102 CHAPTER 4. APPLICATIONS TO ARITHMETIC GROUPS Volume <strong>of</strong> a tetrahedron <strong>of</strong> which its three vertices are at the infinity. A picture here!!! Since in the neighborhood <strong>of</strong> each vertex the sum <strong>of</strong> dihedral angles is π, the opposite dihedral angles are equal: we have then at most 3 distinct dihedral angles. Let they be α, β, γ and by intersecting the dihedral with the tetrahedron <strong>of</strong> the precedent type, we see that Proposition 4.3.8. <strong>The</strong> volume <strong>of</strong> a tetrahedron <strong>of</strong> which the vertices are at the infinity, and <strong>of</strong> the dihedral angles α, β, γ, is equal to V = L(α) + L(β) + L(γ). EXAMPLE. A fundamental domain for Picard group P SL(2, Z[i]). <strong>The</strong> domain defined by the relation (Picard [1]): A picture here!!! x 2 + y 2 + z 2 geq1, x ≤ 1/2, y ≤ 1/2, 0 ≤ x + y is a fundamental domain for P SL(2, Z[i]) in H3. It is the union <strong>of</strong> four equal tetrahedrons all <strong>of</strong> which have a vertex at the infinity. With the above definitions we have δ = π/3 and α = π/4. <strong>The</strong> volume <strong>of</strong> the domain then is V = L(π/4 + π/3) + L(π/4 − π/3) + 2L(π/2 − π/4) = 1/3 · L(3π/4 − π) + Lπ/4) = 1/3 · L(−π/4) + L(π/4) = 2/3 · L(π/4) = (4π 2 ) −1 · D 3/2 K · ζK(2), if K = Q(i) On the other side, for Tamagawa measure we have vol(SL(2, Z(i)\SL(2, C)) = 4π 2 V . Using the same argument to SL(2, R), we can prove the following corollary by compare. Corollary 4.3.9. <strong>The</strong> Tamagawa measure on SL(2, C) is the product <strong>of</strong> the hyperbolic measure on H3 by the Haar measure on SL(2, C) such that vol(SU(2, C)) = 8π 2 . <strong>The</strong>refore, if Γ is a discrete subgroup <strong>of</strong> SL(2, C) <strong>of</strong> finite covolume vol(SL(2, C)/Γ) = � 4π 2 vol( ¯ Γ\H3)if −1 ∈ Γ 8π 2 vol( ¯ Γ\H3) if −1 /∈ Γ . We find again Humbert formula for P SL(2, R) if R is the integer ring <strong>of</strong> an imaginary quadratic field K: vol(P SL(2, R)\H3) = 4π 2 ζK(2)D 3/2 K . Remark 4.3.10. Let H/K be a quaternion algebra satisfying the properties in the beginning <strong>of</strong> chapter IV, and C be a maximal compact subgroup <strong>of</strong> G 1 . <strong>The</strong> .
4.3. EXAMPLES AND APPLICATIONS 103 groups Γ <strong>of</strong> units <strong>of</strong> reduced norm 1 in the R (S)-order <strong>of</strong> H are allowed to define the arithmetic variety : XΓ = Γ\G 1 /C. <strong>The</strong> results <strong>of</strong> Chapter III have then the interesting applications to the study <strong>of</strong> varieties XΓ. We refer reader to the works <strong>of</strong> Ihara, Shimura, SErre, Mumford, Cerednik, Kurahara cited in the bibliography.
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