Fundamental domains for SL 2 ( Z ) and Γ θ

1. H as homogeneous space for G = SL 2 (R) 2. Fundamental domain for Γ = SL 2 (Z) on H 3. Inversion and translation generate SL 2 (Z). 4. Re-enabling the action of SL 2 (R) 5. Fundamental domain for Γ θ and Γ(2) 6. Generators for Γ θ 7. Theta series are modular forms The real line R, being a group, is a homogeneous space in the sense that (of course) it acts transitively on itself. The upper half-plane H is not a group itself, but is acted upon by SL 2 (R) (2-by-2 real matrices with determinant 1) acting by linear fractional transformations a b c d (z) = az + b cz + d We will see that the simplest quotient of the upper half-plane, SL 2 (Z)\H, is topologically a sphere with a point missing. However, in the SL 2 (R)-invariant geometry, the missing point is infinitely far away, so the shape is not a round sphere, but stretched out like a raindrop.