scholarly journals Binary quadratic forms and ray class groups

2019 ◽  
Vol 150 (2) ◽  
pp. 695-720 ◽  
Author(s):  
Ick Sun Eum ◽  
Ja Kyung Koo ◽  
Dong Hwa Shin
Keyword(s):  
Quadratic Forms ◽  
Quadratic Field ◽  
Congruence Subgroup ◽  
Abelian Extension ◽  
Prime Ideals ◽  
Binary Quadratic Forms ◽  
Class Groups ◽  
Extended Form ◽  
Form Class ◽  
Ray Class Field

AbstractLet K be an imaginary quadratic field different from $\open{Q}(\sqrt {-1})$ and $\open{Q}(\sqrt {-3})$. For a positive integer N, let KN be the ray class field of K modulo $N {\cal O}_K$. By using the congruence subgroup ± Γ1(N) of SL2(ℤ), we construct an extended form class group whose operation is basically the Dirichlet composition, and explicitly show that this group is isomorphic to the Galois group Gal(KN/K). We also present an algorithm to find all distinct form classes and show how to multiply two form classes. As an application, we describe Gal(KNab/K) in terms of these extended form class groups for which KNab is the maximal abelian extension of K unramified outside prime ideals dividing $N{\cal O}_K$.

scholarly journals Central extensions and rational quadratic forms

1993 ◽  
Vol 130 ◽  
pp. 177-182 ◽  
Author(s):  
Yoshiomi Furuta ◽  
Tomio Kubota
Keyword(s):  
Central Extension ◽  
Quadratic Forms ◽  
Class Group ◽  
Quadratic Field ◽  
Abelian Extension ◽  
Narrow Sense ◽  
Maximal Extension ◽  
Central Extensions ◽  
Class Field ◽  
Ray Class Field

The purpose of this paper is to characterize by means of simple quadratic forms the set of rational primes that are decomposed completely in a non-abelian central extension which is abelian over a quadratic field. More precisely, let L = Q be a bicyclic biquadratic field, and let K = Q. Denote by the ray class field mod m of K in narrow sense for a large rational integer m. Let be the maximal abelian extension over Q contained in and be the maximal extension contained in such that Gal(/L) is contained in the center of Gal(/Q). Then we shall show in Theorem 2.1 that any rational prime p not dividing d1d2m is decomposed completely in /Q if and only if p is representable by rational integers x and y such that x ≡ 1 and y ≡ 0 mod m as followswhere a, b, c are rational integers such that b2 − 4ac is equal to the discriminant of K and (a) is a norm of a representative of the ray class group of K mod m.Moreover is decomposed completely in if and only if .

THE CUBIC CONGRUENCE x3 + Ax2 + Bx + C ≡ 0 (mod p) AND BINARY QUADRATIC FORMS II

2001 ◽  
Vol 64 (2) ◽  
pp. 273-274 ◽  
Author(s):  
BLAIR K. SPEARMAN ◽  
KENNETH S. WILLIAMS
Keyword(s):  
Quadratic Forms ◽  
Class Group ◽  
Binary Quadratic Forms ◽  
Form Class ◽  
Mod P

It is shown that the splitting modulo a prime p of a given monic, integral, irreducible cubic with non-square discriminant is equivalent to p being represented by forms in a certain subgroup of index 3 in the form class group of discriminant equal to the discriminant of the field defined by the cubic.

REPRESENTATION OF PRIMES BY THE PRINCIPAL FORM OF NEGATIVE DISCRIMINANT $\Delta$ WHEN $h(\Delta)$ IS 4

1994 ◽  
Vol 25 (4) ◽  
pp. 321-334
Author(s):  
KENNETH S. WILLIAMS ◽  
D. LIU
Keyword(s):  
Cyclic Group ◽  
Quadratic Forms ◽  
Class Group ◽  
Positive Definite ◽  
Negative Integer ◽  
Binary Quadratic Forms ◽  
Form Class ◽  
Quartic Polynomial ◽  
Mod P

