Hecke Groups Research Articles

On the transcendance of quasi-periodic Rosen continued fractions

AbstractIn this paper, we consider the two Hecke groups $$G_{4}$$ G 4 and $$G_{6}$$ G 6 and we use the Schmidt Subspace Theorem to establish the transcendence of some quasi-periodic Rosen continued fractions in order to get the exact analogues of the results established with the regular continued fractions.

On Ramanujan's modular equations and Hecke groups

Inspired by the work of Ramanujan, many people have studied generalized modular equations and the numerous identities found by Ramanujan. These identities known as modular equations can be transformed into polynomial equations. There is no developed theory about how to find the degrees of these polynomial modular equations explicitly. In this paper, we determine the degrees of the polynomial modular equations explicitly and study the relation between Hecke groups and modular equations in Ramanujan's theories of signatures 2, 3, and 4.

Automorphic integrals with log-polynomial period functions and arithmetical identities

Building on the works of S. Bochner on equivalence of modular relation with functional equation associated to the Dirichlet series, K. Chandrasekharan and R. Narasimhan obtained new equivalences between the functional equation and some arithmetical identities. Sister Ann M. Heath considered the functional equation in the Hawkins and Knopp context and showed its equivalence to two arithmetical identities associated with entire modular cusp integrals involving rational period functions for the full modular group. In this paper we use techniques of Chandrasekharan and Narasimhan to prove results analogous to those of Sister Ann M. Heath. Specifically, we establish equivalence of two arithmetical identities with a functional equation associated with automorphic integrals involving log-polynomial-period functions on the discrete Hecke groups.

Linear groups related to Fibonacci polynomials

We examine the generators of linear groups associated with the roots of Fibonacci polynomials, matrix representations of the generators, some elements, and group structures. From the viewpoint of the roots of Fibonacci polynomials, we obtain the features of various classes of linear groups under certain circumstances. Furthermore, we analyze the relationships between the roots of Fibonacci polynomials and the new generating matrices. We get some amazing relationships between the roots of Fibonacci polynomials and the extended modular group, the extended Hecke groups, and the extended generalized Hecke groups with geometric viewpoints.

Conjugacy classes and rational period functions for the Hecke groups

In this paper, we establish a one-to-one correspondence between conjugacy classes of any Hecke group and irreducible systems of poles of rational period functions for automorphic integrals on the same group. We use this correspondence to construct irreducible systems of poles and to count poles. We characterize Hecke-conjugation and Hecke-symmetry for poles of rational period functions in terms of the transpose of matrices in conjugacy classes. We construct new rational period functions and families of rational period functions.

Fourier non-uniqueness sets from totally real number fields

Let K be a totally real number field of degree n \geq 2 . The inverse different of K gives rise to a lattice in \mathbb{R}^n . We prove that the space of Schwartz Fourier eigenfunctions on \mathbb{R}^n which vanish on the “component-wise square root” of this lattice, is infinite dimensional. The Fourier non-uniqueness set thus obtained is a discrete subset of the union of all spheres \sqrt{m}S^{n-1} for integers m \geq 0 and, as m \rightarrow \infty , there are \sim c_{K} m^{n-1} many points on the m -th sphere for some explicit constant c_{K} , proportional to the square root of the discriminant of K . This contrasts a recent Fourier uniqueness result by Stoller (2021) Using a different construction involving the codifferent of K , we prove an analogue for discrete subsets of ellipsoids. In special cases, these sets also lie on spheres with more densely spaced radii, but with fewer points on each. We also study a related question about existence of Fourier interpolation formulas with nodes “ \sqrt{\Lambda} ” for general lattices \Lambda \subset \mathbb{R}^n . Using results about lattices in Lie groups of higher rank we prove that if n \geq 2 and a certain group \Gamma_{\Lambda} \leq \operatorname{PSL}_2(\mathbb{R})^n is discrete, then such interpolation formulas cannot exist. Motivated by these more general considerations, we revisit the case of one radial variable and prove, for all n \geq 5 and all real \lambda > 2 , Fourier interpolation results for sequences of spheres \sqrt{2 m/ \lambda}S^{n-1} , where m ranges over any fixed cofinite set of non-negative integers. The proof relies on a series of Poincaré type for Hecke groups of infinite covolume and is similar to the one in Stoller (2021).

