We show that every elliptic modular form of integral weight greater than $1$ can be expressed as linear combinations of products of at most two cusp expansions of Eisenstein series. This removes the obstruction of nonvanishing central $\mathrm{L}$-values present in all previous work. For weights greater than $2$, we refine our result further, showing that linear combinations of products of exactly two cusp expansions of Eisenstein series suffice.
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 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.
In recent work, M. Just and the second author defined a class of "semi-modular forms" on $\mathbb C$, in analogy with classical modular forms, that are "half modular" in a particular sense; and constructed families of such functions as Eisenstein-like series using symmetries related to integer partitions. Looking for further natural examples of semi-modular behavior, here we construct a family of Eisenstein-like series to produce semi-modular forms, using symmetries related to Fibonacci numbers instead of partitions. We then consider other Lucas sequences that yield semi-modular forms.
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.
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.
For any odd integer N, we explicitly write down the Eisenstein cycles in the first homology group of modular curves of level N as linear combinations of Manin symbols. These cycles are, by definition, those over which every integral of holomorphic differential forms vanish. Our result can be seen as an explicit version of the Manin-Drinfeld Theorem. Our method is to characterize such Eisenstein cycles as eigenvectors for the Hecke operators. We make crucial use of expressions of Hecke actions on modular symbols and on auxiliary level 2 structures.
We define Eisenstein series twisted by modular symbols on the group SL(n), generalizing a construction of the first author. We show that, in the case of series attached to the minimal parabolic subgroup, our series converges for all points in a suitable cone. We conclude with examples for SL(2) and SL(3).
We explicitly write down the Eisenstein elements inside the space of modular symbols for Eisenstein series with integer coefficients for the congruence subgroups Γ_0 (pq) with p and q distinct odd primes, giving an answer to a question of Merel in these cases. We also compute the winding elements explicitly for these congruence subgroups. Our results are explicit versions of the Manin-Drinfeld Theorem.
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 introduce a new family of real analytic modular forms on the upper half plane. They are arguably the simplest class of `mixed' versions of modular forms of level one and are constructed out of real and imaginary parts of iterated integrals of holomorphic Eisenstein series. They form an algebra of functions satisfying many properties analogous to classical holomorphic modular forms. In particular, they admit expansions in $q, \overline{q}$ and $\log |q|$ involving only rational numbers and single-valued multiple zeta values. The first non-trivial functions in this class are real analytic Eis…
The purpose of this article is to give a simple and explicit construction of mock modular forms whose shadows are Eisenstein series of arbitrary integral weight, level, and character. As application, we construct forms whose shadows are Hecke's Eisenstein series of weight one associated to imaginary quadratic fields, recovering some results by Kudla, Rapoport and Yang (1999), and Schofer (2009), and forms whose shadows equal $Θ^{2k}(z)$ for $k\in \{1,2,3,4\}$, where $Θ(z)$ denotes Jacobi's theta function.
In this paper, we study the torsion subgroup of $J_0(N)$ over the field generated by those points in the cuspidal group, where $N$ is an odd positive integer. We prove that, considered as Hecke modules, this group and the cuspidal subgroup are both supported at the maximal Eisenstein ideals.
We give a formula for certain values and derivatives of Siegel series and use them to compute Fourier coefficients of derivatives of the Siegel Eisenstein series of weight g/2 and genus g. When g=4, the Fourier coefficient is approximated by a certain Fourier coefficient of the central derivative of the Siegel Eisenstein series of weight 2 and genus 3, which is related to the intersection of 3 arithmetic modular correspondences. Applications include a relation between weighted averages of representation numbers of symmetric matrices.
Borisov and Gunnells observed in 2001 that certain linear relations between products of two holomorphic weight 1 Eisenstein series had the same structure as the relations between periods of modular forms; a similar phenomenon exists in higher weights. We give a conceptual reason for this observation in arbitrary weight. This involves an unconventional way of expanding the Rankin-Selberg convolution of a cusp form with an Eisenstein series. We also prove a partial result towards understanding the action of a Hecke operator on a product of two Eisenstein series.
We show how our p-adic method to compute Galois representations occurring in the torsion of Jacobians of algebraic curves can be adapted to modular curves. The main ingredient is the use of "moduli-friendly" Eisenstein series introduced by Makdisi, which allow us to evaluate modular forms at p-adic of modular curves points and dispenses us of the need for equations of modular curves and for q-expansion computations. The resulting algorithm compares very favourably to the complex-analytic method.
In this paper, we will study the arithmetic of the Eisenstein part of the modular Jacobians. In the first section, we introduce some general preliminaries of the arithmetic theory of modular curves that we will need later. In the second section, we give an example of modular abelian varieties due to Gross and study its properties in some details. In the third section, we define Eisenstein quotients of the modular Jacobians in general and give a criterion of the non-triviality of Heegner points on such Eisenstein quotients. The last two sections return to the concrete examples when the level of…
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).
In this paper we make a systematic study of certain motivic cohomology classes ("Rankin-Eisenstein classes") attached to the Rankin--Selberg convolution of two modular forms of weight $\ge 2$. The main result is the computation of the $p$-adic syntomic regulators of these classes. As a consequence we prove many cases of the Perrin-Riou conjecture for Rankin--Selberg convolutions of cusp forms.
