).

−1 Moreover, G(`3, f) ⊆ G(`2, f) ⊆ K2. Thus to test if mK2m is equal to K1, it −1 is sufficient to test if msm is in K1, for all s ∈ S for some set of generators S of K2 over G(`3, f). (The fact that G(`3, f) and G(`2, f) are normal subgroups of K2 and that K2/G(`2, f) was constructed as a permutation group in [5] simplifies the computation of the cosets). Thus we are finished if we have an algorithm for + testing when an element g in Γ0(f) is in K1, but as the group K1/G(`1, f) was + constructed as a subgroup of Γ0(f) /G(`1, f) in [5], we can simply test to see if + the image of g in Γ0(f) /G(`1, f) is in K1/G(`1, f). In summary we have the following algorithm.

The Algorithm

Input: A square-free integer f, and two congruence subgroups K1 and K2 of + Γ0(f) , of levels `1 and `2 respectively.

Output: Return true if K1 and K2 are conjugate in SL(2, R) and false otherwise.

• I1 ← invariants of K1

• I2 ← invariants of K2

• If I1 6= I2 then return false • Else + • C1 ← conjugates of K1 in Γ0(f) + • C2 ← conjugates of K2 in Γ0(f) • M ← list of possible conjugating matrices • For K in C2 p q • For m = in M 0 r

• `3 ← lcm(pr`1, `2). • S ← a set of generators of K over the normal

subgroup G(`3, f). • S0 ← mSm−1 0 + • If S is not a subset of Γ0(f) then move to the next m • For each L in C1 0 + • If the image of S in Γ0(f) /G(`1, f) is contained in L/G(`1, f) then return true

• If after testing all elements of C2 we have not returned true then return false

4. Conclusions In the previous section an algorithm was given for determining whether two + subgroups of Γ0(f) are conjugate in SL(2, R). Table 2 of [5] records when the + conjugacy classes of subgroups Ci of Γ0(fi) , i = 1, 2, are such that f1 6= f2 and C1 ∩ C2 is not empty. As explained in the last section, combining these two pieces Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.126, on 02 Oct 2021 at 04:58:39, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms268 . https://doi.org/10.1112/S1461157000001510 On Conjugacy Classes of Congruence Subgroups of PSL(2, R).

of information yields the SL(2, R) conjugacy classes of congruence subgroups of genus-zero and genus-one. The algorithm of the previous section was programmed in MAGMA [1]. The + results are presented in Table1. Each entry in this table is a list of the Γ 0(f) conjugacy classes from Table 2 of [5] which are subsets of one SL(2, R) conjugacy + class. Cases with a single Γ0(f) conjugacy class have been omitted. The notation genus is as in Table 2 of [5]: each class has a label level(letter)f , where f is a square- + free integer such that the group is contained in Γ0(f) ; level is the level of the group with respect to this f (as defined in Definition 2.4); genus is its genus; and letter is a letter labelling the group amongst all groups of the same level and genus. Sebbar [8] has shown that there are 15 SL(2, R) conjugacy classes of torsion- free congruence subgroups of genus-zero. As a test of the results of this paper, using the additional data from Table 2 of [5], we also find 15 SL(2, R) conjugacy classes of torsion-free, genus-zero congruence subgroups. Moreover, these give rise to 33 classes of subgroups of the modular group, again in agreement with Sebbar’s results.

References 1. Wieb Bosma, John Cannon and Catherine Playoust, ‘The Magma algebra system. I. The user language.’ J. Symbolic Comput. 24 (1997) 235–265. Computational algebra and number theory (London, 1993). 269

