For any even integer $k \ge 4$, let $\E_k$ be the normalized Eisenstein series of weight $k$ for $\SL_2(\Z)$. Also let $\D$ be the closure of the standard fundamental domain of the Poincaré upper half plane modulo $\SL_2(\Z)$. F.~K.~C.~Rankin and H. P. F. Swinnerton-Dyer showed that all zeros of $\E_k$ in $\D$ are of modulus one. In this article, we study the critical points of $\E_k$, that is to say the zeros of the derivative of $\E_k$. We show that they are simple. We count those belonging to $\D$, prove that they are located on the two vertical edges of $\D$ and produce explicit intervals that separate them. We then count those belonging to $γ\D$, for any $γ\in \SL_2(\Z)$.
In this paper we study iterated Eisenstein τ-integrals and multiple Eisenstein L-series, they are functions on the complex upper half plane and form two Q-algebras. They reduce to iterated Eisenstein integrals and multiple Hecke L-functions with respect to Eisenstein series respectively after analytic extension when τ->0. We give the relations among them and prove the linear independence of their elements. Finally, we explain the connections among double Eisenstein L-functions and holomorphic double modular values.
In this short note, we will construct a harmonic Eisenstein series of weight one, whose image under the $ξ$-operator is a weight one Eisenstein series studied by Hecke.
In this paper we consider certain classes of generalized double Eisenstein series by simple differential calculations of trigonometric functions. In particular, we give four new transformation formula for some double Eisenstein series. We can find that these double Eisenstein series are reducible to infinite series involving hyperbolic functions. Moreover, some interesting new examples are given.
Shimura studied the analytic properties of the non-holomorphic Siegel Eisenstein series and derived a residue formula. Herein, we provide a refinement of his result for several types of Eisenstein series.
We study the multiple Eisenstein series introduced by Gangl, Kaneko and Zagier. We give a proof of (restricted) finite double shuffle relations for multiple Eisenstein series by developing an explicit connection between the Fourier expansion of multiple Eisenstein series and the Goncharov coproduct on Hopf algebras of iterated integrals.
A generalization of Serre's $p$-adic Eisenstein series in the case of Siegel modular forms is studied and a coincidence between a $p$-adic Siegel Eisenstein series and a genus theta series associated with a quaternary quadratic form is proved.
We obtain an conditional result for the regularized fourth moment of some newform Eisenstein series, using a recipe by Conrey, et al. We also discuss the fourth moment of truncated newform Eisenstein series.
Cusp forms for the full modular group can be written as linear combination of double Eisenstein series introduced by Gangl, Kaneko and Zagier. We give an explicit formula for decomposing a Hecke eigenform into double Eisenstein series.
We study the double shuffle relations satisfied by the double zeta values of level 2, and introduce the double Eisenstein series of level 2 which satisfy the double shuffle relations. We connect the double Eisenstein series to modular forms of level 2.
We show that the most standard Eisenstein series such as $E_4(τ)$ or $2E_2(2τ)-E_2(τ)$, and also the function $θ^2(τ)$, are in a natural way the product of two conjugate Eisenstein series of half their weight and double their level, as well as a number of similar elementary identities for $E_6$ and Eisenstein series of levels $2$, $3$, and $4$.
We compute the Fourier expansion of vector valued Eisenstein series for the Weil representation associated to an even lattice. To this end, we define certain twists by Dirichlet characters of the usual Eisenstein series associated to isotropic elements in the discriminant form of the underlying lattice. These twisted functions still form a generating system for the space of Eisenstein series but have better multiplicative properties than the individual Eisenstein series. We adapt a method of Bruinier and Kuss to obtain algebraic formulas for the Fourier coefficients of the twisted Eisenstein s…
It is shown that for a non-unitary twist of a Fuchsian group, which is unitary at the cusps, Eisenstein series converge in some half-plane. It is shown that invariant integral operators provide a spectral decomposition of the space of cusp forms and that Eisenstein series admit a meromorphic continuation.
