Representation of Primes Through Normalizers of Hecke Congruence Subgroups

Köroğlu T., Güler B. Ö., Yazıcı M.

The First International Conference on Mathematics and Applied Data Science (ICMADS’25), Konya, Turkey, 29 - 31 August 2025, vol.1, no.1, pp.254, (Summary Text)

  • Publication Type: Conference Paper / Summary Text
  • Volume: 1
  • City: Konya
  • Country: Turkey
  • Page Numbers: pp.254
  • Karadeniz Technical University Affiliated: Yes

Abstract

The normalizer of the Hecke congruence subgroup  in  is denoted by  consists of matrices of the form , where

(i)              h is the maximal divisor of 24 with , 

(ii)             e exactly divides  (denoted ), 

(iii)           the determinant condition  

holds. This group plays a fundamental role in moonshine theory and arithmetic geometry [1,2].

This study investigates the relationship between prime numbers of a specific form and the structural properties of modular groups, particularly focusing on the normalizer  . We show the conditions under which the action of this normalizer on the extended rationals   is transitive or primitive. For the level   (where   is prime), we demonstrate that the action is not transitive on   but identifies a maximal subset where transitivity holds.

Furthermore, we prove that the action is imprimitive, partitioning  ​ into two distinct blocks. From [3], we analyze the structural properties of directed graphs within these blocks. Then, we establish conditions for the existence of self-paired edges, quadrilaterals, or triangles in these graphs, linking them to the representation of primes by quadratic forms.

This work bridges group theory and number theory through combinatorial and geometric approaches to classical problems. These results demonstrate how group-theoretic frameworks can expose fundamental arithmetic structures, particularly the representation of primes as sums of two squares.

Keywords: Congruence equation, Normalizer, Hecke group