Connected fundamental domains for congruence subgroups

Abstract

Abstract We give explicit sets of right coset representatives for the congruence subgroups Γ 0 ( N ) $\Gamma _0(N)$ normal upper Gamma 0 left parenthesis upper N right parenthesis , Γ 1 ( N ) , $\Gamma _1(N),$ normal upper Gamma 1 left parenthesis upper N right parenthesis comma and Γ ( N ) $\Gamma (N)$ normal upper Gamma left parenthesis upper N right parenthesis , and prove that the corresponding unions of standard modular triangles are connected fundamental domains. The construction is based on a study of the projective line P 1 ( Z / N Z ) ${\mathbb P}^1({\mathbb Z}/N{\mathbb Z})$ double struck upper P Superscript 1 Baseline left parenthesis double struck upper Z divided by upper N double struck upper Z right parenthesis . For every residue class j ∈ Z / N Z $j\in {\mathbb Z}/N{\mathbb Z}$ j element of double struck upper Z divided by upper N double struck upper Z , the number of representatives above j is governed by the simple function W j = min { m ∈ Z