We explicitly write down the Eisenstein elements inside the space of modular symbols for Eisenstein series with integer coefficients for the congruence subgroups $Γ_0(N)$ with $N$ odd square-free. We also compute the winding elements explicitly for these congruence subgroups. This gives an answer to a question of Merel in these cases. Our results are explicit versions of the Manin-Drinfeld Theorem [Thm. 6]. These results are the generalization of the paper [1] results to odd square-free level.
Let $\goth E(\goth p)$ denote the Eisenstein ideal in the Hecke algebra $\Bbb T(\goth p)$ of the Drinfeld modular curve $X_0(\goth p)$ parameterizing Drinfeld modules of rank two over $\Bbb F_q[T]$ of general characteristic with Hecke level $\goth p$-structure, where $\goth p\triangleleft\Bbb F_q[T]$ is a non-zero prime ideal. We prove that the characteristic $p$ of the field $\Bbb F_q$ does not divide the order of the quotient $\Bbb T(\goth p)/\goth E(\goth p)$ and the Eisenstein ideal $\goth E(\goth p)$ is locally principal.
Let $E$ be a level 1, vector valued Eisenstein series of half-integral weight, normalized so that the coefficients are all in $\mathbb{Z}$. We show that there is a level one vector valued cusp form $f$ with the same weight as $E$ and with coefficients in $\mathbb{Z}$, which is congruent to $E$ modulo the constant term of $E$.
Let $p$ be a prime $\ge 5$. We establish explicit rates of overconvergence for members of the "Eisenstein family", notably for the $p$-adic modular function $V(E_{(1,0)}^{\ast})/E_{(1,0)}^{\ast}$ ($V$ the $p$-adic Frobenius operator) that plays a pi\-votal role in Coleman's theory of $p$-adic families of modular forms. The proof goes via an in-depth analysis of rates of overconvergence of $p$-adic modular functions of form $V(E_k)/E_k$ where $E_k$ is the classical Eisenstein series of level $1$ and weight $k$ divisible by $p-1$. Under certain conditions, we extend the latter result to a vast g…
By the theory of Eisenstein series, generating functions of various divisor functions arise as modular forms. It is natural to ask whether further divisor functions arise systematically in the theory of mock modular forms. We establish, using the method of Zagier and Zwegers on holomorphic projection, that this is indeed the case for certain (twisted) "small divisors" summatory functions $σ_ψ^{\mathrm{sm}}(n)$. More precisely, in terms of the weight 2 quasimodular Eisenstein series $E_2(τ)$ and a generic Shimura theta function $θ_ψ(τ)$, we show that there is a constant $α_ψ$ for which $$ \math…
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.
In 1975, Cohen constructed a kind of one-variable modular forms of half-integral weight, says $r+(1/2),$ whose $n$-th Fourier coefficient $H(n)$ only occurs when $(-1)^r n$ is congruent to 0 or 1 modulo 4. The space of modular forms whose Fourier coefficients have the above property is called Kohnen plus space, initially introduced by Kohnen in 1980. Recently, Hiraga and Ikeda generalized the Kohnen plus space to the spaces for half-integral weight Hilbert modular forms with respect to general totally real number fields. If one such Hilbert modular form $f$ of parallel weight $κ+(1/2)$ lying i…
From the theory of modular forms, there are exactly $[(k-2)/6]$ linear relations among the Eisenstein series $E_k$ and its products $E_{2i}E_{k-2i}\ (2\le i \le [k/4])$. We present explicit formulas among these modular forms based on the partial fraction decomposition, and use them to determining a basis of the space of modular forms of weight $k$ on ${\rm SL}_2({\mathbb Z})$.
In this paper we prove one side divisibility of the Iwasawa-Greenberg main conjecture for Rankin-Selberg product of a weight two cusp form and an ordinary CM form of higher weight, using congruences between Klingen Eisenstein series and cusp forms on $\mathrm{GU}(3,1)$, generalizing earlier result of the third-named author to allow non-ordinary cusp forms. The main result is a key input in the third author's proof for Kobayashi's $\pm$-main conjecture for supersingular elliptic curves. The new ingredient here is developing a semi-ordinary Hida theory along an appropriate smaller weight space, …
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 …
Let $\frak{n}$ be a square-free ideal of $\mathbb{F}_q[T]$. We study the rational torsion subgroup of the Jacobian variety $J_0(\frak{n})$ of the Drinfeld modular curve $X_0(\frak{n})$. We prove that for any prime number $\ell$ not dividing $q(q-1)$, the $\ell$-primary part of this group coincides with that of the cuspidal divisor class group. We further determine the structure of the $\ell$-primary part of the cuspidal divisor class group for any prime $\ell$ not dividing $q-1$.
We study the Eisenstein ideal of Drinfeld modular curves of small levels, and the relation of the Eisenstein ideal to the cuspidal divisor group and the component groups of Jacobians of Drinfeld modular curves. We prove that the characteristic of the function field is an Eisenstein prime number when the level is an arbitrary non square-free ideal of $\mathbb{F}_q[T]$ not equal to a square of a prime.