Let $\Delta$ be a negative integer which is congruent to 0 or 1 (mod 4). Let $H(\Delta)$ denote the form class group of classes of positive-definite, primitive integral binary quadratic forms $ax^2 +bxy +cy^2$ of discriminant $\Delta$. If $H(\Delta)$ is a cyclic group of order 4, an explicit quartic polynomial $\rho \Delta(x)$ of the form $x^4-bx^2 +d$ with integral coefficients is determined such that for an odd prime $p$ not dividing $\Delta$, $p$ is represented by the principal form of discriminant $\Delta$ if and only if the congruence $\rho \Delta(x) \equiv 0$ (mod $p$) has four solutions.

The order of binary quadratic forms from representation numbers

2016 ◽  
Vol 12 (03) ◽  
pp. 679-690
Author(s):  
A. G. Earnest ◽  
Robert W. Fitzgerald
Keyword(s):  
Quadratic Form ◽  
Explicit Form ◽  
Quadratic Forms ◽  
Class Group ◽  
Binary Quadratic Form ◽  
Partial Answer ◽  
Binary Quadratic Forms ◽  
Form Class ◽  
The Relationship

We investigate the relationship between the numbers of representations of certain integers by a primitive integral binary quadratic form [Formula: see text] of discriminant [Formula: see text] and the order of the class of [Formula: see text] in the form class group of discriminant [Formula: see text], in the case when this order is even. The explicit form of the solutions obtained is used to give a partial answer to a question regarding which multiples of [Formula: see text] can be parameterized in a particular way.

scholarly journals Ramanujan’s function k(τ)=r(τ)r 2(2τ) and its modularity

2020 ◽  
Vol 18 (1) ◽  
pp. 1727-1741
Author(s):  
Yoonjin Lee ◽  
Yoon Kyung Park
Keyword(s):  
Continued Fraction ◽  
Quadratic Field ◽  
Singular Values ◽  
Modular Function ◽  
Imaginary Quadratic Field ◽  
Modular Equations ◽  
Positive Rational Number ◽  
Congruence Relations ◽  
Symmetry Relations ◽  
Ray Class Field

Abstract We study the modularity of Ramanujan’s function k ( τ ) = r ( τ ) r 2 ( 2 τ ) k(\tau )=r(\tau ){r}^{2}(2\tau ) , where r ( τ ) r(\tau ) is the Rogers-Ramanujan continued fraction. We first find the modular equation of k ( τ ) k(\tau ) of “an” level, and we obtain some symmetry relations and some congruence relations which are satisfied by the modular equations; these relations are quite useful for reduction of the computation cost for finding the modular equations. We also show that for some τ \tau in an imaginary quadratic field, the value k ( τ ) k(\tau ) generates the ray class field over an imaginary quadratic field modulo 10; this is because the function k is a generator of the field of the modular function on Γ 1 ( 10 ) {{\mathrm{\Gamma}}}_{1}(10) . Furthermore, we suggest a rather optimal way of evaluating the singular values of k ( τ ) k(\tau ) using the modular equations in the following two ways: one is that if j ( τ ) j(\tau ) is the elliptic modular function, then one can explicitly evaluate the value k ( τ ) k(\tau ) , and the other is that once the value k ( τ ) k(\tau ) is given, we can obtain the value k ( r τ ) k(r\tau ) for any positive rational number r immediately.

scholarly journals On Elliptic Curves with Complex Multiplication as Factors of the Jacobians of Modular Function Fields

1971 ◽  
Vol 43 ◽  
pp. 199-208 ◽  
Author(s):  
Goro Shimura
Keyword(s):  
Elliptic Curves ◽  
Cusp Form ◽  
Mellin Transform ◽  
Quadratic Field ◽  
Congruence Subgroup ◽  
Function Fields ◽  
Complex Multiplication ◽  
Modular Function ◽  
Imaginary Quadratic Field

1. As Hecke showed, every L-function of an imaginary quadratic field K with a Grössen-character γ is the Mellin transform of a cusp form f(z) belonging to a certain congruence subgroup Γ of SL2(Z). We can normalize γ so that

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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