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 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 Scalar extension of quadratic lattices II

1977 ◽  
Vol 67 ◽  
pp. 159-164 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Rational Numbers ◽  
Positive Definite ◽  
Maximal Order ◽  
Quadratic Space ◽  
Quadratic Lattices ◽  
Totally Real ◽  
Quadratic Lattice

Let k be a totally real algebraic number field, the maximal order of k, and let L (resp. M) be a Z-lattice of a positive definite quadratic space U (resp. V) over the field Q of rational numbers. Suppose that there is an isometry σ from L onto M. We have shown that the assumption implies σ(L) = M in some cases in [2]. Our aim in this paper is to improve the results of [2]. In § 1 we introduce the notion of E-type: Let L be a positive definite quadratic lattice over Z.

scholarly journals On standard L-functions attached to automorphic forms on definite orthogonal groups

1998 ◽  
Vol 152 ◽  
pp. 57-96 ◽  
Author(s):  
Atsushi Murase ◽  
Takashi Sugano
Keyword(s):  
Functional Equation ◽  
Number Field ◽  
Algebraic Number ◽  
Orthogonal Group ◽  
Automorphic Form ◽  
Automorphic Forms ◽  
Algebraic Number Field ◽  
Constant Function ◽  
Orthogonal Groups ◽  
Totally Real

Abstract.We show an explicit functional equation of the standard L-function associated with an automorphic form on a definite orthogonal group over a totally real algebraic number field. This is a completion and a generalization of our previous paper, in which we constructed standard L-functions by using Rankin-Selberg convolution and the theory of Shintani functions under certain technical conditions. In this article we remove these conditions. Furthermore we show that the L-function of f has a pole at s = m/2 if and only if f is a constant function.

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 On the representation of a number as a sum of squares

1978 ◽  
Vol 19 (2) ◽  
pp. 173-197 ◽  
Author(s):  
Karl-Bernhard Gundlach
Keyword(s):  
Positive Integer ◽  
Modular Forms ◽  
Remainder Term ◽  
Modular Group ◽  
Congruence Subgroup ◽  
Hilbert Modular Group ◽  
Totally Positive ◽  
Hilbert Modular ◽  
Image Position ◽  
Totally Real

It is well known that the number Ak(m) of representations of a positive integer m as the sum of k squares of integers can be expressed in the formwhere Pk(m) is a divisor function, and Rk(m) is a remainder term of smaller order. (1) is a consequence of the fact thatis a modular form for a certain congruence subgroup of the modular group, andwithwhere Ek(z) is an Eisenstein series and is a cusp form (as was first pointed out by Mordell [9]). The result (1) remains true if m is taken to be a totally positive integer from a totally real number field K and Ak(m) is the number of representations of m as the sum of k squares of integers from K (at least for 2|k, k>2, and for those cases with 2+k which have been investigated). then are replaced by modular forms for a subgroup of the Hilbert modular group with Fourier expansions of the form (10) (see section 2).

On the elliptic points of the Hilbert modular group of the totally real cyclotomic cubic field ℚ(ζ9)+

2013 ◽  
Vol 143 (5) ◽  
pp. 893-903
Author(s):  
A. Arenas
Keyword(s):  
Modular Group ◽  
Root Of Unity ◽  
Cubic Field ◽  
Hilbert Modular Group ◽  
Hilbert Modular ◽  
Totally Real

We determine explicitly the elliptic points with respect to the Hilbert modular group associated with the totally real cyclotomic cubic field ℚ(ζ + ζ−1), where ζ stands for a primitive 9th root of unity.

scholarly journals On the class number of a unit lattice over a ring of real quadratic integers

1983 ◽  
Vol 92 ◽  
pp. 89-106 ◽  
Author(s):  
Yoshio Mimura
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Class Number ◽  
Orthonormal Basis ◽  
Algebraic Number Field ◽  
Positive Definite ◽  
Quadratic Space ◽  
Totally Real ◽  
Real Quadratic

Let K be a totally real algebraic number field. In a positive definite quadratic space over K a lattice En is called a unit lattice of rank n if En has an orthonormal basis {e1 …, en}. The class number one problem is to find n and K for which the class number of En is one. Dzewas ([1]), Nebelung ([3]), Pfeuffer ([6], [7]) and Peters ([5]) have settled this problem.

scholarly journals On Block Idempotents of Modular Group Rings

1966 ◽  
Vol 27 (2) ◽  
pp. 429-433 ◽  
Author(s):  
Masaru Osima
Keyword(s):  
Prime Number ◽  
Finite Order ◽  
Number Field ◽  
Algebraic Number ◽  
Modular Group ◽  
Algebraic Number Field ◽  
Roots Of Unity ◽  
Group Rings ◽  
Irreducible Characters

We consider a group G of finite order g = pag′ where p is a prime number and (p, g′) = 1. Let Ω be the algebraic number field which contains the p-th roots of unity. Let K1, K2,…, Kn be the classes of conjugate elements in G and the first m(≦n) classes be p-regular. There exist n distinct (absolutely) irreducible characters x1, x2,…, xn of G.

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