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.
Systematic choice of the Hecke eigenforms of half-integral weight is an interesting problem in the theory of modular forms. In this paper, we find all Dedekind-eta products of half-integral weight which are Hecke eigenforms up to weight 15/2 with varying levels. Proof is based on the Shimura lift.
An Analogue of the Dedekind Eta Function for Hecke Groups $H(\sqrt{D})$
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Max Planck Institute for Mathematics（马克斯·普朗克数学研究所）；Columbia University（哥伦比亚大学）；RAND Corporation（兰德公司）；Simion Stoilow Institute of Mathematics of the Romanian Academy（罗马尼亚科学院西蒙·斯托伊洛数学研究所）
The elliptic Apostol-Dedekind sums generate odd Dedekind symbols with Laurent polynomial reciprocity laws
Dedekind符号是经典Dedekind和（符号）的推广。具有Laurent多项式互反律的Dedekind符号空间与模形式空间之间存在自然同构。我们将定义一个新的椭圆模拟的Apostol-Dedekind和。然后，我们将表明，新定义的和生成所有奇Dedekind符号与劳伦多项式互易律。我们的构造是基于Machide关于椭圆型Dedekind-Rademacher和的结果。作为我们结果的应用，我们发现了艾森斯坦系列等式，这些等式推广了Ramanujan、van der Pol、Rankin和Skoruppa的某些公式。
In this paper we study multiple Dedekind symbols and the associated multiple reciprocity functions. There is a bijection between the two sets of them after a normalization. By this bijection we define products of multiple reciprocity functions, and study the relationship to the shuffle property. We construct and calculate shuffled multiple Dedekind symbols and shuffled multiple reciprocity functions from modular forms by regularized iterated integrals. Also we give a decomposition for them.
Let $k=\mathbb{Q}(\sqrt{-3})$, and let $c\in \mathfrak{O}_k$ be a square free algebraic integer such that $c\equiv 1~({\rm mod}~{\langle9\rangle})$. Let $ζ_{k(c^{1/3})}(s)$ be the Dedekind zeta function of the cubic field $k(c^{1/3})$ and $ζ_k(s)$ be the Dedekind zeta function of $k$. For fixed real $σ>1/2$, we obtain asymptotic distribution functions $F_σ$ for the values of the logarithm and the logarithmic derivative of the Artin $L$-functions \begin{equation*} L_c(σ)= \frac{ζ_{k(c^{1/3})}(σ)}{ζ_k(σ)}, \end{equation*} as $c$ varies. Moreover, we express the characteristic function of $F_σ$ e…
We revisit some papers by Eichler and Shimura in order to give an algebraic formulation (based on Farey symbols) for the intersection product on the space of modular symbols, as described by Pollack and Stevens. We define the period homomorphism of an Eisenstein series (Eisenstein-Dedekind-Stevens symbol) and extend the definition of the intersection product to these objects. We construct a computationally convenient basis for the space of Eisenstein series for $Γ$ 0 (N) with rational periods. Given a Farey symbol for a subgroup $Γ$ of the modular group and a subgroup $Γ$ ' of finite index of …
We provide a new and simple automorphic method using Eisenstein series to study the equidistribution of modular symbols modulo primes, which we apply to prove an average version of a conjecture of Mazur and Rubin. More precisely, we prove that modular symbols corresponding to a Hecke basis of weight 2 cusp forms are asymptotically jointly equidistributed mod $p$ while we allow restrictions on the location of the cusps. As an application, we obtain a residual equidistribution result for Dedekind sums. Furthermore, we calculate the variance of the distribution and show a surprising bias with con…
