A Comprehensive Overview on the Formation of Homomorphic Copies in Coset Graphs for the Modular Group

This work deals with the well-known group-theoretic graphs called coset graphs for the modular group G and its applications. The group action of G on real quadratic fields forms infinite coset graphs. These graphs are made up of closed paths. When M acts on the finite field Zp, the coset graph appears through the contraction of the vertices of these infinite graphs. Thus, finite coset graphs are composed of homomorphic copies of closed paths in infinite coset graphs. In this work, we have presented a comprehensive overview of the formation of homomorphic copies.

Dihedral group and classification of G-circuits of length 10

In this paper, we classify G-circuits of length 10 with the help of the location of the reduced numbers lying on G-circuit. The reduced numbers play an important role in the study of modular group action on P S L ( 2 , Z ) $PSL(2,\mathbb{Z})$ -subset of Q ( m ) \ Q $Q(\sqrt{m}){\backslash}Q$ . For this purpose, we define new notions of equivalent, cyclically equivalent, and similar G-circuits in P S L ( 2 , Z ) $PSL(2,\mathbb{Z})$ -orbits of real quadratic fields. In particular, we classify P S L ( 2 , Z ) $PSL(2,\mathbb{Z})$ -orbits of Q ( m ) \ Q $Q(\sqrt{m}){\backslash}Q$ = ⋃ k ∈ N Q * k 2 m $={\bigcup }_{k\in N}{Q}^{{\ast}}\left(\sqrt{{k}^{2}m}\right)$ containing G-circuits of length 10 and determine that the number of equivalence classes of G-circuits of length 10 is 41 in number. We also use dihedral group to explore cyclically equivalence classes of circuits and use cyclic group to explore similar G-circuits of length 10 corresponding to 10 of these circuits. By using cyclically equivalent classes of circuits and similar circuits, we obtain the exact number of G-orbits and the structure of G-circuits corresponding to cyclically equivalent classes. This study also helps us in classifying the reduced numbers lying in the P S L ( 2 , Z ) $PSL(2,\mathbb{Z})$ -orbits.

Prime geodesics and averages of the Zagier L-series

The Zagier L-series encode data of real quadratic fields. We study the average size of these L-series, and prove asymptotic expansions and omega results for the expansion. We then show how the error term in the asymptotic expansion can be used to obtain error terms in the prime geodesic theorem.

Indefinite zeta functions

We define generalised zeta functions associated with indefinite quadratic forms of signature $$(g-1,1)$$ ( g - 1 , 1 ) —and more generally, to complex symmetric matrices whose imaginary part has signature $$(g-1,1)$$ ( g - 1 , 1 ) —and we investigate their properties. These indefinite zeta functions are defined as Mellin transforms of indefinite theta functions in the sense of Zwegers, which are in turn generalised to the Siegel modular setting. We prove an analytic continuation and functional equation for indefinite zeta functions. We also show that indefinite zeta functions in dimension 2 specialise to differences of ray class zeta functions of real quadratic fields, whose leading Taylor coefficients at $$s=0$$ s = 0 are predicted to be logarithms of algebraic units by the Stark conjectures.