Normalizers of Γ_Δ(N): the group M_Δ, finiteness, and a coset count
Formal statements and proofs, in Lean 4 with Mathlib, about the normalizers in SL₂(ℝ) of the groups Γ_Δ(N) between Γ₁(N) and Γ₀(N), and the quotients Nor(Γ_Δ(N))/Γ_Δ(N), which for −1 ∈ Δ are the modular automorphism groups of the curves X_Δ(N): the matrices allowed by the generalized Newman bound h_Δ form a group M_Δ = D⁻¹Γ₀(n_Δ)⁺D, D = diag(h_Δ, 1), which contains Nor(Γ_Δ(N)) and contains Γ₀(N) with finite index, so Nor(Γ_Δ(N))/Γ_Δ(N) is finite for every Δ; its order is [(ℤ/N)ˣ : Δ] times the number of pairs (σ, H), with σ in a fixed set of representatives of Γ₀(N)\SL₂(ℤ) and H a Hermite…
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