2. J. H. Conway, ‘Understanding groups like Γ0(N).’ ‘Groups, difference sets, and the Monster (Columbus, OH, 1993),’ (de Gruyter, Berlin, 1996), vol. 4 of Ohio State Univ. Math. Res. Inst. Publ. pp. 327–343, pp. 327–343. 265 3. David A. Cox and Walter R. Parry, ‘Genera of congruence subgroups in Q-quaternion algebras.’ J. Reine Angew. Math. 351 (1984) 66–112. 264 4. David A. Cox and Walter R. Parry, ‘Genera of congruence subgroups in Q-quaternion algebras.’ Unabridge preprint. 264 5. C. J. Cummins, ‘Congruence subgroups of groups commensurable with PSL(2, Z) of genus 0 and 1.’ Experiment. Math. 13 (2004) 361–382. 264, 265, 266, 267, 268, 269 6. C. J. Cummins,‘fundamental domains for congruence subgroups of genus 0 and 1, .’ submitted for publication . 264 7. Heinz Helling, ‘Bestimmung der Kommensurabilitätsklasse der Hilbertschen Modulgruppe.’ Math. Z. 92 (1966) 269–280. 265 8. Abdellah Sebbar, ‘Classification of torsion-free genus zero congruence groups.’ Proc. Amer. Math. Soc. 129 (2001) 2517–2527 (electronic). 269 9. J. G. Thompson, ‘A finiteness theorem for subgroups of PSL(2, R) which are commensurable with PSL(2, Z).’ ‘The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979),’ (Amer. Math. Soc., Providence, R.I., 1980), vol. 37 of Proc. Sympos. Pure Math. pp. 533–555, pp. 533–555. 264 10. P. Zograf, ‘A spectral proof of Rademacher’s conjecture for congruence subgroups of the modular group.’ J. Reine Angew. Math. 414 (1991) 113–116. 264

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Table 1: PSL(2, R) conjugacy classes of congruence subgroups of genus-zero and genus-one.

Classes Classes Classes Classes

0 0 0 0 0 0 0 0 0 2B1 1B2 8C1 4D2 14C1 2D7 4H3 4I3 8C3 0 0 2L6 4I6 0 0 0 0 0 0 0 2C1 4B1 2D2 8I1 4K2 14D1 14C7 0 0 0 4J3 8D3 4J6 0 0 0 0 0 0 0 3B1 1B3 8J1 8K1 16E1 15C1 3C5 0 0 0 0 0 0 8L2 8U2 8V2 4K3 8F3 4M6 0 0 0 0 0 3D1 9B1 3D3 15D1 15E5 0 0 0 0 0 8L1 16F1 8W2 6E3 3C6 0 0 0 0 4C1 2C2 16C1 8J2 0 0 0 0 8M1 8N2 7A3 1C21 0 0 0 0 0 4E1 8D1 2F2 18C1 6F3 0 0 0 0 0 0 4F2 8P1 8Q1 16H1 7B3 7B21 0 0 0 0 0 16I1 32A1 8AB2 24A1 12E2 0 0 0 0 0 0 0 0 0 4F1 8E1 4G2 8Y2 8Z2 16E2 12F3 24B3 12D6 0 0 26A1 26C13 0 0 0 0 0 0 0 0 4G1 8H1 16D1 8R1 8AA2 12G3 12H3 12I3 0 0 0 0 0 0 0 4J2 8M2 28A1 28C7 24A3 12A6 12B6 0 0 0 9C1 3C3 12C6 0 0 0 0 5B1 1B5 30A1 30C5 0 0 0 0 0 9H1 9I1 27A1 12N3 6F6 0 0 0 0 0 0 5D1 5C5 9F3 9G3 36A1 12O3 0 0 0 12R3 12S3 24D3 0 0 0 0 0 0 0 0 0 0 5G1 25A1 5H5 9J1 9E3 48A1 24C2 24E3 12M6 12N6 0 12O6 0 0 0 0 0 0 0 5H1 25B1 5I5 10B1 10D5 3B2 1C6 0 0 0 21A3 21B3 21B21 0 0 0 0 0 0 0 0 6C1 2D3 10C1 5D2 2C5 3H2 9B2 3E6 0 0 0 1E10 2B5 1B10 0 0 0 0 6D1 3C2 5B2 1D10 0 0 0 0 0 0 10F1 5G2 10N5 2D5 4B5 2G10 0 0 0 0 0 0 6F1 3F2 2F3 5C10 6D2 2E6 0 0 0 1E6 3B5 1B15 0 0 0 0 0 10G1 20A1 10D2 6M2 2M6 0 0 0 0 0 0 0 6G1 3G2 10Q5 20E5 10G10 4C5 2D10 0 0 0 9F2 9G2 9C6 0 0 0 0 0 0 0 6H1 12D1 6H2 12B1 12C3 4E5 2I10 0 0 10B2 2F10 0 0 0 0 0 0 0 6I1 12E1 6L2 12C1 6E2 6B5 2D15 0 0 0 0 0 2H3 4G3 2K6 10E2 10J10 0 0 0 0 12H1 6N2 8A5 4D10 0 0 0 0 0 6J1 18D1 6G3 12D2 4E6 0 0 0 0 0 0 12I1 12J1 24B1 10F5 10G5 5A10 0 0 0 0 0 0 0 0 6K1 18F1 3I2 12G2 12H2 12P3 12I2 12L6 0 0 0 0 0 0 0 0 0 9E2 6H3 3G6 12Q3 24C3 12I6 10H5 20A5 10B10 0 0 0 12J6 2C3 1D6 0 0 0 0 0 0 6L1 12G1 6O2 10I5 20B5 10A10 0 0 0 0 0 13A1 1B13 2E3 4B3 2H6 0 0 0 0 0 7C1 1B7 10O5 20G5 10F10 0 0 0 0 0 13B1 13D13 2G3 4C3 2G6 0 0 0 0 0 7G1 7B7 10P5 20F5 10E10 0 0 0 0 13C1 13E13 4D3 2F6

