Notes: Elliptic Points and Cusps

which we call the fundamental domain. By previous results, this set also (essentially) represents equivalence classes of complex elliptic curves under isomorphism (τ ∈ D ↔ C/Λτ ). Lemma 1.1. The map π : D → Y (1) is a surjection, where π is the obvious projection π(τ) = SL2(Z)τ . Proof. This lemma states that D generates all of H under the action of SL2(Z); that is, it suffices to show that every τ ∈ H is SL2(Z)-invariant to some point of D. Repeatedly apply the translation map τ &→ τ ± 1, given by [ 1 ±1 0 1 ] , to trans-