Large Sieves and Cusp Forms of Weight One

10.1017/s0027763000001033 ◽

1988 ◽

Vol 111 ◽

pp. 115-129 ◽

Cited By ~ 1

Author(s):

Yoshio Tanigawa ◽

Hirofumi Ishikawa

Keyword(s):

Cusp Forms ◽

Dimension Formula ◽

Weight One ◽

The purpose of this paper is to study the dimension formula for cusp forms of weight one, following the series of Hiramatsu [2] and Hiramatsu-Akiyama [3]. We define as usual the subgroup Γ0(N) of SL2(Z) by.

Cusp forms of weight one, quartic reciprocity and elliptic curves

10.1017/s0027763000021413 ◽

1985 ◽

Vol 98 ◽

pp. 117-137 ◽

Cited By ~ 9

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]).

The modular equation and modular forms of weight one

10.1017/s0027763000000271 ◽

1985 ◽

Vol 100 ◽

pp. 145-162 ◽

Cited By ~ 1

Author(s):

Toyokazu Hiramatsu ◽

Yoshio Mimura

Keyword(s):

Modular Forms ◽

Quadratic Field ◽

Congruence Subgroup ◽

Algebraic Integer ◽

Complex Multiplication ◽

Cusp Forms ◽

Imaginary Quadratic Fields ◽

Modular Equation ◽

Class Invariants ◽

Weight One

This is a continuation of the previous paper [8] concerning the relation between the arithmetic of imaginary quadratic fields and cusp forms of weight one on a certain congruence subgroup. Let K be an imaginary quadratic field, say K = with a prime number q ≡ − 1 mod 8, and let h be the class number of K. By the classical theory of complex multiplication, the Hubert class field L of K can be generated by any one of the class invariants over K, which is necessarily an algebraic integer, and a defining equation of which is denoted byΦ(x) = 0.

ON THE DIMENSION OF THE SPACE OF CUSP FORMS OF OCTAHEDRAL TYPE

International Journal of Number Theory ◽

10.1142/s1793042110003198 ◽

2010 ◽

Vol 06 (04) ◽

pp. 767-783 ◽

Cited By ~ 1

Author(s):

SATADAL GANGULY

Keyword(s):

Galois Representations ◽

Character Formula ◽

Cusp Forms ◽

Holomorphic Cusp Forms ◽

Weight One ◽

For a prime q ≡ 3 ( mod 4) and the character [Formula: see text], we consider the subspace of the space of holomorphic cusp forms of weight one, level q and character χ that is spanned by forms that correspond to Galois representations of octahedral type. We prove that this subspace has dimension bounded by [Formula: see text] upto multiplication by a constant that depends only on ε.

Petersson norm of cusp forms associated to real quadratic fields

Forum Mathematicum ◽

10.1515/forum-2017-0227 ◽

2018 ◽

Vol 30 (5) ◽

pp. 1097-1109

Author(s):

Yingkun Li

Keyword(s):

Quadratic Fields ◽

Cusp Forms ◽

Real Quadratic Fields ◽

Weight One ◽

Real Quadratic ◽

Petersson Norm

AbstractIn this article, we compute the Petersson norm of a family of weight one cusp forms constructed by Hecke and express it in terms of the Rademacher symbol and the regulator of real quadratic fields.

On some dimension formula for automorphic forms of weight one III

10.1017/s0027763000001069 ◽

1988 ◽

Vol 111 ◽

pp. 157-163 ◽

Cited By ~ 2

Author(s):

Toyokazu Hiramatsu ◽

Shigeki Akiyama

Keyword(s):

Zeta Function ◽

Linear Space ◽

Fuchsian Group ◽

Fundamental Domain ◽

Automorphic Forms ◽

Cusp Forms ◽

Dimension Formula ◽

Weight One ◽

Let Γ be a fuchsian group of the first kind and assume that Γ does not contain the element . Let S1(Γ) be the linear space of cusp forms of weight 1 on the group Γ and denote by d1 the dimension of the space S1(Γ). When the group Γ has a compact fundamental domain, we have obtained the following (Hiramatsu [3]):(*) ,where ς*(s) denotes the Selberg type zeta function defined by.

The Selberg trace formula for modular correspondences

10.1017/s0027763000001823 ◽

1990 ◽

Vol 117 ◽

pp. 93-123

Author(s):

Shigeki Akiyama ◽

Yoshio Tanigawa

Keyword(s):

Zeta Function ◽

Kernel Function ◽

Trace Formula ◽

Conjugacy Classes ◽

Hecke Operators ◽

Cusp Forms ◽

Selberg Trace Formula ◽

Weight One ◽

In Selberg [11], he introduced the trace formula and applied it to computations of traces of Hecke operators acting on the space of cusp forms of weight greater than or equal to two. But for the case of weight one, the similar method is not effective. It only gives us a certain expression of the dimension of the space of cusp forms by the residue of the Selberg type zeta function. Here the Selberg type zeta function appears in the contribution from the hyperbolic conjugacy classes when we write the trace formula with a certain kernel function ([3J, [4], [7], [8], [9], [12]).

On Some Dimension Formula for Automorphic Forms of Weight One, II

10.1017/s0027763000000829 ◽

1987 ◽

Vol 105 ◽

pp. 169-186 ◽

Cited By ~ 5

Author(s):

Toyokazu Hiramatsu

Keyword(s):

Linear Space ◽

Unitary Representation ◽

Fuchsian Group ◽

Automorphic Forms ◽

Cusp Forms ◽

Interesting Problem ◽

Dimension Formula ◽

Weight One ◽

Roch Theorem ◽

Let Γ be a fuchsian group of the first kind and assume that Γ contains the element and let x be a unitary representation of Γ of degree 1 such that X(—I) = — 1. Let S1(Γ,X) be the linear space of cusp forms of weight one on the group Γ with character X. We shall denote by d1 the dimension of the linear space S1(Γ, X). It is not effective to compute the number dl by means of the Riemann-Roch theorem. Because of this reason, it is an interesting problem in its own right to determine the number d1 by some other method (for example,).