The classical Kronecker limit formula gives the constant term of the non-holomorphic Eisenstein series E(z,s) for SL(2,Z) at s=1 in terms of the Dedekind eta function. Here we compute the analagous formula for an Eisenstein series twisted by modular symbols (periods of weight two holomorphic cusp forms) for general Fuchsian groups of the first kind.
Poincaré and Eisenstein series are building blocks for every type of modular forms. We define Poincaré series for Jacobi forms of lattice index and state some of their basic properties. We compute the Fourier expansions of Poincaré and Eisenstein series and give an explicit formula for the Fourier coefficients of the trivial Eisenstein series. For even weight and fixed index, finite linear combinations of Fourier coefficients of non-trivial Eisenstein series are equal to finite linear combinations of Fourier coefficients of the trivial one.
We show that certain $p$-adic Eisenstein series for quaternionic modular groups of degree 2 become "real" modular forms of level $p$ in almost all cases. To prove this, we introduce a $U(p)$ type operator. We also show that there exists a $p$-adic Eisenstein series of the above type that has transcendental coefficients. Former examples of $p$-adic Eisenstein series for Siegel and Hermitian modular groups are both rational (i.e., algebraic).
We find nice representatives for the 0-dimensional cusps of the degree $n$ Siegel upper half-space under the action of $Γ_0(\stufe)$. To each of these we attach a Siegel Eisenstein series, and then we make explicit a result of Siegel, realizing any integral weight average Siegel theta series of arbitrary level $\stufe$ and Dirichlet character $χ_L$ modulo $\stufe$ as a linear combination of Siegel Eisenstein series.
We obtain an asymptotic for the regularized fourth moment of the Eisenstein series for the full modular group, in agreement with the Random Wave Conjecture.
The purpose of this work is to produce a converse theorem for adelic Eisenstein series on the double metaplectic cover of the group $SL_2(\mathbb{A})$. We show that the double Dirichlet series, which satisfy the natural functional equations required in our theorem, are in fact Mellin transforms at infinity of metaplectic Eisenstein series.
We construct a basis for the space of half-integral weight Siegel Eisenstein series of level 4N where N is odd and square-free. Then we restrict our attention to those Eisenstein series generated from elements of $Γ_0(4)$, commenting on why this restriction is necessary for our methods. We directly apply to these forms all Hecke operators attached to odd primes, and we realize the images explicitly as linear combinations of Siegel Eisenstein series. Using this information, we diagonalize the subspace of Eisenstein series generated from elements of $Γ_0(4)$, obtaining a multiplicity-one result.
Higher-order non-holomorphic Eisenstein series associated to a Fuchsian group $Γ$ are defined by twisting the series expansion for classical non-holomorphic Eisenstein series by powers of modular symbols. Their functional identities include multiplicative and additive factors, making them distinct from classical Eisenstein series. Here we prove the meromorphic continuation of these series and establish their functional equations. In addition, we construct high rank vector bundles $\cal V$ from certain unipotent representations $π$ of $Γ$ and show that higher-order non-holomorphic Eisenstein se…
We study a certain class of q-analogues of multiple zeta values, which appear in the Fourier expansion of multiple Eisenstein series. Studying their algebraic structure and their derivatives we propose conjectured explicit formulas for the derivatives of double and triple Eisenstein series.
The aim of this paper is to generalize the Maass relation for generalized Cohen-Eisenstein series of degree two and of degree three. Here the generalized Cohen-Eisenstein series are certain Siegel modular forms of half-integral weight, and generalized Maass relations are certain relations among Fourier-Jacobi coefficients of them.
We find explicit change-of-basis formulas between Eisenstein series attached to cusps, and newform Eisenstein series attached to pairs of primitive Dirichlet characters. As a consequence, we prove a Bruggeman-Kuznetsov formula for newforms of square-free level and trivial nebentypus. We also derive, in explicit form, the Fourier expansion, Hecke eigenvalues, Atkin-Lehner pseudo-eigenvalues, and other data associated to these Eisenstein series, with arbitrary integer weight, level, and nebentypus.
