Thomas Brazelton, Victoria Jakicic
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.
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The first negative Fourier coefficient of an Eisenstein series newform
本文针对Linowitz和Thompson提出的关于艾森斯坦级数新形式傅里叶系数首个负号出现平均时机的猜想进行了修正，并证明了修正后的猜想，推进了新形式傅里叶系数符号变化统计问题的研究。
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.
Newform Eisenstein Congruences of Local Origin
本文提出一个关于新形式与新艾森斯坦级数之间艾森斯坦同余的普遍猜想，并探讨其与布洛赫-卡托猜想的关系。
In this paper we prove a number of theorems that determine the extent to which the signs of the Hecke eigenvalues of an Eisenstein newform determine the newform. We address this problem broadly and provide theorems of both individual and statistical nature. Many of these results are Eisenstein series analogues of well-known theorems for cusp forms. For instance, we determine how often the p-th Fourier coefficients of an Eisenstein newform begin with a fixed sequence of signs \varepsilon_p = {\pm 1, 0}. Moreover, we prove the following variant of the strong multiplicity-one theorem: an Eisenste…
Let $p\geq 5$ be prime. For elliptic modular forms of weight 2 and level $Γ_0(N)$ where $N>6$ is squarefree, we bound the depth of Eisenstein congruences modulo $p$ (from below) by a generalized Bernoulli number with correction factors and show how this depth detects the local non-principality of the Eisenstein ideal. We then use admissibility results of Ribet and Yoo to give an infinite class of examples where the Eisenstein ideal is not locally principal. Lastly, we illustrate these results with explicit computations and give an interesting commutative algebra application related to Hilbert-…
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.
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.
On the Tamagawa number conjecture for newforms at Eisenstein primes
本文推广[CGLS22]至更高权模形式，在周期假设下证明Eisenstein素数处秩0的Tamagawa数公式，并讨论秩1部分结果及条件性p-逆定理。
On the anticyclotomic Iwasawa theory of newforms at Eisenstein primes of semistable reduction
该研究去除了已有结果的限制条件，将半稳定约化艾森斯坦素数下新形式的反分圆岩泽理论推广到更高权情形，证明了相关岩泽主猜想与强BSD猜想的p部分，得到p逆定理并应用于Goldfeld猜想。
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.
Let F be a genus two Siegel newform and g a classical newform, both of squarefree levels and of equal weight l. We prove a pullback formula for certain Eisenstein series -- thus generalizing a construction of Shimura -- and use this to derive an explicit integral representation for the degree eight L-function L(s, F X g). This integral representation involves the pullback of a simple Siegel-type Eisenstein series on the unitary group GU(3,3). As an application, we prove a reciprocity law -- predicted by Deligne's conjecture -- for the critical special values L(m, F X g) where m is an integer, …
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.
We prove congruences between cuspidal newforms and Eisenstein series of prime level, which generalize Ramanujan's congruence. Such congruences were recently found by Billerey and Menares, and we refine them by specifying the Atkin-Lehner eigenvalue of the newform involved. We show that similar refinements hold for the level raising congruences between cuspidal newforms of different levels, due to Ribet and Diamond. The proof relies on studying the new subspace and the Eisenstein subspace of the space of period polynomials for the congruence subgroup $Γ_0(N)$, and on a version of Ihara's lemma.
Siegel Eisenstein Series with Paramodular Level
本文由导子为 $N$ 的本原 Dirichlet 特征构造级 $N^2$、权 $k\geq4$ 的 paramodular Siegel Eisenstein 级数，计算其 Fourier 展开，并证明它是生成不可约自守表示的 paramodular 新形式。
We express all the newforms of weight $2$ and levels $30$, $33$, $35$, $38$, $40$, $42$, $44$, $45$ as linear combinations of eta quotients and Eisenstein series, and list their corresponding strong Weil curves. Let $p$ denote a prime and $E (\zz_p)$ denote the the group of algebraic points of an elliptic curve $E$ over $\zz_p$. We give a generating function for the order of $E (\zz_p)$ for certain strong Weil curves in terms of eta quotients and Eisenstein series. We then use our generating functions to deduce congruence relations for the order of $E (\zz_p)$ for those strong Weil curves.
Let $k > 3$ be an integer and $p$ a prime with $p > 2k-2$. Let $f$ be a newform of weight $2k-2$ and level 1 so that $f$ is ordinary at $p$ and $\barρ_{f}$ is irreducible. Under some additional hypotheses we prove that $ord_{p}(L_{alg}(k,f)) \leq ord_{p}(# S)$ where $S$ is the Pontryagin dual of the Selmer group associated to $ρ_{f} \otimes ε^{1-k}$ with $ε$ the $p$-adic cyclotomic character. We accomplish this by first constructing a congruence between the Saito-Kurokawa lift of $f$ and a non-CAP Siegel cusp form. Once this congruence is established, we use Galois representations to obtain th…
In a recent work of Darmon, Pozzi and Vonk, the authors consider a particular $p$-adic family of Hilbert Eisenstein series $E_k(1,\brch)$ associated with an odd character $\brch$ of the narrow ideal class group of a real quadratic field $F$ and compute the first derivative of a certain one-variable twisted triple product $p$-adic $L$-series attached to $E_k(1,\brch)$ and an elliptic newform $f$ of weight $2$ on $Γ_0(p)$. In this paper, we generalize their construction to include the cyclotomic variable and thus obtain a two-variable twisted triple product $p$-adic $L$-series. Moreover, when $f…
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.
L-series of Eisenstein series vanishing at critical values
本文利用有理函数、余切和与周期多项式的关系，刻画了L-级数在临界值处消失的爱森斯坦级数，并在权较大时确定了具有同时消失性质的子空间。
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 prove quantum unique ergodicity for a subspace of the continuous spectrum spanned by the degenerate Eisenstein Series on GL(n).
In this monograph, we study in detail a special class of $GL_2$-real analytic 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.
In this paper we study congruences between Siegel Eisenstein series and Siegel cusp forms for Sp_4(Z).
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.
Klingen Eisenstein series congruences and modularity
本研究构造Klingen Eisenstein级数与Siegel尖点形式的模ℓ同余，证明相关Bloch-Kato塞尔默群非零性，推导通用形变环的R=dvr定理，并给出非循环塞尔默群与非主Eisenstein理想的条件。
On explicit Fourier expansions of theta lifts to ${\rm SO}(3,n+1)$ arising from elliptic newforms of level one
利用退化 Whittaker 函数和 Eisenstein 级数的显式计算，得到了到特殊正交群 $G={\rm SO}(3,n+1)$ 的 theta 提升的 Fourier 展开的显式公式，其中 $n\ge 3$ 且 $G$ 在所有有限位分裂，这些提升是来自 $\SL_2(\Z)$ 的椭圆新形式的 Hecke 本征、非尖点、平方可积自守形式。
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 $ζ(1+it)$.
In this paper we present a new method to study Fourier coefficients of holomorphic and non-holomorphic Eisenstein series simultaneously.
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 $\mathbb{H}$.
On the zeros of odd weight Eisenstein series
本文研究了奇权重埃西恩系列在SL₂(ℤ)变换下的基本域中零点的计数问题。
Using results from the theory of modular forms, we reprove and extend a result of Romik about lacunary recurrence relations for Eisenstein series.
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.
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