Arithmetic and geometry of the Hecke groups
Cheng Lien Lang, Mong Lung Lang

TL;DR
This paper investigates the arithmetic and geometric properties of Hecke groups, deriving conditions for subgroups with specific characteristics related to genus, cusps, and conjugacy classes, with special cases for odd q.
Contribution
It provides explicit criteria linking subgroup properties of Hecke groups to their geometric and arithmetic invariants, extending understanding of their structure.
Findings
Derived conditions for subgroup existence based on genus, cusps, and conjugacy classes.
Established relations between subgroup indices and geometric invariants.
Simplified criteria for odd q cases where certain conditions are redundant.
Abstract
We study the arithmetic and geometry properties of the Hecke group . In particular, we prove that has a subgroup of index , genus with cusps, and (resp. ) conjugacy classes of elements that are conjugates of (resp. ) if and only if (i) , and (ii) is a multiple of , (iii) . In the case is odd, (ii) is a consequence of (i).
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