This work is an example driven overview article of recent works on the connection of multiple zeta values, modular forms and q-analogues of multiple zeta values given by multiple Eisenstein series.
The Fourier coefficient of a second order Eisenstein series is described as a shifted convolution sum. This description is used to obtain the spectral decomposition of and estimates for the shifted convolution sum.
New expressions are given for the Fourier expansions of non-holomorphic Eisenstein series with weight $k$. Among other applications, this leads to non-holomorphic analogs of formulas of Ramanujan, Grosswald and Berndt containing Eichler integrals of holomorphic Eisenstein series.
In this work, we define a new type of Eisenstein-like series by using Pell-Lucas numbers and call them the Pell-Lucas-Eisenstein Series. Firstly, we show that the Pell-Lucas-Eisenstein series are convergent on their domain. Afterwards we prove that they satisfy some certain functional equations. Proofs follows from some on calculations on Pell-Lucas numbers.
We give effective lower bounds for $L(1,χ)$ via Eisenstein series on $Γ_0(q) \backslash \mathbb{H}$. The proof uses the Maass-Selberg relation for truncated Eisenstein series and sieve theory in the form of the Brun-Titchmarsh inequality. The method follows closely the work of Sarnak in using Eisenstein series to find effective lower bounds for $ζ(1+it)$.
We evaluate the action of Hecke operators on Siegel Eisenstein series of degree 2, square-free level and arbitrary character, without using knowledge of their Fourier coefficients. From this we construct a basis of simultaneous eigenforms for the full Hecke algebra, and we compute their eigenvalues. As well, we obtain Hecke relations among the Eisenstein series. Using these Hecke relations in the case that $\stufe$ is square-free and the character is trivial, we generate a basis for the space of Eisenstein series.
We study the number of two-dimensional sublattices of $\mathbb{Z^4}$ of a fixed covolume and construct the associated Dirichlet series. The latter is shown to be related to Eisenstein series on both $GL_4$ and its metaplectic double cover.
The real analytic Eisenstein series is a special function that has been studied classically. Its generalization to the case of many variables has been studied extensively. Moreover, the analytic properties of certain Eisenstein series on the Siegel modular groups have also been investigated. The purpose of this study is to provide concrete forms of the residue of E_0^{(m)}(z,s) at s=m/2.
We study nearly holomorphic Siegel Eisenstein series of general levels and characters on $\mathbb{H}_{2n}$, the Siegel upper half space of degree $2n$. We prove that the Fourier coefficients of these Eisenstein series (once suitably normalized) lie in the ring of integers of $\mathbb{Q}_p$ for all sufficiently large primes $p$. We also prove that the pullbacks of these Eisenstein series to $\mathbb{H}_n \times \mathbb{H}_n$ are cuspidal under certain assumptions.
We prove a sup norm bound on the real-analytic Eisenstein series, of the form $E(z, 1/2 + iT) \ll T^{3/8 + \varepsilon}$, uniformly for $z$ in a fixed compact subset of $\mathbb{H}$.
Siegel modular forms in the space of the mod $p$ kernel of the theta operator are constructed by the Eisenstein series in some odd-degree cases. Additionally, a similar result in the case of Hermitian modular forms is given.
The zeros of classical Eisenstein series satisfy many intriguing properties. Work of F. Rankin and Swinnerton-Dyer pinpoints their location to a certain arc of the fundamental domain, and recent work by Nozaki explores their interlacing property. In this paper we extend these distribution properties to a particular family of Eisenstein series on Gamma(2) because of its elegant connection to a classical Jacobi elliptic function cn(u) which satisfies a differential equation. As part of this study we recursively define a sequence of polynomials from the differential equation mentioned above that …
We obtain pullback formulas for Klingen Eisenstein series with arbitrary levels, with respect to both Siegel congruence and paramodular subgroups, in degree two. Pullback results are used, along with the Fourier series expansion of Klingen Eisenstein series given by Mizumoto, to prove certain identities involving critical values of $ L$-functions attached to normalized elliptic modular forms of weight $ k $ and full level.
