scholarly journals The dimension of the space of cusp forms on the Siegel upper half plane of degree two related to a quaternion unitary group

1981 ◽  
Vol 33 (1) ◽  
pp. 125-145 ◽  
Author(s):  
Tsuneo ARAKAWA
Keyword(s):  
Half Plane ◽  
Unitary Group ◽  
Cusp Forms ◽  
Upper Half Plane ◽  
Space Of Cusp Forms

A note on cusp forms and representations of SL2(𝔽𝑝)

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Zhe Chen
Keyword(s):  
Finite Field ◽  
Half Plane ◽  
Congruence Subgroup ◽  
Holomorphic Functions ◽  
Principal Congruence ◽  
Cusp Forms ◽  
Cohomology Space ◽  
Principal Congruence Subgroup ◽  
Upper Half Plane ◽  
Space Of Cusp Forms

AbstractCusp forms are certain holomorphic functions defined on the upper half-plane, and the space of cusp forms for the principal congruence subgroup \Gamma(p), 𝑝 a prime, is acted on by \mathrm{SL}_{2}(\mathbb{F}_{p}). Meanwhile, there is a finite field incarnation of the upper half-plane, the Deligne–Lusztig (or Drinfeld) curve, whose cohomology space is also acted on by \mathrm{SL}_{2}(\mathbb{F}_{p}). In this note, we compute the relation between these two spaces in the weight 2 case.

scholarly journals Rankin-Selberg method for Siegel cusp forms

1990 ◽  
Vol 120 ◽  
pp. 35-49 ◽  
Author(s):  
Tadashi Yamazaki
Keyword(s):  
Holomorphic Function ◽  
Half Plane ◽  
Cusp Form ◽  
Functional Equations ◽  
Modular Group ◽  
Cusp Forms ◽  
The Real ◽  
Upper Half Plane ◽  
Siegel Cusp Form

Let Gn (resp. Γn) be the real symplectic (resp. Siegel modular) group of degree n. The Siegel cusp form is a holomorphic function on the Siegel upper half plane which satisfies functional equations relative to Γn and vanishes at the cusps.

scholarly journals On construction of holomorphic cusp forms of half integral weight

1975 ◽  
Vol 58 ◽  
pp. 83-126 ◽  
Author(s):  
Takuro Shintani
Keyword(s):  
Half Plane ◽  
Cusp Forms ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Holomorphic Cusp Forms ◽  
Upper Half Plane

In [10], G.Shimura gave a method of constructing holomorphic cusp forms of even integral weight from given forms of half integral weight. In this paper, we try to present an inverse construction. To state our main result, some notational preliminaries are necessary. We denote by the complex upper half plane.

scholarly journals Mixed cusp forms and parabolic cohomology

1997 ◽  
Vol 62 (3) ◽  
pp. 279-289
Author(s):  
Min Ho Lee
Keyword(s):  
Half Plane ◽  
Fuchsian Group ◽  
Cusp Forms ◽  
Holomorphic Map ◽  
Cohomology Space ◽  
Upper Half Plane

AbstractLet Sk, l(Γ, ω, χ) be the space of mixed cusp forms of type (k, l) associated to a Fuchsian group Γ, a holomorphic map ω: ℋ → ℋ of the upper half plane into itself and a homomorphism χ: Γ → SL(2, R) such that ω and χ are equivariant. We construct a map from Sk, l(Γ, ω, χ) to the parabolic cohomology space of Γ with coefficients in some Γ-module and prove that this map is injective.

scholarly journals CONSTRUCTING SIMULTANEOUS HECKE EIGENFORMS

2010 ◽  
Vol 06 (05) ◽  
pp. 1117-1137 ◽  
Author(s):  
T. SHEMANSKE ◽  
S. TRENEER ◽  
L. WALLING
Keyword(s):  
Hecke Operators ◽  
Cusp Forms ◽  
Hecke Eigenforms ◽  
Linearly Independent ◽  
Integral Weight ◽  
Trivial Character ◽  
Free Level ◽  
Fixed Space ◽  
Space Of Cusp Forms

It is well known that newforms of integral weight are simultaneous eigenforms for all the Hecke operators, and that the converse is not true. In this paper, we give a characterization of all simultaneous Hecke eigenforms associated to a given newform, and provide several applications. These include determining the number of linearly independent simultaneous eigenforms in a fixed space which correspond to a given newform, and characterizing several situations in which the full space of cusp forms is spanned by a basis consisting of such eigenforms. Part of our results can be seen as a generalization of results of Choie–Kohnen who considered diagonalization of "bad" Hecke operators on spaces with square-free level and trivial character. Of independent interest, but used herein, is a lower bound for the dimension of the space of newforms with arbitrary character.

scholarly journals On Complex Explicit Formulae Connected with the Möbius Function of an Elliptic Curve

2014 ◽  
Vol 57 (2) ◽  
pp. 381-389
Author(s):  
Adrian Łydka
Keyword(s):  
Functional Equation ◽  
Meromorphic Function ◽  
Elliptic Curve ◽  
Explicit Formula ◽  
Half Plane ◽  
Complex Plane ◽  
Möbius Function ◽  
Mobius Function ◽  
Upper Half Plane ◽  
Explicit Formulae

AbstractWe study analytic properties function m(z, E), which is defined on the upper half-plane as an integral from the shifted L-function of an elliptic curve. We show that m(z, E) analytically continues to a meromorphic function on the whole complex plane and satisfies certain functional equation. Moreover, we give explicit formula for m(z, E) in the strip |ℑz| < 2π.

scholarly journals The dimension formula of the space of cusp forms of weight one for Γ0(p)

1988 ◽  
Vol 111 ◽  
pp. 115-129 ◽  
Author(s):  
Yoshio Tanigawa ◽  
Hirofumi Ishikawa
Keyword(s):  
Cusp Forms ◽  
Dimension Formula ◽  
Weight One ◽  
Space Of Cusp Forms

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.

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