A new proof on the free product structure of Hecke groups

A new proof on the free product structure of Hecke groups

On the roots of Fibonacci polynomials

In this paper, we investigate Fibonacci polynomials as complex hyperbolic functions. We examine the roots of these polynomials. Also, we give some exciting identities about images of the roots of Fibonacci polynomials under another member of the Fibonacci polynomials class. Finally, we obtain some excellent relationships between the roots of Fibonacci polynomials and the modular group, Hecke groups and generalized Hecke groups with geometric interpretations.

Erratum to: A note on the McShane’s identity for Hecke groups

Due to a typesetting mistake, the PDF version of this article was published with an error in Theorem A.

The finiteness of derivation-invariant prime ideals and the algebraic independence of the Eisenstein series

We prove that for a graded algebra $$\mathcal {A}$$ with a derivation $$D_{\mathcal {A}}$$ satisfying certain conditions, and a bi-graded algebra $$\mathcal {A}[q]$$ with an extended derivation D of $$D_{\mathcal {A}}$$ , there are only finitely many $$D_{\mathcal {A}}$$ - and D-invariant (or differential with respect to $$D_{\mathcal {A}}$$ and D) principal prime ideals of $$\mathcal {A}$$ and of $$\mathcal {A}[q]$$ , respectively. As its application, we prove the algebraic independence of the values of certain Eisenstein series for the arithmetic Hecke groups H(m) for $$m=4,6, \infty $$ , which is an extension of Nesterenko’s result for the Eisenstein series for SL $$_2(\mathbb {Z})$$ .

Triangle Groups: Automorphic Forms and Nonlinear Differential Equations

We study the relations governing the ring of quasiautomorphic forms associated to triangle groups with a single cusp, thereby extending our earlier results on Hecke groups. The Eisenstein series associated to these triangle groups are shown to satisfy Ramanujan-like identities. These identities in turn allow us to associate a nonlinear differential equation to each triangle group. We show that they are solved by the quasiautomorphic weight-2 Eisenstein series associated to the triangle group and its orbit under the group action. We conclude by discussing the Painlev\'e property of these nonlinear differential equations.

Aspects of Hecke Symmetry:Anomalies, Curves, and Chazy Equations

We study various relations governing quasi-automorphic forms associated to discrete subgroups of ${\rm SL}(2,\mathbb{R}) $ called Hecke groups. We show that the Eisenstein series associated to a Hecke group ${\rm H}(m)$ satisfy a set of $m$ coupled linear differential equations, which are natural analogues of the well-known Ramanujan identities for quasi-modular forms of ${\rm SL}(2,\mathbb{Z})$. Each Hecke group is then associated to a (hyper-)elliptic curve, whose coefficients are determined by an anomaly equation. For the $m=3$ and $4$ cases, the Ramanujan identities admit a natural geometric interpretation as a Gauss-Manin connection on the parameter space of the elliptic curve. The Ramanujan identities also allow us to associate a nonlinear differential equation of order $ m $ to each Hecke group. These equations are higher-order analogues of the Chazy equation, and we show that they are solved by the quasi-automorphic Eisenstein series $E_2^{(m)}$ associated to ${\rm H}(m) $ and its orbit under the Hecke group. We conclude by demonstrating that these nonlinear equations possess the Painlev\'e property.

Convergence of Poincare series on Hecke groups of large width

In earlier work, we described a relation between the parameters associated with multiplier systems of complex weight on the discrete Hecke groups $G_{\lambda }$ when $1 \leq \lambda \lt 2$, and consequently showed that parabolic Poincare series of nonreal weight on the modular group are not absolutely convergent anywhere. In the current paper we establish an analogous divergence result for all Hecke groups with $\lambda > 2$.

Power and free normal subgroups of generalized Hecke groups

Let [Formula: see text] and [Formula: see text] be integers such that [Formula: see text] [Formula: see text] and let [Formula: see text] be generalized Hecke group associated to [Formula: see text] and [Formula: see text] Generalized Hecke group [Formula: see text] is generated by [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] In this paper, for positive integer [Formula: see text] we study the power subgroups [Formula: see text] of generalized Hecke groups [Formula: see text]. Also, we give some results about free normal subgroups of generalized Hecke groups [Formula: see text]

