The $\lambda $-quiddities of size n are n-tuples of elements of a fixed set, solutions of a matrix equation appearing in the study of Coxeter’s friezes. Their number and properties are closely linked to the structure and the cardinality of the chosen set. Our main objective is an explicit formula giving the number of $\lambda $-quiddities of odd size, and a lower and upper bound for the number of $\lambda $-quiddities of even size, over the rings ${\mathbb {Z}}/2^{m}{\mathbb {Z}}$ ($m \geq 2$). We also give explicit formulae for the number of $\lambda $-quiddities of size n over ${\mathbb {Z}}/8{\mathbb {Z}}$.

We prove the following theorem. Suppose that $F\,=\,\left( {{f}_{1}},\,{{f}_{2}} \right)$ is a 2-dimensional, vector-valued modular form on $\text{S}{{\text{L}}_{2}}\left( \mathbb{Z} \right)$ whose component functions ${{f}_{1}}$, ${{f}_{2}}$ have rational Fourier coefficients with bounded denominators. Then ${{f}_{1}}$ and ${{f}_{2}}$ are classical modular forms on a congruence subgroup of the modular group.

It is well known that if a convex hyperbolic polygon is constructed as a fundamental domain for a subgroup of SL(2,ℝ), then its translates by the group form a locally finite tessellation and its side-pairing transformations form a system of generators for the group. Such a hyperbolically convex fundamental domain for any discrete subgroup can be obtained by using Dirichlet’s and Ford’s polygon constructions. However, these two results are not well adapted for the actual construction of a hyperbolically convex fundamental domain due to their nature of construction. A third, and most important and practical, method of obtaining a fundamental domain is through the use of a right coset decomposition as described below. If Γ2 is a subgroup of Γ1 such that Γ1=Γ2⋅{L1,L2,…,Lm} and 𝔽 is the closure of a fundamental domain of the bigger group Γ1, then the set is a fundamental domain of Γ2. One can ask at this juncture, is it possible to choose the right coset suitably so that the set ℛ is a convex hyperbolic polygon? We will answer this question affirmatively for Hecke modular groups.

The Modular Group M is PSL2(Z) the group of linear fractional transformations with integral entries and determinant one. M has been of great interest in many diverse fields of Mathematics, including Number Theory, Automorphic Function Theory and Group Theory. In this paper we give an effective algorithm to determine, for each integer d, a complete set of representatives for the trace classes in trace d. This algorithm depends on the combinatorial group theoretic structure of M. It has been subsequently extended by Sheingorn to the general Hecke groups. The number h(d) of trace classes in trace d is equal to the ideal class number of the field The algorithm mentioned above then provides a new straightforward computational procedure for determining h(d). Finally as an outgrowth of the algorithm we present a wide generalization of the Fermat Two-Square theorem. This last result can also be derived from classical work of Gauss.