The basis for space of cusp forms and Petersson trace formula

2012 ◽  
Author(s):  
Ming-ho Ng
Keyword(s):  
Trace Formula ◽  
Cusp Forms ◽  
Space Of Cusp Forms

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 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 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.

Computations with modular forms and Galois representations

2011 ◽  
Author(s):  
Johan Bosman
Keyword(s):  
Modular Forms ◽  
Galois Representations ◽  
Cusp Forms ◽  
Space Of Cusp Forms

This chapter discusses several aspects of the practical side of computing with modular forms and Galois representations. It starts by discussing computations with modular forms, and from there work towards the computation of polynomials that give the Galois representations associated with modular forms. Throughout, the chapter denotes the space of cusp forms of weight k, group Γ‎₁(N), and character ε‎ by Sₖ(N, ε‎).

scholarly journals Petersson products of bases of spaces of cusp forms and estimates for Fourier coefficients

2018 ◽  
Vol 14 (08) ◽  
pp. 2277-2290 ◽  
Author(s):  
Rainer Schulze-Pillot ◽  
Abdullah Yenirce
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Orthogonal Basis ◽  
Cusp Forms ◽  
Integral Weight ◽  
Space Of Cusp Forms

We prove a bound for the Fourier coefficients of a cusp form of integral weight which is not a newform by computing an explicit orthogonal basis for the space of cusp forms of given integral weight and level.

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