The finite subgroups of maximal arithmetic kleinian groups

Arithmetic Fuchsian Groups of Genus Zero

“…for some fixed k. But then, the first term on the right hand side of (8) (8) ; 0) 3,13 (0; 6, 2 (7) ; 0) 3,17 (0; 2 (10) ; 0) 3,19 (0; 6, 2 (9) ; 0) 3,23 (0; 2 (12) ; 0) 3,31 (0; 6, 2 (13) ; 0) 7,11 (0; 2 (12) ; 0) (10) ; 0) 13 (0; 2 (11) ; 0) 17 (0; 2 (13) ; 0) 19 (0; 2 (14) ; 0) 2,3,5,11 ∅ (0; 6, 2 (4) ; 0) 7 (0; 6, 2 (9) ; 0) 13 (0; 6, 2 (14) ; 0) 2,3,5,13 ∅ (0; 2 (6) (14) ; 0) 2,3,7,11 ∅ (0; 4, 2 (5) ; 0) 5 (0; 4, 2 (10) ; 0) 2,3,7,13 ∅ (0; 2 (7) ; 0) 5 (0; 2 (13) ; 0) 2,3,7,17 ∅ (0; 2 (8) ; 0) 2,3,7,19 ∅ (0; 4, 2 (7) ; 0) 2,3 7,23 ∅ (0; 4, 2 (8) ; 0) 2,3 7,29 ∅ (0; 2 (11) ; 0) 2,3,7,31 ∅ (0; 4, 2 (10) ; 0) 2,3,7,41 ∅ (0; 2 (14) ; 0) 2,3,7,47 ∅ (0; 4, 2 (14) ; 0) 2,3,11,13 ∅ (0; 2 (9) ; 0) 2,3,11,17 ∅ (0; 6, 2 (9) ; 0) 2,3,11,19 ∅ (0; 4, 2 (10) ; 0) 2,3,11,29 ∅ (0; 6, 2 (14) ; 0) 2,3,13,17 ∅ (0; 2 (12) : 0) 2,3,13,23 ∅ (0; 2 (15) ; 0) 2,3,17,19 ∅ (0; 2 (16) ; 0) 2,5,7,11 ∅ (0; 2 (9) ; 0) 3 (0; 2 (14) ; 0) 2,5,7,13 ∅ (0; 2 (10) ; 0) 2,5,7,17 ∅ (0; 2 (12) : 0) Table 3. Cocompact groups with four ramified primes.…”

“…Then r = 3, 4 or 6 and there exists an embedding σ : Q(e 2πi/r ) → B such that N (E) + contains a conjugate of σ(u) where u = 1 + e 2πi/r (see [8,27]). Conditions for the existence of such an embedding are given in Theorem 2.2 and are incorporated into the formula (6) below.…”

“…It should be noted that necessary and sufficient conditions for N (E) + to contain images of u have been given in [8]. In these cases of groups defined over Q, these conditions are incorporated into the definition of F and the formula for s(u) at (6).…”

“…The description of these maximal groups was reformulated in [8] (see also [27] Chapter 11) using Eichler orders as follows:…”

“…(0; 6, 2 (3) ; 0) 7 (0; 3, 2 (4) ; 0) 11 (0; 2 (6) ; 0) 13 (0; 3, 2 (5) ; 0) 17 (0; 2 (7) ; 0) 19 (0; 3, 2 (6) ; 0) 23 (0; 2 (8) ; 0) 29 (0; 2 (9) ; 0) 31 (0; 3, 2 (8) ; 0) 37 (0; 3, 2 (9) ; 0) 41 (0; 2 (11) ; 0) 53 (0; 2 (13) ; 0) 61 (0; 3, 2 (13) ; 0) 2,7 ∅ (0; 4, 2 (3) ; 0) 3,5 (0; 2 (7) ; 0) 3,13 (0; 2 (11) ; 0) 3,19 (0; 2 (14) ; 0) 3 (0; 2 (5) ; 0) 5 (0; 4, 2 (4) ; 0) 11 (0; 2 (7) ; 0) 13 (0; 4, 2 (6) ; 0) 19 (0; 2 (9) ; 0) 23 (0; 2 (10) ; 0) 29 (0; 4, 2 (10) ; 0) 37 (0; 4, 2 (12) ; 0) 43 (0; 2 (15) ; 0)…”