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Classes Classes Classes Classes

0 0 0 1 1 1 1 1 1 1 15C5 15D5 5A15 8G1 16G1 8AC2 12L1 6L2 16D1 8N2 1 8V2 0 0 0 1 1 1 1 15G5 15H5 15A15 12M1 6K2 16F1 8W2 1 1 1 8H1 16I1 8AE2 0 0 0 1 1 1 1 20H5 20I5 10I10 12N1 6J2 16J1 8AB2 1 1 1 8I1 16H1 8AD2 0 0 1 1 1 1 1 1 2C7 1B14 12O1 24G1 12N2 16K1 32C1 16Q2 1 1 1 8K1 16M1 32E1 0 0 0 1 1 1 1 1 1 1 1 1 2E7 4A7 2D14 8AH2 8AI2 16W2 12P1 36B1 12K3 16L1 32D1 16R2

0 0 1 1 1 1 1 1 1 6B7 3B14 9A1 3A3 12Q1 24I1 6P2 17A1 1A17 1 1 1 12V2 4H3 8J3 0 0 1 1 1 1 1 1 1 14B7 7C14 9D1 27A1 3B3 2H6 4U6 17B1 17A17 1 9D3 0 0 1 1 1 1 2B11 1B22 12S1 12O3 17C1 17B17 1 1 9E1 9B3 0 0 1 1 1 1 3A11 1B33 12U1 6R2 18D1 6E3 1 1 1 9J1 27C1 9H3 0 0 1 1 1 1 1 1 4B11 2C22 9I3 12V1 24K1 12Y2 18E1 9D2

0 0 1 1 1 1 1 1 1 2C15 1E30 10A1 2B5 12W1 24L1 12AC2 18I1 9I2 1 1 1 12AD2 12T3 24O3 0 0 1 1 1 1 1 1 2E15 1F30 10B1 5A2 12U6 12V6 18K1 6L3

