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

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**Extra resources for Non-Commutative Algebraic Geometry**

**Example text**

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 } .