Modular structures and extended-modular-group-structures after Hecke pairs

The simplices and the complexes arsing form the grading of the fundamental (desymmetrized) domain of arithmetical groups and non-arithmetical groups, as well as their extended (symmetrized) ones are described also for oriented manifolds in dim > 2. The conditions for the definition of fibers are summarized after Hamiltonian analysis, the latters can in some cases be reduced to those for sections for graded groups, such as the Picard groups and the Vinberg group.The cases for which modular structures rather than modular-group-structure measures can be analyzed for non-arithmetic groups, i.e. also in the cases for which Gelfand triples (rigged spaces) have to be substituted by Hecke couples, as, for Hecke groups, the existence of intertwining operators after the calculation of the second commutator within the Haar measures for the operators of the correspondingly-generated C* algebras is straightforward. The results hold also for (also non-abstract) groups with measures on (manifold) boundaries. The Poincaré invariance of the representation of Wigner-Bargmann (spin 1/2) particles is analyzed within the Fock-space interaction representation.The well-posed-ness of initial conditions and boundary ones for the connected (families of) equations is discussed. As an example, Picard-related equations can be classified according to the genus of the modular curve(s) attached to the solutions(s).From the Hamiltonian analysis, further results in the contraction of the congruence (extended sub-)groups for non-arithmetical groups for the construction of tori is provided as an alternative to the free diffeomorphism group.In addition, the presence of Poincaré complexes is found compatible with non-local interactions, i.e. both lattices interactions or spin-like ones.

Some results on Hecke and extended Hecke groups

Let $q\geq 3$ be a prime number and let $\overline{H}(\lambda _{q})$ be the extended Hecke group associated with $q.$ In this paper, we determine the presentation of the commutator subgroup ($H$($\lambda _{q})\alpha )^{\prime } $ of the normal subgroup $H$($\lambda _{q})\alpha $, where $H$($\lambda _{q})\alpha $ is a subgroup of index $2$ in $\overline{H}$($\lambda _{q}).$ Next we discuss the commutator subgroup ($\overline{H}_{2})^{\prime }$($% \lambda _{q})$ of the principal congruence subgroup $\overline{H}_{2}$($% \lambda _{q})$ of $\overline{H}$($\lambda _{q})$. Then we show that some quotient groups of $\overline{H}$($\lambda _{q})$ are generalized $M^{\ast }- $groups. Finally, we prove some results related to some normal subgroups of $\overline{H}$($\lambda _{q})$, especially in the case $q=5.$

Commutator Subgroups of Generalized Hecke and Extended Generalized Hecke Groups

Abstract Let p and q be integers such that 2 ≤ p ≤ q; p + q > 4 and let Hp,q be the generalized Hecke group associated to p and q: The generalized Hecke group Hp,q is generated by X(z) = -(z-λp)-1 and Y (z) = -(z+ λq)-1 where λp = 2 cos ≤ π/p and λq = 2 cos π/q . The extended generalized Hecke group H̅p,q is obtained by adding the reection R(z) = 1/z̅ to the generators of generalized Hecke group Hp,q: In this paper, we study the commutator subgroups of generalized Hecke groups Hp,q and extended generalized Hecke groups H̅p,q.

Geodesic Rosen Continued Fractions

We describe how to represent Rosen continued fractions by paths in a class of graphs that arise naturally in hyperbolic geometry. This representation gives insight into Rosen's original work about words in Hecke groups, and it also helps us to identify Rosen continued fraction expansions of shortest length.

Arithmetic and geometry of the Hecke groups

We study the arithmetic and geometry properties of the Hecke group Gq. In particular, we prove that Gq has a subgroup X of index d, genus g with v∞ cusps, and τ2 (resp. vri) conjugacy classes of elliptic elements that are conjugates of S (resp. Rq/ri) if and only if (i) 2g−2+τ2/2+∑i=1kvri(1−1/ri)+v∞=d(1/2−1/q), and (ii) m0=4g−4+τ2+2v∞+∑i=1kvri(2−q/ri)≥0 is a multiple of q−2. Note that if q is odd (resp. prime), then m0/(q−2)∈Z (resp. N∪{0}) is a consequence of (i).

On Normal Subgroups of Generalized Hecke Groups

Abstract We consider the generalized Hecke groups Hp,q generated by X(z) = -(z -λp)-1, Y (z) = -(z +λq)-1 with and where 2 ≤ p ≤ q < ∞, p+q > 4. In this work we study the structure of genus 0 normal subgroups of generalized Hecke groups. We construct an interesting genus 0 subgroup called even subgroup, denoted by . We state the relation between commutator subgroup H′p,q of Hp,q defined in [1] and the even subgroup. Then we extend this result to extended generalized Hecke groups H̅p,q.