0 0 0 1 1 1 1 1 1 2F15 4A15 2D30 10D1 10C5 14C1 7E2 19A1 1A19

0 0 1 1 1 1 1 1 2A23 1B46 10F1 5C2 14D1 7B2 19B1 19A19

1 1 1 1 1 1 1 1 1 1 1 1 6C1 12B1 6C2 10G1 20E1 10F2 14E1 7F2 2A7 20B1 20C1 20E5 1 1 1 1 2D5 4D5 2H10 1B14 1 1 1 1 1 6D1 18C1 6D3 20D1 4C5 1 1 1 1 1 10I1 5D2 14L1 7N2 14C7 1 1 1 1 1 1 1 6E1 12K1 6M2 7B14 20F1 10E2 4E5 1 1 1 1 10K1 20J1 10L2 2G10 1 1 1 1 1 1 1 1 6F1 12T1 18J1 10F5 20W5 10L10 15B1 5B3 1 1 1 1 1 36C1 6Q2 18F2 20I1 20V5 1 1 1 1 1 1 1 1 6K3 12P3 6I6 11A1 1A11 15C1 5C3 3B5 1 1 1 1 1B15 20K1 20L1 40A1 1 1 1 1 1 1 1 1 7B1 49A1 7A7 11D1 11A11 40B1 20G2 20H2 1 1 1 1 1 15E1 3C5 20AA5 40G5 20L10 1 1 1 1 8A1 4B2 12A1 4B3 1 1 1 1 1 1 15G1 5F3 15E5 20M1 10K2 20Y5 1 1 1 1 1 1 1 8B1 16A1 4D2 12C1 6E2 5A15 10K10 1 8H2 1 1 1 1 1 1 1 1 12F1 6I2 4C3 15H1 15F5 21A1 21B1 21A7 1 1 1 8C1 4E2 2G6 1 1 1 1 1 1 15I1 15D3 15I5 21C1 7A3 3C7 1 1 1 1 1 1 1 8D1 16C1 8T2 12I1 4G3 15A15 1B21

1 1 1 1 1 1 1 1 1 1 1 8F1 16E1 32A1 12J1 24E1 12H2 16B1 8G2 21I1 21C3 21C7 1 1 1 1 4F2 8X2 16I2 21B21

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Classes Classes Classes Classes

