scholarly journals Finite arithmetic subgroups of GLn, V

1997 ◽  
Vol 146 ◽  
pp. 131-148 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Number Field ◽  
Rational Number ◽  
Galois Extension ◽  
Finite Subgroup ◽  
Finite Arithmetic

Abstract.Let K be a finite Galois extension of the rational number field Q and G a Gal(K/Q)-stable finite subgroup of GLn(OK). We have shown that G is of A-type in several cases under some restrictions on K. In this paper, we show that it is true for n = 2 without any restrictions on K.

scholarly journals Finite arithmetic subgroups of GLn, II

1980 ◽  
Vol 77 ◽  
pp. 137-143 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Number Field ◽  
Rational Number ◽  
Galois Extension ◽  
Finite Subgroup ◽  
Maximal Order ◽  
Finite Arithmetic ◽  
Totally Real

In [1] ∼ [6] the following question was treated: Let k be a totally real Galois extension of the rational number field Q, O the maximal order of k and G a finite subgroup of GL(n, O) which is stable under the operation of G(k/Q). Then does G ⊂ GL(n, Z) hold?

scholarly journals Finite arithmetic subgroups of GLn, IV

1996 ◽  
Vol 142 ◽  
pp. 183-188 ◽  
Author(s):  
Yoshiyuki Kitaoka ◽  
Hiroshi Suzuki
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Rational Number ◽  
Finite Subgroup ◽  
The Third ◽  
Finite Arithmetic

In this paper, we improve a result of the third paper of this series, that is we showTHEOREM. Let K be a nilpotent extension of the rational number field Q with Galois group Γ, and G a Γ- stable finite subgroup of GLn(0K). Then G is of A-type.

scholarly journals Some Remarks to Ono’s theorem on a generalization of Gauss’ genus theory

1988 ◽  
Vol 111 ◽  
pp. 131-142 ◽  
Author(s):  
Ryuji Sasaki
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Algebraic Number ◽  
Rational Number ◽  
Galois Extension ◽  
Multiplicative Group ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Genus Theory

Let K\k be a finite Galois extension of finite algebraic number fields with Galois group g. We denote by Gm the multiplicative group defined over the rational number field Q and put.

scholarly journals Gaussian Integers

2013 ◽  
Vol 21 (2) ◽  
pp. 115-125
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Daichi Mizushima ◽  
Yasunari Shidama
Keyword(s):  
Number Field ◽  
Rational Number ◽  
Rational Numbers ◽  
Gaussian Integer ◽  
Algebraic Integers ◽  
Quotient Field ◽  
Gaussian Integers ◽  
Integer Ring

Summary Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic.

scholarly journals Automorphy factors for a Hilbert modular group

1988 ◽  
Vol 30 (2) ◽  
pp. 231-236
Author(s):  
Shigeaki Tsuyumine
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Rational Number ◽  
Modular Group ◽  
Algebraic Number Field ◽  
Maximal Order ◽  
Hilbert Modular Group ◽  
Hilbert Modular ◽  
Totally Real

Let K be a totally real algebraic number field of degree n > 1, and let OK be the maximal order. We denote by гk, the Hilbert modular group SL2(OK) associated with K. On the extent of the weight of an automorphy factor for гK, some restrictions are imposed, not as in the elliptic modular case. Maass [5] showed that the weight is integral for K = ℚ(√5). It was shown by Christian [1] that for any Hilbert modular group it is a rational number with the bounded denominator depending on the group.

scholarly journals On Galois groups of class two extensions over the rational number field

1979 ◽  
Vol 75 ◽  
pp. 121-131 ◽  
Author(s):  
Susumu Shirai
Keyword(s):  
Nilpotent Group ◽  
Galois Group ◽  
Number Field ◽  
Direct Product ◽  
Rational Number ◽  
Nilpotency Class ◽  
Abelian Extension ◽  
Galois Groups ◽  
Classification Theory ◽  
Ray Class Field

Let Q be the rational number field, K/Q be a maximal Abelian extension whose degree is some power of a prime l, and let f(K) be the conductor of K/Q; if l = 2, let K be complex, and if in addition f(K) ≡ 0 (mod 2), let f(K) ≡ 0 (mod 16). Denote by (K) the Geschlechtermodul of K over Q and by K̂ the maximal central l-extension of K/Q contained in the ray class field mod (K) of K. A. Fröhlich [1, Theorem 4] completely determined the Galois group of K̂ over Q in purely rational terms. The proof is based on [1, Theorem 3], though he did not write the proof in the case f(K) ≡ 0 (mod 16). Moreover he gave a classification theory of all class two extensions over Q whose degree is a power of l. Hence we know the set of fields of nilpotency class two over Q, because a finite nilpotent group is a direct product of all its Sylow subgroups. But the theory becomes cumbersome, and it is desirable to reconstruct a more elementary one.

scholarly journals Cusp forms of weight one, quartic reciprocity and elliptic curves

1985 ◽  
Vol 98 ◽  
pp. 117-137 ◽  
Author(s):  
Noburo Ishii
Keyword(s):  
Positive Integer ◽  
Elliptic Curves ◽  
Galois Group ◽  
Cusp Form ◽  
Rational Number ◽  
Galois Extension ◽  
Dihedral Group ◽  
Cusp Forms ◽  
Two Dimensional ◽  
Weight One

Let m be a non-square positive integer. Let K be the Galois extension over the rational number field Q generated by and . Then its Galois group over Q is the dihedral group D4 of order 8 and has the unique two-dimensional irreducible complex representation ψ. In view of the theory of Hecke-Weil-Langlands, we know that ψ defines a cusp form of weight one (cf. Serre [6]).

scholarly journals On a Certain Function Analogous to log|η(z)|

1970 ◽  
Vol 40 ◽  
pp. 193-211 ◽  
Author(s):  
Tetsuya Asai
Keyword(s):  
Eisenstein Series ◽  
Number Field ◽  
Algebraic Number ◽  
Rational Number ◽  
Modular Group ◽  
Classical Case ◽  
Algebraic Number Field ◽  
View Point ◽  
Limit Formula

The purpose of this paper is to give the limit formula of the Kronecker’s type for a non-holomorphic Eisenstein series with respect to a Hubert modular group in the case of an arbitrary algebraic number field. Actually, we shall generalize the following result which is well-known as the first Kronecker’s limit formula. From our view-point, this classical case is corresponding to the case of the rational number field Q.

scholarly journals On maximal orders of division quaternion algebras over the rational number field with certain optimal embeddings

1982 ◽  
Vol 88 ◽  
pp. 181-195 ◽  
Author(s):  
Tomoyoshi Ibukiyama
Keyword(s):  
Number Field ◽  
Rational Number ◽  
Quaternion Algebra ◽  
Quaternion Algebras ◽  
Maximal Orders

In this paper, we shall give explicit Z-basis of certain maximal orders of definite quaternion algebras over the rational number field Q (See Theorems below). We shall also give some remarks on symmetric maximal orders in Ponomarev [9] and Hashimoto [6] (Proposition 4.3). More precise contents are as follows. Let D be a division quaternion algebra over Q.

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