Selberg trace formula for the Hilbert modular group of a real quadratic algebraic number field

1982 ◽  
Vol 19 (6) ◽  
pp. 1637-1652 ◽  
Author(s):  
P. G. Zograf
Keyword(s):  
Number Field ◽  
Trace Formula ◽  
Algebraic Number ◽  
Modular Group ◽  
Algebraic Number Field ◽  
Selberg Trace Formula ◽  
Hilbert Modular Group ◽  
Hilbert Modular ◽  
Real Quadratic

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 SYMMETRIES FOR BORCHERDS LIFTS ON HILBERT MODULAR GROUPS AND HIRZEBRUCH–ZAGIER DIVISORS

2013 ◽  
Vol 24 (08) ◽  
pp. 1350065 ◽  
Author(s):  
BERNHARD HEIM ◽  
ATSUSHI MURASE
Keyword(s):  
Modular Group ◽  
Quadratic Field ◽  
Hecke Operators ◽  
The Other ◽  
Real Quadratic Field ◽  
Hilbert Modular Group ◽  
Modular Groups ◽  
Hilbert Modular ◽  
The One ◽  
Real Quadratic

We show certain symmetries for Borcherds lifts on the Hilbert modular group over a real quadratic field. We give two different proofs, the one analytic and the other arithmetic. The latter proof yields an explicit description of the action of Hecke operators on Borcherds lifts.

Regular Elliptic Classes and the Stable Relative Trace Formula

1992 ◽  
Vol 35 (2) ◽  
pp. 230-236
Author(s):  
K. F. Lai
Keyword(s):  
Number Field ◽  
Trace Formula ◽  
Algebraic Number ◽  
Reductive Group ◽  
Algebraic Number Field ◽  
Double Cosets ◽  
Relative Trace Formula

AbstractWe study the relative trace formula of a reductive group over an algebraic number field. Following Langlands we stabilize the geometric side of the relative trace formula contributed by the elliptic regular double cosets.

The Additive Characters of the Witt Ring of an Algebraic Number Field

1988 ◽  
Vol 40 (3) ◽  
pp. 546-588 ◽  
Author(s):  
P. E. Conner ◽  
Noriko Yui
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Quadratic Character ◽  
Additive Character ◽  
Witt Ring ◽  
Root Numbers ◽  
Class Invariants ◽  
Local Root ◽  
Real Quadratic

For an algebraic number field K there is a similarity between the additive characters defined on the Witt ring W(K), [20], [11], [17], [14, p. 131], and the local root numbers associated to a real orthogonal representation of the absolute Galois group of K, [18], [5]. Using results of Deligne and of Serre, [16], we shall derive in (5.3) a formula expressing the value, at a prime in K, of the additive character on a Witt class in terms of the rank modulo 2, the stable Hasse-Witt invariant and the local root number associated to the real quadratic character defined by the square class of the discriminant. Thus we are able to separate out the contributions made to the value of the additive character by each of the standard Witt class invariants.

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.

RESTRICTIONS OF HILBERT MODULAR FORMS

2009 ◽  
Vol 05 (01) ◽  
pp. 67-80
Author(s):  
NAJIB OULED AZAIEZ
Keyword(s):  
Modular Forms ◽  
Modular Group ◽  
Quadratic Field ◽  
Hilbert Modular Forms ◽  
Real Quadratic Field ◽  
Modular Surface ◽  
Modular Curve ◽  
Hilbert Modular Group ◽  
Hilbert Modular ◽  
Real Quadratic

Let Γ ⊂ PSL (2, ℝ) be a discrete and finite covolume subgroup. We suppose that the modular curve [Formula: see text] is "embedded" in a Hilbert modular surface [Formula: see text], where ΓK is the Hilbert modular group associated to a real quadratic field K. We define a sequence of restrictions (ρn)n∈ℕ of Hilbert modular forms for ΓK to modular forms for Γ. We denote by Mk1, k2(ΓK) the space of Hilbert modular forms of weight (k1, k2) for ΓK. We prove that ∑n∈ℕ ∑k1, k2∈ℕ ρn(Mk1, k2(ΓK)) is a subalgebra closed under Rankin–Cohen brackets of the algebra ⊕k∈ℕ Mk(Γ) of modular forms for Γ.

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