We define certain higher-dimensional Dedekind sums that generalize the classical Dedekind-Rademacher sums, and show how to compute them effectively using a generalization of the continued-fraction algorithm.
We present two applications. First, we show how to express special values of partial zeta functions associated to totally real number fields in terms of these sums via the Eisenstein cocycle introduced by the second author. Hence we obtain a polynomial-time algorithm for computing these special values. Second, we show how to use our techniques to compute certain special values of the Witten zeta-function, and compute some explicit examples.
In this paper, a transformation formula under modular substitutions is derived for a large class of generalized Eisenstein series. Appearing in the transformation formulae are generalizations of Dedekind sums involving the periodic Bernoulli function. Reciprocity theorems are proved for these Dedekind sums. Furthermore, as an application of the transformation formulae, relations between various infinite series and evaluations of several infinite series are deduced. Finally, we consider these sums for some special cases.
For primitive non-trivial Dirichlet characters $χ_1$ and $χ_2$, we study the weight zero newform Eisenstein series $E_{χ_1,χ_2}(z,s)$ at $s=1$. The holomorphic part of this function has a transformation rule that we express in finite terms as a generalized Dedekind sum. This gives rise to the explicit construction (in finite terms) of elements of $H^1(Γ_0(N), \mathbb{C})$. We also give a short proof of the reciprocity formula for this Dedekind sum.
Let $E(z,s)$ be the non-holomorphic Eisenstein series for the modular group $SL(2,{\mathbb Z})$. The classical Kronecker limit formula shows that the second term in the Laurent expansion at $s=1$ of $E(z,s)$ is essentially the logarithm of the Dedekind eta function. This eta function is a weight $1/2$ modular form and Dedekind expressed its multiplier system in terms of Dedekind sums. Building on work of Goldstein, we extend these results from the modular group to more general Fuchsian groups $Γ$. The analogue of the eta function has a multiplier system that may be expressed in terms of a map …
We introduce an integral version of the Eisenstein cocycle. As applications we prove a conjecture of Gross regarding the "order of vanishing" of Stickelberger elements relative to an abelian tower of fields and give a cohomological construction of the conjectural Gross--Stark units.
We generalize Sczech's Eisenstein cocycle for $\mathrm{GL}(n)$ over totally real extensions of $\mathbb{Q}$ to finite extensions of imaginary quadratic fields. By evaluating the cocycle on certain cycles, we parametrize complex values of Hecke L-functions previously considered by Colmez, giving a cohomological interpretation of his algebraicity result on special values of the L-functions.
The values of the so-called {\em Dedekind--Rademacher cocycle} at certain real quadratic arguments are shown to be global $p$-units in the narrow Hilbert class field of the associated real quadratic field, as predicted by conjectures of Darmon, Dasgupta, and Vonk. The strategy for proving this result combines an approach of Darmon-Pozzi-Vonk with one crucial extra ingredient: the study of infinitesimal deformations of irregular Hilbert Eisenstein series of weight one in the anti-parallel direction, building on the techniques in earlier work of Betina, Dimitrov, and Pozzi.
We define an integral version of Sczech's Eisenstein cocycle on GLn by smoothing at a prime ell. As a result we obtain a new proof of the integrality of the values at nonpositive integers of the smoothed partial zeta functions associated to ray class extensions of totally real fields. We also obtain a new construction of the p-adic L-functions associated to these extensions. Our cohomological construction allows for a study of the leading term of these p-adic L-functions at s=0. We apply Spiess's formalism to prove that the order of vanishing at s=0 is at least equal to the expected one, as co…
We define a cocycle on Gln using Shintani's method. It is closely related to cocycles defined earlier by Solomon and Hill, but differs in that the cocycle property is achieved through the introduction of an auxiliary perturbation vector Q. As a corollary of our result we obtain a new proof of a theorem of Diaz y Diaz and Friedman on signed fundamental domains, and give a cohomological reformulation of Shintani's proof of the Klingen-Siegel rationality theorem on partial zeta functions of totally real fields. Next we prove that the cohomology class represented by our Shintani cocycle is essenti…
Newform Dedekind sums are a class of crossed homomorphisms that arise from newform Eisenstein series. We initiate a study of the kernel of these newform Dedekind sums. Our results can be loosely described as showing that these kernels are neither "too big" nor "too small." We conclude with an observation about the Galois action on Dedekind sums that allows for significant computational efficiency in the numerical calculation of Dedekind sums.
