Abstract
This chapter discusses the structure of the icosahedral modular group. An icosahedral modular group means the Hilbert modular group PSL(2,O) where O is the ring of integers in the number field Q(√5). This naming comes from the fact that Hirzebruch studied the irreducible action over H × H of the principal congruence subgroup Γ = Γ(2) of SL(2,O) associated with the prime ideal (2) ⊆ O and showed that thecom pactified quotient H × H/Γ is equivariantly birational to the projective plane P2(C) acted nontrivially by the icosahedral group. Γ can be regarded as a subgroup of PSL(2,O) because −1 ∉ Γ. It acts even freely on H × H. The chapter discusses the Hubert modular groups for which the group-structure is combinatorially described.
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