On congruence equations arising from suborbital graphs

Abstract

In this paper we deal with congruence equations arising from suborbital graphs of the normalizer of Γ0(m) in P SL(2, R) . We also propose a conjecture concerning the suborbital graphs of the normalizer and the related congruence equations. In order to prove the existence of solution of an equation over prime finite field, this paper utilizes the Fuchsian group action on the upper half plane and Farey graphs properties.

Hexagonal cell graphs of the normalizer with signature $(2, 6, \infty)$

“…The subgroup Γ 0 (N ) = {g ∈ Γ : c ≡ 0 (mod N )} is a well-known congruence subgroup of the classical modular group Γ. The normalizer of Γ 0 (N ) in P SL(2, R) turns to be a very important group in the study of moonshine and for this reason has been studied by many authors (see, for example, [4,5,8,13,14]). It consists exactly of the matrices…”

Solutions to Pell's Equation via Suborbital Graphs

“…al. studied on the solutions of congruence equations that come from the action of the normalizer of Γ 0 (N ) via suborbital graphs [4,2]. With the similar notion, it is obtained some relationships Fibonacci numbers and suborbital graphs in the study [6,17].…”

Normalizer Maps Modulo N

“…Hence, we can regard the elements of PSL(2, R) as the matrices The study by Jones, Singerman and Wicks [1] is a pioneering study concerning these groups and has enabled the analytical examination of graphs. Many researchers have conducted studies [2][3][4][5][6][7] that reveal the relationship of many groups of graphswith the methods and results presented in this study. In particular, because of the interesting nature of the normalizer Γ B (N) of Γ 0 (N) in PSL(2, R) [8][9][10] and its complexity relative to the modular group, researchers have studied normalizer-related graphs under various conditions [2,[11][12][13].…”