In this article, we obtain a transformation formula for the higher power of odd zeta values, which generalizes Ramanujan's formula for odd zeta values. We have also investigated many important applications, which in turn provide generalizations of the transformation formula of the Eisenstein series, Dedekind eta function etc.
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 polynomial reciprocity laws. Our construction is based on Machide's result on his elliptic Dedekind-Rademacher sums. As an application of our results, we discover Eisenstein series identities which generalize certain …
We study the image of a generalized Dedekind sum relating to the weight zero Eisenstein series $E_{χ_1,χ_2}$. We show that the image is a lattice of full rank inside a number field determined by the characters $χ_1$ and $χ_2$. We also give a generalization of Knopp's identity for the classical Dedekind sum.
A systematic procedure for generating cubic multisections of Eisenstein series is given. The relevant series are determined from Fourier expansions for Eisenstein series by restricting the congruence class of the summation index modulo three. We prove that the resulting series are rational functions of η(τ) and η(3τ), where ηis the Dedekind eta function. A more general treatment of cubic dissection formulas is given by describing the dissection operators in terms of linear transformations. These operators exhibit properties that mirror those of similarly defined quintic operators.
In this paper we define multiple Dedekind zeta values (MDZV), using a new type of iterated integrals, called iterated integrals on a membrane. One should consider MDZV as a number theoretic generalization of Euler's multiple zeta values. Over imaginary quadratic fields MDZV capture, in particular, multiple Eisenstein series (Gangl, Kaneko and Zagier). We give an analogue of multiple Eisenstein series over real quadratic field and an alternative definition of values of multiple Eisenstein-Kronecker series (Goncharov). Each of them is a special case of multiple Dedekind zeta values. MDZV are int…
We compute the p-adic L-function (in the sense of Pollack-Stevens and Bellaiche) of the "evil twin" Eisenstein series. We use Solomon's Shintani cocycle to construct a p-adic Shintani modular symbol, and show how to specialize this modular symbol to overconvergent modular symbols with the same p-adic L-function as evil/critical-slope Eisenstein series. This gives a new proof of recent results of Bellaiche and Dasgupta.
This paper provides an introduction to Eisenstein measures, a powerful tool for constructing certain $p$-adic $L$-functions. First seen in Serre's realization of $p$-adic Dedekind zeta functions associated to totally real fields, Eisenstein measures provide a way to extend the style of congruences Kummer observed for values of the Riemann zeta function (so-called {\em Kummer congruences}) to certain other $L$-functions. In addition to tracing key developments, we discuss some challenges that arise in more general settings, concluding with some that remain open.
Atkin and Garvan introduced the functions $N_k(n)$ and $M_k(n)$, which denote the $k$-th moments of ranks and cranks in the theory of partitions. Let $e_{2r}(n)$ be the $n$-th Fourier coefficient of $E_{2r}(τ)/η(τ)$, where $E_{2r}(τ)$ is the classical Eisenstein series of weight $2r$ and $η(τ)$ is the Dedekind eta function. Via the theory of quasi-modular forms, we find that for $k \leq 5$, $N_k(n)$ and $M_k(n)$ can be expressed using $e_{2r}(n)$ ($2\leq r \leq k$), $p(n)$ and $N_2(n)$. For $k>5$, additional functions are required for such expressions. For $r\in \{2, 3, 4, 5, 7\}$, by studying…
We study the arithmetic of degree $N-1$ Eisenstein cohomology classes for locally symmetric spaces associated to $\mathrm{GL}_N$ over an imaginary quadratic field $k$. Under natural conditions we evaluate these classes on $(N-1)$-cycles associated to degree $N$ extensions $F/k$ as linear combinations of generalised Dedekind sums. As a consequence we prove a remarkable conjecture of Sczech and Colmez expressing critical values of $L$-functions attached to Hecke characters of $F$ as polynomials in Kronecker--Eisenstein series evaluated at torsion points on elliptic curves with multiplication by …
In this paper, transformation formulas for a large class of Eisenstein series defined by \[ G(z,s;A_α,B_β;r_{1},r_{2})=\sum\limits_{m,n=-\infty}^{\infty }\ \hspace{-0.19in}^{^{\prime}}\frac{f(αm)f^{\ast}(βn)} {((m+r_{1})z+n+r_{2})^{s}},\text{ }\operatorname{Re}(s)>2,\text{ }\operatorname{Im}(z)>0 \] are investigated for $s=1-r$, $r\in\mathbb{N}$. Here $\left\{ f(n)\right\}$ and $\left\{ f^{\ast}(n)\right\}$, $-\infty<n<\infty$ are sequences of complex numbers with period $k>0$, and $A_α=\left\{ f(αn)\right\} $ and $B_β=\left\{ f^{\ast}(βn)\right\}$, $α,β\in\mathbb{Z}$. Appearing in the transfo…
We employ the spectral theory of Eisenstein series to prove that the Hardy sums, integer-valued analogs of the classical Dedekind sums, are uniformly distributed in $\mathbf{Z} / m \mathbf{Z}$ for any integer $m > 1$.
