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By F.M.J. van Oystaeyen, A.H.M.J. Verschoren

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Claim: (i) The map φ : Μ - SL(2, Ζ) given by M ( . bc is a monomorphism of groups. The image of the map φ is the subgroup Γ : = { Δ G SL(2, Z) A t = A ~ mod 2 } = 1 Χ Γ χ (2) l X ~ . (ii) Τ M^a bj T _ 1 is always an element of Γ12 . For (i) we remark that a lengthy, but straightforward calculation shows the correctness of φ to be a homomorphism. = II2 implies a — l,b = c = d — 0, thus φ is injective. Observe that φ is well-defined and the inverse map is given by α β \ α ^ w ere + δ 2 α = α ' — δ 2 = β ' - η 2 c = ß ' + j 2 = For (ii) a computation yields ( α ΤΜ,α μ Γ - 1 U d) + + c —d 0 b 2 (d \ - a + c) b — d d - c and from this we derive: ΓΜ/« b\T \c 2c 2(6 - c) b - d 1 c + d —c —c a — b — c + d - b € Γι 2 2(6 - c) 4(c - 6) \ 2(6 - a) d a + b — c — d j a + b + c + d = 1 mod 2 .

First note that all fixed varieties are disjoint with the exception of the inclusions Ci C Ho for i = 1,2. This is a consequence of the property that every isotropy group is cyclic generated. The invariance group of the curve C\ resp. C2 is the normalizer of the group resp. 8. Since both cyclic groups contain only one involution, namely Jo, every element in the normalizer must commute with Jo, hence is of type IQ (Δι,Δ2) G Ι \ η . This shows clearly that the images of the curves Ci (i = 1,2) are isomorphic to the modular curve of level η .

4. The matrices of finite order in Γι^ , which commute with the involution Io are given by (up to sign and conjugacy): I4, Iq, R, S, T. )) , hh = —is . Next we introduce the following subgroups Γι (2) := {<7 e SL(2, Ζ) | <7 = (* Γ 4 (2) := { ρ G Γ (2) I ρ ξ (* mod 2 } mod 4 } where Γ (2) denotes as usual the principal congruence subgroup of level 2. From this we conclude for the centralizer of h the characterization: C r i , A h ) = { Ι 5 ( Δ ι , Δ 2 ) I Δ ι , Δ 2 G Γι (2), 6 + /3 ξ 0 mod 4 } .

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