Hidenori Katsurada, Rainer Schulze-Pillot
We derive explicit formulas for the action of the Hecke operator $T(p)$ on the genus theta series of a positive definite integral quadratic form and prove a theorem on the generation of spaces of Eisenstein series by genus theta series. We also discuss connections of our results with Kudla's matching principle for theta integrals.
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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 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 formulate a weak Gorenstein property for the Eisenstein component of the p-adic Hecke algebra associated to modular forms. We show that this weak Gorenstein property holds if and only if a weak form of Sharifi's conjecture and a weak form of Greenberg's conjecture hold.
We relate non-critical special values $p$-adic $L$-functions associated to algebraic Hecke characters of an imaginary quadratic number field with class number one to $p$-adic Coleman function called the $p$-adic Eisenstein-Kronecker series, when the conductors of the algebraic Hecke characters are divisible by $p$.
We evaluate the action of Hecke operators on Siegel Eisenstein series of arbitrary degree, level and character. For square-free level, we simultaneously diagonalise the space with respect to all the Hecke operators, computing the eigenvalues explicitly, and obtain a multiplicity-one result. For arbitrary level, we simultaneously diagonalise the space with respect to the Hecke operators attached to primes not dividing the level, again computing the eigenvalues explicitly.
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.
In this paper, we have investigated the mod p kernel of the theta operator for Hermitian modular forms when the base field is the Eisenstein field.
Using the modularity technique of Wiles, we study the Hecke algebra of weight 2 and prime level N localized at the Eisenstein primes. On the way, we recover some results of Mazur ("Modular Curves and the Eisenstein Ideal") from a deformation theoretic point of view. Combining some of our results with a theorem of Merel, we obtain new information about the p-part of the class groups of Q(N^(1/p)), where p and N are prime, and N = 1 mod p.
We compute the intertwining relation between the Hecke operators and the Siegel lowering operators on Siegel modular forms of arbitrary level $N$ and character $χ$ by using formulas for the action of the Hecke operators on Fourier expansions. Using an explicit description of the Satake compactification of $Γ_0^{(n)}(N) \backslash \mathfrak{H}_n$ when $N$ is squarefree we extend this to give intertwining relations for each cusp. As an application we give formulas for the action of Hecke operators on the space of Klingen--Eisenstein series of squarefree level $N$, for primes $p \nmid N$.
Let $N$ and $p$ be primes such that $p$ divides the numerator of $\frac{N-1}{12}$. In this paper, we study the rank $g_p$ of the completion of the Hecke algebra acting on cuspidal modular forms of weight $2$ and level $Γ_0(N)$ at the $p$-maximal Eisenstein ideal. We give in particular an explicit criterion to know if $g_p \geq 3$, thus answering partially a question of Mazur. In order to study $g_p$, we develop the theory of \textit{higher Eisenstein elements}, and compute the first few such elements in four different Hecke modules. This has applications such as generalizations of the Eichler …
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.
For certain algebraic Hecke characters chi of an imaginary quadratic field F we define an Eisenstein ideal in a p-adic Hecke algebra acting on cuspidal automorphic forms of GL_2/F. By finding congruences between Eisenstein cohomology classes (in the sense of G. Harder) and cuspidal classes we prove a lower bound for the index of the Eisenstein ideal in the Hecke algebra in terms of the special L-value L(0,chi). We further prove that its index is bounded from above by the order of the Selmer group of the p-adic Galois character associated to chi^{-1}. This uses the work of R. Taylor et al. on a…
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.
Towards Keating-Snaith's conjecture for cubic Hecke $L$-functions over the Eisenstein field
本文基于Radziwill和Soundararajan的工作，在广义黎曼假设下通过扩展David和Güloğlu的三次Hecke L函数零点1级密度扭曲估计，得到艾森斯坦域上稀疏三次Hecke L函数族的Keating-Snaith猜想条件下界。
We consider the Kudla-Millson theta series associated to a quadratic space of signature $(N,N)$. By combining a `see-saw' argument with the Siegel-Weil formula, we show that its (regularized) integral along a torus attached to a totally real field of degree $N$ is the diagonal restriction of an Eisenstein series. It allows us to express the Fourier coefficients of the diagonal restriction as intersection numbers, which generalizes a result of Darmon-Pozzi-Vonk to totally real fields.