We examine the zeros of newform Eisenstein series $E_{χ_1,χ_2,k}(z)$ of weight $k$ on $Γ_0(q_1 q_2)$, where $χ_1$ and $χ_2$ are primitive characters modulo $q_1$ and $q_2$, respectively. We determine the location and distribution of a significant fraction of the zeros of these Eisenstein series for $k$ sufficiently large.
In this paper, we compute constant terms of Eisenstein series defined over a totally real field, at various cusps. In his paper published in 2003, M. Ohta computed the constant terms of Eisenstein series of weight two over the field of rational numbers, at all equivalence classes of cusps. As for Eisenstein series defined over a totally real field, S. Dasgupta, H. Darmon and R. Pollack calculated the constant terms at particular (not all) equivalence classes of cusps in 2011. We compute constant terms of Eisenstein series defined over a general totally real field at all equivalence classes of …
We develop some of the finer details of the location of the zeros of the weight two Eisenstein series. These zeros are the same as the zeros of the derivative of the Ramanujan delta function.
W. Luo and P. Sarnak have proved the quantum unique ergodicity property for Eisenstein series on $\rm{PSL}(2,\mathbb{Z}) \backslash H$. We extend their result to Eisenstein series on $\rm{PSL}(2,O) \backslash H^n$, where $O$ is the ring of integers in a totally real field of degree $n$ over $Q$ with narrow class number one, using the Eisenstein series considered by I. Efrat. We also give an expository treatment of the theory of Hecke operators on non-holomorphic Hilbert modular forms.
We calculate the constant terms of certain Hilbert modular Eisenstein series at all cusps. Our formula relates these constant terms to special values of Hecke $L$-series. This builds on previous work of Ozawa, in which a restricted class of Eisenstein series were studied. Our results have direct arithmetic applications---in separate work we apply these formulas to prove the Brumer-Stark conjecture away from $p=2$ and to give an exact analytic formula for Brumer-Stark units.
We form real-analytic Eisenstein series twisted by Manin's noncommutative modular symbols. After developing their basic properties, these series are shown to have meromorphic continuations to the entire complex plane and satisfy functional equations in some cases. This theory neatly contains and generalizes earlier work in the literature on the properties of Eisenstein series twisted by classical modular symbols.
We locate all of the zeros of the Eisenstein series associated with the Fricke groups $Γ_0^{*}(2)$ and $Γ_0^{*}(3)$ in their fundamental domains by applying and expanding the method of F. K. C. Rankin and H. P. F. Swinnerton-Dyer (``{\it On the zeros of Eisenstein series}'', 1970).
We prove an asymptotic formula expressing the regularized fourth moment of all newform Eisenstein series of primitive nebentypus as a mean value of $L$-functions. This is an analogue of the recent work of Djanković and Khan in the spectral aspect.
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 a 2005 paper, Yang constructed families of Hilbert Eisenstein series, which when restricted to the diagonal are conjectured to span the underlying space of elliptic modular forms. One approach to these conjectures is to show the non-vanishing of an inner product of elliptic eigenforms with the restrictions of Eisenstein series. In this paper, we compute this inner product locally by using explicit values of new vectors in the Waldspurger model.
We compute the $p$-adic $L$-functions of evil Eisenstein series, showing that they factor as products of two Kubota--Leopoldt $p$-adic $L$-functions times a logarithmic term. This proves in particular a conjecture of Glenn Stevens.
Let $Γ$ be a Fuchsian group of the first kind acting on the hyperbolic upper half plane $\mathbb H$, and let $M = Γ\backslash \mathbb H$ be the associated finite volume hyperbolic Riemann surface. If $γ$ is parabolic, there is an associated (parabolic) Eisenstein series, which, by now, is a classical part of mathematical literature. If $γ$ is hyperbolic, then, following ideas due to Kudla-Millson, there is a corresponding hyperbolic Eisenstein series. In this article, we study the limiting behavior of parabolic and hyperbolic Eisenstein series on a degenerating family of finite volume hyperbol…
