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 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 Period polynomials and explicit formulas for Hecke operators on Γ0(2)

2009 ◽  
Vol 146 (2) ◽  
pp. 321-350 ◽  
Author(s):  
SHINJI FUKUHARA ◽  
YIFAN YANG
Keyword(s):  
Vector Space ◽  
Explicit Form ◽  
Congruence Subgroup ◽  
Bernoulli Polynomials ◽  
Affirmative Answer ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Isomorphism Theorem ◽  
Explicit Formulas ◽  
Space Of Cusp Forms

AbstractLet Sw+2(Γ0(N)) be the vector space of cusp forms of weight w + 2 on the congruence subgroup Γ0(N). We first determine explicit formulas for period polynomials of elements in Sw+2(Γ0(N)) by means of Bernoulli polynomials. When N = 2, from these explicit formulas we obtain new bases for Sw+2(Γ0(2)), and extend the Eichler–Shimura–Manin isomorphism theorem to Γ0(2). This implies that there are natural correspondences between the spaces of cusp forms on Γ0(2) and the spaces of period polynomials. Based on these results, we will find explicit form of Hecke operators on Sw+2(Γ0(2)). As an application of main theorems, we will also give an affirmative answer to a speculation of Imamoglu and Kohnen on a basis of Sw+2(Γ0(2)).

Composition operators on weighted spaces of holomorphic functions on the upper half plane

2018 ◽  
Vol 122 (1) ◽  
pp. 141
Author(s):  
Wolfgang Lusky
Keyword(s):  
Half Plane ◽  
Composition Operator ◽  
Unit Disc ◽  
Composition Operators ◽  
Bounded Operator ◽  
Holomorphic Functions ◽  
Weighted Spaces ◽  
Weight Functions ◽  
Spaces Of Holomorphic Functions ◽  
Upper Half Plane

We consider moderately growing weight functions $v$ on the upper half plane $\mathbb G$ called normal weights which include the examples $(\mathrm{Im} w)^a$, $w \in \mathbb G$, for fixed $a > 0$. In contrast to the comparable, well-studied situation of normal weights on the unit disc here there are always unbounded composition operators $C_{\varphi }$ on the weighted spaces $Hv(\mathbb G)$. We characterize those holomorphic functions $\varphi \colon \mathbb G \rightarrow \mathbb G$ where the composition operator $C_{\varphi } $ is a bounded operator $Hv(\mathbb G) \rightarrow Hv(\mathbb G)$ by a simple property which depends only on $\varphi $ but not on $v$. Moreover we show that there are no compact composition operators $C_{\varphi }$ on $Hv(\mathbb G)$.

ON KAZHDAN CONSTANTS OF FINITE INDEX SUBGROUPS IN SLn(ℤ)

2012 ◽  
Vol 22 (03) ◽  
pp. 1250026
Author(s):  
UZY HADAD
Keyword(s):  
Finite Index ◽  
Congruence Subgroup ◽  
Infinite Family ◽  
Principal Congruence ◽  
The Other ◽  
Index Subgroup ◽  
Generating Set ◽  
Principal Congruence Subgroup ◽  
Finite Index Subgroup ◽  
Finite Index Subgroups

We prove that for any finite index subgroup Γ in SL n(ℤ), there exists k = k(n) ∈ ℕ, ϵ = ϵ(Γ) > 0, and an infinite family of finite index subgroups in Γ with a Kazhdan constant greater than ϵ with respect to a generating set of order k. On the other hand, we prove that for any finite index subgroup Γ of SL n(ℤ), and for any ϵ > 0 and k ∈ ℕ, there exists a finite index subgroup Γ′ ≤ Γ such that the Kazhdan constant of any finite index subgroup in Γ′ is less than ϵ, with respect to any generating set of order k. In addition, we prove that the Kazhdan constant of the principal congruence subgroup Γn(m), with respect to a generating set consisting of elementary matrices (and their conjugates), is greater than [Formula: see text], where c > 0 depends only on n. For a fixed n, this bound is asymptotically best possible.

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.

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