In this paper, we explicitly construct mock modular forms whose shadows are Eisenstein series of arbitrary integral and half-integral weight, level and character at the cusps $\infty$ and $0$. As an application, we give explicit construction of harmonic weak Maass forms which are Hecke eigenforms and are the preimages of $Θ^k, k \in \{ 3, 5, 7\}$ under the shadow operator, where $Θ$ is the classical Jacobi theta function.
We derive a precise relation of poles of Eisenstein series associated to the cuspidal datum $χ\otimesσ$ and the lowest occurrence of theta lifts of a cuspidal automorphic representation $σ$ of a unitary group, where $χ$ is a conjugate self-dual character. A key ingredient of the proof is the computation of period integrals of truncated Eisenstein series.
Let $p$ and $\ell$ be primes such that $p > 3$ and $p \mid \ell-1$ and $k$ be an even integer. We use deformation theory of pseudo-representations to study the completion of the Hecke algebra acting on the space of cuspidal modular forms of weight $k$ and level $Γ_0(\ell)$ at the maximal Eisenstein ideal containing $p$. We give a necessary and sufficient condition for the $\mathbb{Z}_p$-rank of this Hecke algebra to be greater than $1$ in terms of vanishing of the cup products of certain global Galois cohomology classes. We also recover some of the results proven by Wake and Wang-Erickson for …
Let p be a prime number and C be the p-adic tame level 1 eigencurve introduced by Coleman-Mazur. We prove that C is smooth at the evil Eisenstein points and we give necessary and sufficient conditions for etaleness of the map to the weight space at these points in terms of p-adic zeta values. A key step is the determination at these points of the schematic reducibility locus of the pseudo-character carried by C restricted to a decomposition group at p. Then, the smoothness appears to be a consequence of the fact that the Dirichlet L-functions only have simple zeros at integers.
We use pseudodeformation theory to study Mazur's Eisenstein ideal. Given prime numbers $N$ and $p>3$, we study the Eisenstein part of the $p$-adic Hecke algebra for $Γ_0(N)$. We compute the rank of this Hecke algebra (and, more generally, its Newton polygon) in terms of Massey products in Galois cohomology, answering a question of Mazur and generalizing a result of Calegari-Emerton. We also also give new proofs of Merel's result on this rank and of Mazur's results on the structure of the Hecke algebra.
Given a Siegel theta series and a prime p not dividing the level of the theta series, we apply to the theta series the n+1 Hecke operators that generate the local Hecke algebra at p. We show that the average theta series is an eigenform and we compute the eigenvalues.
Orthogonal Eisenstein Series and Theta Lifts
本文证明了向量值非全纯 Eisenstein 级数的加性 Borcherds 提升是 $O(2, l)$ 的正交非全纯 Eisenstein 级数，据此证明了其亚纯延拓、Fourier 展开与函数方程，并给出了提升满射的充分条件。
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 consider the action of Hecke-type operators on Hilbert-Siegel theta series attached to lattices of even rank. We show that average Hilbert-Siegel theta series are eigenforms for these operators, and we explicitly compute the eigenvalues.
Congruences between Klingen-Eisenstein series and cusp forms on $\mathrm{U}_{n,n}$
研究Hermitian Klingen-Eisenstein级数与U_{n,n}上cusps形式的Hecke特征值一致性，证明Hermitian自守形式空间的有理性和Hecke特征值的整性。
Quantum variance for cubic moment of Hecke--Maass cusp forms and Eisenstein series
本文分别给出了Hecke--Maass尖点形式和Eisenstein级数三次矩方差的上界，基于大筛不等式和Lindelöf均值型界，并建立了特殊点附近GL(2)相关L-函数四次矩和八次矩的精确上界。
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.
We develop a setting in which one can evaluate certain Hecke-Rogers series in terms of false theta functions. We apply our setting to recent false theta function identities of Chan and Kim as well as Andrews and Warnaar.
In this paper, we give an overview of our previous paper concerning the investigation of the algebraic and $p$-adic properties of Eisenstein-Kronecker numbers using Mumford's theory of algebraic theta functions.
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 $\goth E(\goth p)$ 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.