We discuss computation of the special values of partial zeta functions associated to totally real number fields. The main tool is the \emph{Eisenstein cocycle} $Ψ$, a group cocycle for $GL_{n} (\Z )$; the special values are computed as periods of $Ψ$, and are expressed in terms of generalized Dedekind sums. We conclude with some numerical examples for cubic and quartic fields of small discriminant.
We construct, out of modular symbols, 1-traces that are invariant with respect to the actions of the Hopf algebra $\Hc_{1}$ on the crossed product $\Ac_{Q}$ of the algebra of modular forms of all levels by $\GL^+ (2,Q)$ investigated in earlier work. This provides a conceptual explanation for the construction of the Euler cocycle representing the image of the universal Godbillon-Vey class under the characteristic map of noncommutative Chern-Weil theory which we developed in our earlier work. We then refine the construction to produce secondary data by transgression. For the action determined by…
For the generalized Dedekind sums s_{ij}(p,q) defined in association with the x^{i}y^{j}-coefficient of the Todd power series of the lattice cone in R^2 generated by (1,0) and (q,p), we associate an exponential sum. We obtain this exponential sum using the cocycle property of the Todd series of 2d cones and the nonsingular cone decomposition along with the continued fraction of q/p. Its Weil bound is given for the modulus q, applying the purity theorem of the cohomology of the related l-adic sheaf due to Denef and Loeser. The Weil type bound of Denef and Loeser fulfills the Weyl's equidistribu…
We consider generalized Dedekind sums in dimension $n$, for fixed $n$-tuple of natural numbers, defined as sum of products of values of periodic Bernoulli functions. This includes the higher dimensional Dedekind sums of Zagier and Apostol-Carlitz' generalized Dedekind sums as well as the original Dedekind sums. These are realized as coefficients of Todd series of lattice cones and satisfy reciprocity law from the cocycle property of Todd series. Using iterated residue formula, we compute the coefficient of the decomposition of of the Todd series corresponding to a nonsingular decomposition of …
The temperature inversion properties of the internal energy, E, on odd spheres, and its derivatives, together with their expression in elliptic terms, as expounded in previous papers, are extended to the integrals of E, thence making contact with the theory of modular forms with rational period functions. I point out that the period functions of (holomorphic) Eisenstein series computed by Zagier were already available since the time of Ramanujan and I give a rederivation by contour integration. Removing both the Planck and Casimir terms gives a fully subtracted form of the series which allows …
Recent work of Bettin and Conrey on the period functions of Eisenstein series naturally gave rise to the Dedekind-like sum \[ c_{a}\left(\frac{h}{k}\right) \ = \ k^{a}\sum_{m=1}^{k-1}\cot\left(\frac{πmh}{k}\right)ζ\left(-a,\frac{m}{k}\right), \] where $a\in\mathbb{C}$, $h$ and $k$ are positive coprime integers, and $ζ(a,x)$ denotes the Hurwitz zeta function. We derive a new reciprocity theorem for these Bettin--Conrey sums, which in the case of an odd negative integer $a$ can be explicitly given in terms of Bernoulli numbers. This, in turn, implies explicit formulas for the period functions ap…
An exact transformation, which we call the \emph{master identity}, is obtained for the first time for the series $\sum_{n=1}^{\infty}σ_{a}(n)e^{-ny}$ for $a\in\mathbb{C}$ and Re$(y)>0$. New modular-type transformations when $a$ is a non-zero even integer are obtained as its special cases. The precise obstruction to modularity is explicitly seen in these transformations. These include a novel companion to Ramanujan's famous formula for $ζ(2m+1)$. The Wigert-Bellman identity arising from the $a=0$ case of the master identity is derived too. When $a$ is an odd integer, the well-known modular tran…
