scholarly journals ON BINARY QUADRATIC FORMS AND THE HECKE GROUPS

2009 ◽  
Vol 05 (08) ◽  
pp. 1401-1418 ◽  
Author(s):  
WENDELL RESSLER
Keyword(s):  
Fixed Points ◽  
Continued Fractions ◽  
Quadratic Forms ◽  
Modular Group ◽  
Reduction Theory ◽  
Binary Quadratic Forms ◽  
Hecke Groups ◽  
Hecke Group ◽  
Hyperbolic Fixed Points

We present a reduction theory for certain binary quadratic forms with coefficients in ℤ[λ], where λ is the minimal translation in a Hecke group. We generalize from the modular group Γ(1) = PSL(2,ℤ) to the Hecke groups and make extensive use of modified negative continued fractions. We also define and characterize "reduced" and "simple" hyperbolic fixed points of the Hecke groups.

scholarly journals ASYMPTOTIC FORMULAS FOR CLASS NUMBER SUMS OF INDEFINITE BINARY QUADRATIC FORMS ON ARITHMETIC PROGRESSIONS

2012 ◽  
Vol 09 (01) ◽  
pp. 27-51 ◽  
Author(s):  
YASUFUMI HASHIMOTO
Keyword(s):  
Class Number ◽  
Quadratic Forms ◽  
Modular Group ◽  
Conjugacy Classes ◽  
Arithmetic Progressions ◽  
Asymptotic Formulas ◽  
Binary Quadratic Forms ◽  
Congruence Subgroups ◽  
Arithmetic Properties ◽  
Prime Geodesic Theorem

It is known that there is a one-to-one correspondence between equivalence classes of primitive indefinite binary quadratic forms and primitive hyperbolic conjugacy classes of the modular group. Due to such a correspondence, Sarnak obtained the asymptotic formula for the class number sum in order of the fundamental unit by using the prime geodesic theorem for the modular group. In the present paper, we propose asymptotic formulas of the class number sums over discriminants on arithmetic progressions. Since there are relations between the arithmetic properties of the discriminants and the conjugacy classes in the finite groups given by the modular group and its congruence subgroups, we can get the desired asymptotic formulas by arranging the Tchebotarev-type prime geodesic theorem. While such asymptotic formulas were already given by Raulf, the approaches are quite different, the expressions of the leading terms of our asymptotic formulas are simpler and the estimates of the remainder terms are sharper.

Trace Classes and Quadratic Forms in the Modular Group

1994 ◽  
Vol 37 (2) ◽  
pp. 202-212 ◽  
Author(s):  
Benjamin Fine
Keyword(s):  
Quadratic Forms ◽  
Modular Group ◽  
Function Theory ◽  
Computational Procedure ◽  
Hecke Groups ◽  
Linear Fractional Transformations ◽  
Classical Work ◽  
Complete Set ◽  
Combinatorial Group ◽  
The Ideal

AbstractThe Modular Group M is PSL2(Z) the group of linear fractional transformations with integral entries and determinant one. M has been of great interest in many diverse fields of Mathematics, including Number Theory, Automorphic Function Theory and Group Theory. In this paper we give an effective algorithm to determine, for each integer d, a complete set of representatives for the trace classes in trace d. This algorithm depends on the combinatorial group theoretic structure of M. It has been subsequently extended by Sheingorn to the general Hecke groups. The number h(d) of trace classes in trace d is equal to the ideal class number of the field The algorithm mentioned above then provides a new straightforward computational procedure for determining h(d). Finally as an outgrowth of the algorithm we present a wide generalization of the Fermat Two-Square theorem. This last result can also be derived from classical work of Gauss.

Conjugacy Classes and Binary Quadratic Forms for the Hecke Groups

2013 ◽  
Vol 56 (3) ◽  
pp. 570-583 ◽  
Author(s):  
Giabao Hoang ◽  
Wendell Ressler
Keyword(s):  
Lower Bound ◽  
Quadratic Forms ◽  
Conjugacy Classes ◽  
Class Numbers ◽  
Block Length ◽  
Binary Quadratic Forms ◽  
Hecke Groups ◽  
Canonical Class

Abstract.In this paper we give a lower bound with respect to block length for the trace of non-elliptic conjugacy classes of the Hecke groups. One consequence of our bound is that there are finitely many conjugacy classes of a given trace in anyHecke group. We show that another consequence of our bound is that class numbers are finite for related hyperbolic ℤ[λ]-binary quadratic forms. We give canonical class representatives and calculate class numbers for some classes of hyperbolic ℤ[λ]-binary quadratic forms.

scholarly journals On S-class number relations of algebraic tori in Galois extensions of global fields

1991 ◽  
Vol 124 ◽  
pp. 133-144 ◽  
Author(s):  
Masanori Morishita
Keyword(s):  
Class Number ◽  
Quadratic Forms ◽  
Euler Number ◽  
Number Fields ◽  
Binary Quadratic Forms ◽  
Algebraic Number Fields ◽  
Algebraic Tori ◽  
Genus Theory ◽  
Class Number Formula ◽  
Number Formula

As an interpretation and a generalization of Gauss’ genus theory on binary quadratic forms in the language of arithmetic of algebraic tori, Ono [02] established an equality between a kind of “Euler number E(K/k)” for a finite Galois extension K/k of algebraic number fields and other arithmetical invariants associated to K/k. His proof depended on his Tamagawa number formula [01] and Shyr’s formula [Sh] which follows from the analytic class number formula of a torus. Later, two direct proofs were given by Katayama [K] and Sasaki [Sa].

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