1 1 1 1 1 1 1 1 1 24F1 12F2 13A2 1B26 8L3 4R6 3D5 9B5 3B15

1 1 1 1 1 1 1 1 24J1 12X2 13B2 13B26 8M3 4T6 4B5 2C10

1 1 1 1 1 1 1 1 1 1 26A1 2C13 15D2 3F10 8N3 16F3 8P6 4F5 8B5 4K10

1 1 1 1 1 1 1 1 1 1 26B1 26F13 15E2 5C6 8O3 16G3 8Q6 4G5 8C5 4J10

1 1 1 1 1 1 1 1 1 27B1 9C3 17A2 1C34 8P3 16H3 8V6 6D5 3D10

1 1 1 1 1 1 1 1 1 30D1 6E5 17B2 17C34 10B3 5B6 6H5 3G10 2F15 1 1F30 1 1 1 1 1 1 1 1 30E1 30K5 20B2 20C2 20C10 12A3 24B3 12A6 1 1 7A5 1A35 1 1 1 1 1 1 1 32B1 16G2 20D2 4I10 12D3 24A3 12B6 1 1 7B5 7A35 1 1 1 1 1 1 1 1 1 39A1 39B1 39A13 20I2 20J2 20P10 12H3 24G3 12H6 1 1 1 12K6 8A5 4F10 1 1 1 1 1 1 52A1 52B1 52B13 24J2 24L2 8M6 1 1 1 1 1 12I3 24H3 12I6 8E5 4N10 1 1 1 1 1 1 1 5E2 25A2 5A10 24K2 24M2 24E6 12J6 1 1 1 10D5 20H5 10E10 1 1 1 1 1 1 1 1 6B2 2B6 24N2 24O2 24H6 12L3 24I3 24J3 1 1 1 1 1 1 24K3 12M6 12N6 10E5 20I5 10F10 1 1 1 1 1 1 6O2 18B2 6E6 24P2 8L6 12O6 1 1 16A5 8F10 1 1 1 1 1 1 1 7A2 1A14 4A3 2D6 12M3 24L3 48A3 1 1 1 1 1 12L6 24G6 20A5 20B5 10A10 1 1 1 1 7G2 7A14 4D3 2E6 1 1 1 1 1 1 12Q3 24N3 12T6 20AB5 40H5 20M10 1 1 1 1 1 9A2 3C6 4E3 8C3 2F6 1 1 1 1 1 1 1 4M6 12S3 24P3 12W6 20AC5 40I5 20K10 1 1 9B2 3D6 1 1 1 1 1 1 1 1 4F3 8D3 4N6 13A3 1A39 20J5 20Q5 10I10 1 1 9G2 9D6 1 1 1 1 1 1 1 4I3 8K3 16E3 13B3 13F39 20K5 10C10 1 1 1 1 9H2 9E6 4V6 8O6 1 1 1 1 1 14A3 2D21 20L5 20O5 10H10 1 1 1 1 1 9J2 27A2 9H6 5A3 1A15 1 1 1 1 1 14B3 2E21 20M5 20N5 10J10 1 1 1 1 10B2 2B10 6C3 3B6 1 1 1 1 18J3 9G6 20P5 10D10 1 1 1 1 1 11A2 1A22 6F3 12F3 6D6 1 1 1 1 1 1 21A3 21B3 21A21 20R5 40C5 20D10 1 1 1 1 1 11B2 11A22 6H3 12G3 6C6 1 1 1 1 1 1 21D3 21E3 21D21 20S5 40D5 20E10 1 1 1 1 12E2 4K6 6J3 3E6 1 1 1 1 1 1 42A3 42B3 42B21 20X5 20Z5 10M10 1 1 1 1 12U2 4S6 8E3 4J6 1 1 1 1 1 1 2C5 4A5 2E10 30C5 30D5 10A15 1 1 1 1 12W2 4X6 8F3 4E6

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Classes Classes

1 1 1 1 1 1 30I5 30J5 10C15 33A11 33B11 33A33

1 1 1 1 1 40A5 40B5 20B10 2B13 1A26

1 1 1 1 1 40E5 40F5 20J10 3A14 1B42

1 1 1 1 5A6 1C30 3D14 1F42

1 1 1 1 1 5D6 1D30 2E15 4C15 2M30

1 1 1 1 1 5I6 5C30 2G15 4B15 2N30

1 1 1 1 1 7A6 1C42 2H15 4G15 2R30

1 1 1 1 7B6 7A42 4D15 2I30

1 1 1 1 1 2B7 4B7 2E14 4E15 2H30

1 1 1 1 3A7 1A21 4H15 2Q30

1 1 1 1 4C7 2C14 6D15 3C30

1 1 1 1 4D7 2D14 10D15 5D30

1 1 1 1 1 4F7 8A7 2G14 2A17 1B34 1 4E14 1 1 3B17 1A51 1 1 1 6E7 12A7 6E14 1 1 2B19 1A38 1 1 6F7 3E14 1 1 2C21 1A42 1 1 1 14D7 28F7 14C14 1 1 3B22 1A66 1 1 1 28A7 28B7 28C7 1 1 1 1 1 1 56A7 28A14 28B14 2C23 4A23 2C46 1 28C14 1 1 3A23 1A69 1 1 3A10 1B30 1 1 2C31 1A62 1 1 1 2B11 4B11 2B22 1 1 2C33 1C66 1 1 1 2C11 4A11 2C22 1 1 2C35 1C70 1 1 1 4D11 8C11 4C22 1 1 3A35 1C105 1 1 5A11 1A55 1 1 2E39 1B78 1 1 5B11 5A55 1 1 2A47 1A94 1 1 6A11 2D33 1 1 2C55 1A110

Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.126, on 02 Oct 2021 at 04:58:39, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms273 . https://doi.org/10.1112/S1461157000001510 On Conjugacy Classes of Congruence Subgroups of PSL(2, R).

C. J. Cummins [email protected] Department of Mathematics and Statistics Concordia University Montr´eal Qu´ebec Canada