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 CONGRUENCES CONCERNING LEGENDRE POLYNOMIALS III

2013 ◽  
Vol 09 (04) ◽  
pp. 965-999 ◽  
Author(s):  
ZHI-HONG SUN
Keyword(s):  
Positive Integer ◽  
Quadratic Forms ◽  
Legendre Polynomials ◽  
Binary Quadratic Forms ◽  
Modulo P ◽  
Mod P

Suppose that p is an odd prime and d is a positive integer. Let x and y be integers given by p = x2+dy2 or 4p = x2+dy2. In this paper we determine x( mod p) for many values of d. For example, [Formula: see text] where x is chosen so that x ≡ 1 ( mod 3). We also pose some conjectures on supercongruences modulo p2 concerning binary quadratic forms.

scholarly journals On the least prime ideal and Siegel zeros

2016 ◽  
Vol 12 (08) ◽  
pp. 2201-2229 ◽  
Author(s):  
Asif Zaman
Keyword(s):  
Prime Ideal ◽  
Number Field ◽  
Quadratic Forms ◽  
Class Group ◽  
Binary Quadratic Forms ◽  
Integral Ideal ◽  
Siegel Zero ◽  
Siegel Zeros ◽  
Special Case ◽  
Exceptional Character

Let [Formula: see text] be a number field, [Formula: see text] be an integral ideal, and [Formula: see text] be the associated narrow ray class group. Suppose [Formula: see text] possesses a real exceptional character [Formula: see text], possibly principal, with a Siegel zero [Formula: see text]. For [Formula: see text] satisfying [Formula: see text] [Formula: see text], we establish an effective [Formula: see text]-uniform Linnik-type bound with explicit exponents for the least norm of a prime ideal [Formula: see text]. A special case of this result is a bound for the least rational prime represented by certain binary quadratic forms.

scholarly journals Václav Šimerka: quadratic forms and factorization

2013 ◽  
Vol 16 ◽  
pp. 118-129
Author(s):  
F. Lemmermeyer
Keyword(s):  
Quadratic Forms ◽  
Class Group ◽  
Binary Quadratic Forms ◽  
Carmichael Numbers

AbstractIn this article we show that the Czech mathematician Václav Šimerka discovered the factorization of $\frac{1}{9} (1{0}^{17} - 1)$ using a method based on the class group of binary quadratic forms more than 120 years before Shanks and Schnorr developed similar algorithms. Šimerka also gave the first examples of what later became known as Carmichael numbers.

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 $N{\cal O}_K$.

Sign in / Sign up

Export Citation Format

Share Document