scholarly journals The Selberg trace formula for modular correspondences

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 ◽  
Space Of Cusp Forms

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]).

scholarly journals On some p-adic properties of the eichler-selberg trace formula

1975 ◽  
Vol 56 ◽  
pp. 45-52 ◽  
Author(s):  
Masao Koike
Keyword(s):  
Trace Formula ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Selberg Trace Formula ◽  
Congruence Relations ◽  
Space Of Cusp Forms

In this paper we shall prove some congruence relations mod pα between the traces of Hecke operators T(pm) which act on the space of cusp forms of different weights satisfying some congruences mod pα – pα-1. The method of the proof is very simple and is applicable to all the cases where the trace formula for Hecke operators are already known.

scholarly journals On some dimension formula for automorphic forms of weight one III

1988 ◽  
Vol 111 ◽  
pp. 157-163 ◽  
Author(s):  
Toyokazu Hiramatsu ◽  
Shigeki Akiyama
Keyword(s):  
Zeta Function ◽  
Linear Space ◽  
Fuchsian Group ◽  
Fundamental Domain ◽  
Automorphic Forms ◽  
Cusp Forms ◽  
Dimension Formula ◽  
Weight One ◽  
Space Of Cusp Forms

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.

scholarly journals On p-adic properties of the Eichler-Selberg trace formula II

1976 ◽  
Vol 64 ◽  
pp. 87-96 ◽  
Author(s):  
M. Koike
Keyword(s):  
Prime Number ◽  
Trace Formula ◽  
Cusp Forms ◽  
Identity Operator ◽  
Selberg Trace Formula ◽  
Hecke Operator ◽  
Space Of Cusp Forms

Let be the space of cusp forms of weight k with respect to SL(2, Z). Let p be a prime number and let Tk(p) be the Hecke operator of degree p acting on as a linear endomorphism. Put Hk(X) = det (I – Tk(p)X + pk-lX2I), where I is the identity operator on . Hk(X) is a polynomial with coefficients of rational integers, which is called the Hecke polynomial.

scholarly journals On Shimura’s trace formula

1977 ◽  
Vol 66 ◽  
pp. 183-202 ◽  
Author(s):  
Shinji Niwa
Keyword(s):  
Trace Formula ◽  
Hecke Algebra ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Special Cases ◽  
Space Of Cusp Forms

In [1], G. Shimura gave a very practical formula of the traces of the Hecke operators acting on the space of cusp forms of rational weight and there he emphasized that the traces are effectively computable. We shall practice the computation in some special cases and discuss the structure of the Hecke algebra, which is not necessarily semi-simple.

scholarly journals Trace formula of certain Hecke operators for Γ0(qν)

1979 ◽  
Vol 76 ◽  
pp. 1-33 ◽  
Author(s):  
Hiroshi Saito ◽  
Masatoshi Yamauchi
Keyword(s):  
Trace Formula ◽  
Congruence Subgroup ◽  
Hecke Operators ◽  
The Other ◽  
Cusp Forms ◽  
Other Hand ◽  
Space Of Cusp Forms

Let SK(Γo(qν)) be the space of cusp forms of weight K with respect to the congruence subgroup Γ0(qν), and SK(Γ0(qν)) its subspace of all new forms in SK(Γ0(qν)), where q is a prime such that q ≥ 3. Now for f ∈ SK(Γ(qν)), put Then it is known that W induces an automorphism of SK(Γ0(qν)). On the other hand, for the character χ of (Z/qZ)× of order 2, let δx denote the “twisting operator” with respect to χ which was defined in [15] by Shimura, namely, for .

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 A Selberg trace formula for hypercomplex analytic cusp forms

2015 ◽  
Vol 148 ◽  
pp. 398-428 ◽  
Author(s):  
D. Grob ◽  
R.S. Kraußhar
Keyword(s):  
Trace Formula ◽  
Cusp Forms ◽  
Selberg Trace Formula

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.

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)).

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

2010 ◽  
Vol 06 (04) ◽  
pp. 767-783 ◽  
Author(s):  
SATADAL GANGULY
Keyword(s):  
Galois Representations ◽  
Character Formula ◽  
Cusp Forms ◽  
Holomorphic Cusp Forms ◽  
Weight One ◽  
Space Of Cusp Forms

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.

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