Connected fundamental domains for congruence subgroups

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