We show, for levels of the form $N = p^a q^b N'$ with $N'$ squarefree, that in weights $k \geq 4$ every cusp form $f \in \mathcal{S}_k(N)$ is a linear combination of products of certain Eisenstein series of lower weight. In weight $k=2$ we show that the forms $f$ which can be obtained in this way are precisely those in the subspace generated by eigenforms $g$ with $L(g, 1) \neq 0$. As an application of such representations of modular forms we can calculate Fourier expansions of modular forms at arbitrary cusps and we give several examples of such expansions in the last section.
In this article, we set up a strategy to prove one divisibility towards the main Iwasawa conjecture for the Selmer groups attached to the twisted adjoint modular Galois representations associated to Hida families. This conjecture asserts the equality of the p-adic L-function interpoling the critical values of the symmetric square of the modular forms in these families and the characteristic ideal of the associated Selmer group. The idea is to introduce a third characteristic ideal containing informations on the congruences between cuspidal Siegel modular forms of genus 2 and the Klingen type E…
We show that all Eichler integrals, and more generally all "generalized second order modular forms" can be expressed as linear combinations of corresponding generalized second order Eisenstein series with coefficients in classical modular forms. We determine the Fourier series expansions of generalized second order Eisenstein series in level one, and provide tail estimates via convexity bounds for additively twisted $\mathrm{L}$-functions. As an application, we illustrate a bootstrapping procedure that yields numerical evaluations of, for instance, Eichler integrals from merely the associated …
Consider a subgroup of finite index of modular group. We give an analytic criterion for a cuspidal divisor to be torsion in the Jacobian of the corresponding modular curve. By BelyI theorem, such a criterion would apply to any curve over a number field. Our main tool is the explicit description, in terms of modular symbols, of what we call Eisenstein cycles. The latter are representations of relative homology classes over which integration of any holomorphic differential forms vanishes. Our approach relies in an essential way on the specific case , where we can consider convenient generalized …
We construct a p-adic Eisenstein measure with values in the space of vector-weight p-adic automorphic forms on certain unitary groups. This measure allows us to p-adically interpolate special values of certain vector-weight C-infinity automorphic forms, including Eisenstein series, as their weights vary. We also explain how to extend our methods to the case of Siegel modular forms and how to recover Nicholas Katz's p-adic families of Eisenstein series for Hilbert modular forms.
We identify a class of "semi-modular" forms invariant on special subgroups of $GL_2(\mathbb Z)$, which includes classical modular forms together with complementary classes of functions that are also nice in a specific sense. We define an Eisenstein-like series summed over integer partitions, and use it to construct families of semi-modular forms.
We generalize the work of Ohta on the congruence modules attached to elliptic Eisenstein series to the setting of Hilbert modular forms. Our work involves three parts. In the first part, we construct Eisenstein series adelically and compute their constant terms by computing local integrals. In the second part, we prove a control theorem for one-variable ordinary $Λ$-adic Hilbert modular forms following Hida's work on the space of multivariable ordinary $Λ$-adic Hilbert cusp forms. In part three, we compute congruence modules related to Hilbert Eisenstein series through an analog of Ohta's meth…
Let $k \geq 2$ and $N$ be positive integers and let $χ$ be a Dirichlet character modulo $N$. Let $f(z)$ be a modular form in $M_k(Γ_0(N),χ)$. Then we have a unique decomposition $f(z)=E_f(z)+S_f(z)$, where $E_f(z) \in E_k(Γ_0(N),χ)$ and $S_f(z) \in S_k(Γ_0(N),χ)$. In this paper we give an explicit formula for $E_f(z)$ in terms of Eisenstein series. Then we apply our result to certain families of eta quotients and to representations of positive integers by $2k$-ary positive definite quadratic forms in order to give an alternative version of Siegel's formula for the weighted average number of re…
We prove the equality of several $τ$-recurrent sequences, which were first considered by Pellarin, and which have close connections to Drinfeld vectorial modular forms. Our result has several consequences: an $A$-expansion for the $l^\text{th}$ power ($1 \leq l \leq q$) of the deformation of the weight 2 Eisenstein series; relations between Drinfeld modular forms with $A$-expansions; a new proof of relations between special values of Pellarin $L$-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.
Let $N$ be a non-squarefree positive integer and let $\ell$ be an odd prime such that $\ell^2$ does not divide $N$. Consider the Hecke ring $\mathbb{T}(N)$ of weight $2$ for $Γ_0(N)$, and its rational Eisenstein primes of $\mathbb{T}(N)$ containing $\ell$, defined in Section 3. If $\mathfrak{m}$ is such a rational Eisenstein prime, then we prove that $\mathfrak{m}$ is of the form $(\ell, ~\mathcal{I}^D_{M, N})$, where the ideal $\mathcal{I}^D_{M, N}$ of $\mathbb{T}(N)$ is also defined in Section 3. Furthermore, we prove that $\mathcal{C}(N)[\mathfrak{m}] \neq 0$, where $\mathcal{C}(N)$ is the …
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.
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.
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.
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.