Hecke action on tamely ramified Eisenstein series over $\mathbb{P}^1$
本文研究有理函数域在三个位点驯分歧的自守函数空间，提出 Eisenstein 级数生成的仿射 Hecke 代数三模的生成元与关系猜想，并在 PGL(2) 和 SL(3) 情形给出证明。
Theta correspondence and the Borisov-Gunnells relations
本文通过几何θ对应研究模形式之间的关系，揭示了埃西恩系列之间联系的几何证明，并与李定理建立联系。
By developing a connection between partial theta functions and Appell-Lerch sums, we find and prove a formula which expresses Hecke-type double sums in terms of Appell-Lerch sums and theta functions. Not only does our formula prove classical Hecke-type double sum identities such as those found in work Kac and Peterson on affine Lie Algebras and Hecke modular forms, but once we have the Hecke-type forms for Ramanujan's mock theta functions our formula gives straightforward proofs of many of the classical mock theta function identities. In particular, we obtain a new proof of the mock theta conj…
Let K/F be a quadratic extension of number fields. After developing a theory of the Eisenstein series over F, we prove a formula which expresses a partial zeta function of K as a certain integral of the Eisenstein series. As an application, we obtain a limit formula of Kronecker's type which relates the 0-th Laurent coefficients at s=1 of zeta functions of K and F.
We obtain four Hecke-type double sums for three of Ramanujan's third order mock theta functions. We discuss how these four are related to the new mock theta functions of Andrews' work on $q$-orthogonal polynomials and Bringmann, Hikami, and Lovejoy's work on unified Witten-Reshetikhin-Turaev invariants of certain Seifert manifolds. We then prove identities between these new mock theta functions by first expressing them in terms of the universal mock theta function.
We prove an algebraic property of the elements defining Hecke operators on period polynomials associated with modular forms, which implies that the pairing on period polynomials corresponding to the Petersson scalar product of modular forms is Hecke equivariant. As a consequence of this proof, we derive two indefinite theta series identities which can be seen as analogues of Jacobi's formula for the theta series associated with the sum of four squares.
Köhler's Conjecture on Hecke Theta Series of Weight One
通过从二次域特征诱导的二维Galois表示推导，证明了Günter Köhler关于Hecke theta级数中二次域歧义的猜想。
Orthogonal Eisenstein Series at Harmonic Points and Modular Forms of Singular Weight
本文利用theta提升研究正交非全纯Eisenstein级数在调和点的行为，证明奇异权下其在$s=1$处的留数生成全纯正交模形式，并给出构造满射的充分条件。
It is known that average Siegel theta series lie in the space of Siegel Eisenstein series. Also, every lattice equipped with an even integral quadratic form lies in a maximal lattice. Here we consider average Siegel theta series of degree 2 attached to maximal lattices; we construct maps for which the average theta series is an eigenform. We evaluate the action of these maps on Siegel Eisenstein series of degree 2 (without knowing their Fourier coefficients), and then realise the average theta series as an explicit linear combination of the Eisenstein series.
We investigate certain Eisenstein congruences, as predicted by Harder, for level p paramodular forms of genus 2. We use algebraic modular forms to generate new evidence for the conjecture. In doing this we see explicit computational algorithms that generate Hecke eigenvalues for such forms.
We use pseudodeformation theory to study the analogue of Mazur's Eisenstein ideal with certain squarefree levels. Given a prime number $p>3$ and a squarefree number $N$ satisfying certain conditions, we study the Eisenstein part of the $p$-adic Hecke algebra for $Γ_0(N)$, and show that it is a local complete intersection and isomorphic to a pseudodeformation ring. We also show that in certain cases, the Eisenstein ideal is not principal and that the cuspidal quotient of the Hecke algebra is not Gorenstein. As a corollary, we prove that "multiplicity one" fails for the modular Jacobian in these…
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.
