Arithmetic groups are a rich class of groups where connections between topology and number theory are showcased in a particularly striking way. One construction of these groups is motivated by the modular group, PSL2(Z). The group of orientation preserving isometries of the hyperbolic upper half plane, H, is isomorphic to PSL2(R). Since Z is a discrete subgroup of R it follows that PSL2(Z) is d...

In this paper, we introduce $rho$-altering $J$-type mappings in modular spaces. We prove some fixed point theorems for $rho$-altering and $rho$-altering $J$-type mappings in modular spaces. We also furnish illustrative examples to express relationship between these mappings. As a consequence, the results are applied to the existence of solution of an integral equation arising from an ODE ...

Most mathematicians have encountered modular functions. For example, when the group theorists discovered the monster group, they were surprised to find that the degrees of its irreducible representations were already encoded in the q-coefficients of the j-function. The theory of Shimura varieties grew out of the applications of modular functions and modular forms to number theory. Roughly speak...

In this article we report on a surprising relation between the transfer operators for the congruence subgroups Γ0(n) and the Hecke operators on the space of period functions for the modular group PSL(2,Z). For this we study special eigenfunctions of the transfer operators with eigenvalues ∓1, which are also solutions of the Lewis equations for the groups Γ0(n) and which are determined by eigenf...

Abstract. After Zagier proved that the traces of singular moduli j(z) are Fourier coefficients of a weakly holomorphic modular form, various properties of the traces of the singular values of modular functions mostly on the full modular group PSL2(Z) have been investigated such as their exact formulas, limiting distribution, duality, and congruences. The purpose of this paper is to generalize t...

Problem statement: In this study we propose a group re-keying protocol based on modular polynomial arithmetic over Galois Field GF(2). Common secure group communications requires encryption/decryption for group re-keying process, especially when a group member is leaving the group. Approach: This study proposes secret keys multiplication protocol based on modular polynomial arithmetic (SKMP), w...

Arithmetic subgroups are finite index subgroups of the modular group. Classically, congruence arithmetic subgroups, which can be described by congruence relations, are playing important roles in group theory and modular forms. In reality, the majority of arithmetic subgroups are noncongruence. These groups as well as their modular forms are central players of this survey article. Differences be...