Invariants of algebraic tori of degree and weight 2

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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Cyclic Polynomials Arising from Kummer Theory of Norm Algebraic Tori

Lecture Notes in Computer Science - Algorithmic Number Theory ◽

10.1007/11792086_8 ◽

2006 ◽

pp. 102-113 ◽

Cited By ~ 3

Author(s):

Masanari Kida

Keyword(s):

Algebraic Tori ◽

Kummer Theory

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Corrigenda On a classification of the function fields of algebraic tori: (Nagoya Math. J. 56 (1975), 85–104)

Nagoya Mathematical Journal ◽

10.1017/s0027763000019012 ◽

1980 ◽

Vol 79 ◽

pp. 187-190 ◽

Cited By ~ 2

Author(s):

Shizuo Endo ◽

Takehiko Miyata

Keyword(s):

Direct Product ◽

Cyclic Group ◽

Prime Divisor ◽

Dihedral Group ◽

Quaternion Group ◽

Algebraic Tori ◽

Root Of Unity ◽

Generalized Quaternion Group ◽

Generalized Quaternion

There are some errors in Theorems 3.3 and 4.2 in [2]. In this note we would like to correct them.1) In Theorem 3.3 (and [IV]), the condition (1) must be replaced by the following one;(1) П is (i) a cyclic group, (ii) a dihedral group of order 2m, m odd, (iii) a direct product of a cyclic group of order qf, q an odd prime, f ≧ 1, and a dihedral group of order 2m, m odd, where each prime divisor of m is a primitive qf-1(q — 1)-th root of unity modulo qf, or (iv) a generalized quaternion group of order 4m, m odd, where each prime divisor of m is congruent to 3 modulo 4.

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Integral structures in algebraic tori

Izvestiya Mathematics ◽

10.1070/im1995v059n05abeh000038 ◽

1995 ◽

Vol 59 (5) ◽

pp. 881-897 ◽

Cited By ~ 1

Author(s):

V E Voskresenskii ◽

T V Fomina

Keyword(s):

Algebraic Tori ◽

Integral Structures

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Invariant Parallels, Invariant Meridians and Limit Cycles of Polynomial Vector Fields on Some 2-Dimensional Algebraic Tori in $$\mathbb R ^3$$ R 3

Journal of Dynamics and Differential Equations ◽

10.1007/s10884-013-9315-4 ◽

2013 ◽

Vol 25 (3) ◽

pp. 777-793 ◽

Cited By ~ 3

Author(s):

Jaume Llibre ◽

Salomón Rebollo-Perdomo

Keyword(s):

Limit Cycles ◽

Vector Fields ◽

Polynomial Vector ◽

Polynomial Vector Fields ◽

Algebraic Tori

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On some class number relations for Galois extensions

Nagoya Mathematical Journal ◽

10.1017/s0027763000002579 ◽

1987 ◽

Vol 107 ◽

pp. 121-133 ◽

Cited By ~ 6

Author(s):

Takashi Ono

Keyword(s):

Number Field ◽

Algebraic Number ◽

Class Number ◽

Quadratic Forms ◽

Algebraic Number Field ◽

Narrow Sense ◽

Binary Quadratic Forms ◽

Finite Degree ◽

Algebraic Tori ◽

Number Of Classes

Let k be an algebraic number field of finite degree over Q, the field of rationals, and K be an extension of finite degree over k. By the use of the class number of algebraic tori, we can introduce an arithmetical invariant E(K/k) for the extension K/k. When k = Q and K is quadratic over Q, the formula of Gauss on the genera of binary quadratic forms, i.e. the formula where = the class number of K in the narrow sense, the number of classes is a genus of the norm form of K/Q and tK = the number of distinct prime factors of the discriminant of K, may be considered as an equality between E(K/Q) and other arithmetical invariants of K.

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