We prove that products of at most two vector valued Eisenstein series that originate in level 1 span all spaces of cusp forms for congruence subgroups. This can be viewed as an analogue in the level aspect to a result that goes back to Rankin, and Kohnen and Zagier, which focuses on the weight aspect. The main feature of the proof are vector valued Hecke operators. We recover several classical constructions from them, including classical Hecke operators, Atkin-Lehner involutions, and oldforms. As a corollary to our main theorem, we obtain a vanishing condition for modular forms reminiscent of …
Degeneration of monodromy in the theory of Eisenstein congruences for classical groups
针对酉群与拟分裂典型群的p进自守伽罗瓦表示族，研究人员提出新的赫克理论方法以控制其单值性增长，推进了艾森斯坦同余理论中单值性退化问题的研究。
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 consider the genus of $20$ classes of unimodular Hermitian lattices of rank $12$ over the Eisenstein integers. This set is the domain for a certain space of algebraic modular forms. We find a basis of Hecke eigenforms, and guess global Arthur parameters for the associated automorphic representations, which recover the computed Hecke eigenvalues. Congruences between Hecke eigenspaces, combined with the assumed parameters, recover known congruences for classical modular forms, and support new instances of conjectured Eisenstein congruences for $\mathbb{U}(2,2)$ automorphic forms.
We address some questions posed by Goss related to the modularity of Drinfeld modules of rank 1 defined over the field of rational functions in one variable with coefficients in a finite field. For each positive characteristic valued Dirichlet character, we introduce certain projection operators on spaces of Drinfeld modular forms with character of a given weight and type such that when applied to a Hecke eigenform return a Hecke eigenform whose eigensystem has been twisted by the given Dirichlet character. Unlike the classical case, however, the effect on Goss' $u$-expansions for these eigenf…
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.
Joint cubic moment of Eisenstein series and Hecke-Maass cusp forms
本文研究谱参数趋于无穷时自守形式的联合三次矩，分别给出 Eisenstein 级数对角情形的上界估计，以及非对角情形和 Hecke-Maass 尖点形式混合情形的渐近消失结果。
We show that if an Eisenstein component of the $p$-adic Hecke algebra associated to modular forms is Gorenstein, then it is necessary that the plus-part of a certain ideal class group is trivial. We also show that this condition is sufficient whenever a conjecture of Sharifi holds.
We study the arithmetic of Eisenstein cohomology classes (in the sense of G. Harder) for symmetric spaces associated to GL_2 over imaginary quadratic fields. We prove in many cases a lower bound on their denominator in terms of a special L-value of a Hecke character providing evidence for a conjecture of Harder that the denominator is given by this L-value. We also prove under some additional assumptions that the restriction of the classes to the boundary of the Borel-Serre compactification of the spaces is integral. Such classes are interesting for their use in congruences with cuspidal class…
This note contains an attempt to relate Hecke's presentation of an ideal class zeta function in a real quadratic field as an integral of the nonholomorphic Eisenstein series along the loop on modular curve and Zagier's decomposition of this zeta function into the sum of zeta functions of indefinite quadratic forms. (See D.Zagier, A Kronecker limit formula for real quadratic field.) It is a preliminary version.
Hecke Eigenform Product Identities for Full-Level Hilbert Modular Forms
本研究针对次数不小于2的全实域上的全水平希尔伯特赫克本征形式，结合相关结果分类两类乘积恒等式，证明运用了多项数论方法，还建立了含非平凡窄类特征的权1艾森斯坦因子与尖点本征因子的恒等式。
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.
Generalizing math.AG/0003076 we express universal triple Massey products between line bundles on elliptic curves in terms of Hecke's indefinite theta series. We show that all Hecke's indefinite theta series arise in this way.
A general formula for Hecke-type false theta functions
本文提出一个通用公式，将Hecke型双和分解为θ函数和虚假θ函数，涵盖Matsusaka的分解，并与$b^2-ac>0$情况下的分解结构相似。
Eisenstein series for O(2,n+2)
本研究刻画了O(2,n+2)的艾森斯坦级数为特殊赫克特征形式，证明其属于相关马厄空间，并在偶幺模格条件下推导出其傅里叶系数的显式公式。
We present a general conjecture on congruences between Hecke eigenvalues of parabolically induced and cuspidal automorphic representations of split reductive groups, modulo divisors of critical values of certain $L$-functions. We examine the consequences in several special cases, and use the Bloch-Kato conjecture to further motivate a belief in the congruences.
Modular Heights of Unitary Shimura Varieties I: Derivatives of Eisenstein Series
本文通过比较混合 Eisenstein-theta 级数导数的全纯投影与除子生成级数的算术度，证明了 CM 数域上酉 Shimura 簇的模高度公式。
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