Hecke Operators on Weighted Dedekind Symbols
نویسنده
چکیده
Dedekind symbols generalize the classical Dedekind sums (symbols). The symbols are determined uniquely by their reciprocity laws up to an additive constant. There is a natural isomorphism between the space of Dedekind symbols with polynomial (Laurent polynomial) reciprocity laws and the space of cusp (modular) forms. In this article we introduce Hecke operators on the space of weighted Dedekind symbols. We prove that these newly introduced operators are compatible with Hecke operators on the space of modular forms. As an application, we present formulae to give Fourier coefficients of Hecke eigenforms. In particular we give explicit formulae for generalized Ramanujan’s tau functions.
منابع مشابه
Explicit Formulas for Hecke Operators on Cusp Forms, Dedekind Symbols and Period Polynomials
Let Sw+2 be the vector space of cusp forms of weight w + 2 on the full modular group, and let S∗ w+2 denote its dual space. Periods of cusp forms can be regarded as elements of S∗ w+2. The Eichler-Shimura isomorphism theorem asserts that odd (or even) periods span S w+2 . However, periods are not linearly independent; in fact, they satisfy the Eichler-Shimura relations. This leads to a natural ...
متن کاملSe p 20 09 Motives for elliptic modular groups
In order to study the arithmetic structure of elliptic modular groups which are the fundamental groups of compactified modular curves with cuspidal base points, these truncated Malcev Lie algebras and their direct sums are considered as elliptic modular motives. Our main result is a new theory of Hecke operators on these motives which gives a congruence relation to the Galois action, and a moti...
متن کامل0 Ju l 2 00 7 PERIOD POLYNOMIALS AND EXPLICIT FORMULAS FOR HECKE OPERATORS ON Γ
Let 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 natu...
متن کاملThe Elliptic Apostol-dedekind Sums Generate Odd Dedekind Symbols with Laurent Polynomial Reciprocity Laws
Abstract. Dedekind symbols are generalizations of the classical Dedekind sums (symbols). There is a natural isomorphism between the space of Dedekind symbols with Laurent polynomial reciprocity laws and the space of modular forms. We will define a new elliptic analogue of the Apostol-Dedekind sums. Then we will show that the newly defined sums generate all odd Dedekind symbols with Laurent poly...
متن کاملModular Symbols and Hecke Operators
We survey techniques to compute the action of the Hecke operators on the cohomology of arithmetic groups. These techniques can be seen as generalizations in different directions of the classical modular symbol algorithm, due to Manin and Ash-Rudolph. Most of the work is contained in papers of the author and the author with Mark McConnell. Some results are unpublished work of Mark McConnell and ...
متن کامل