G. Gotsbacher
The automorphic cohomology of a connected reductive algebraic group defined over Q decomposes as a direct algebraic sum of cuspidal and Eisenstein cohomology. In the present paper we construct regular Eisenstein cohomology classes for congruence subgroups of a rational form G of Q-rank 2 of SO(n,2).
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In this paper we study congruences between Siegel Eisenstein series and Siegel cusp forms for Sp_4(Z).
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.
Mass formulas and Eisenstein congruences in higher rank
本文用质量公式将GL(2)权2 Eisenstein同余推广到高阶秩，构造拟分裂酉群上的非内窥Eisenstein同余，并给出相关猜想及补充结果。
In this paper, based on Serre's $p$-adic family of Eisenstein series, we prove a general family of congruences for Eisenstein series $G_k$ in the form $$ \sum_{i=1}^n g_i(p)G_{f_i(p)}\equiv g_0(p)\mod p^N, $$ where $f_1(t),\ldots,f_n(t)\in\mathbb{Z}[t]$ are non-constant integer polynomials with positive leading coefficients and $g_0(t),\ldots,g_n(t)\in\mathbb{Q}(t)$ are rational functions. This generalizes the classical von Staudt-Clausen's and Kummer's congruences of Eisenstein series, and also yields some new congruences.
A recent article of Berndt and Yee found congruences modulo 3^k for certain ratios of Eisenstein series. For all but one of these, we show there are no simple congruences a(pn+c) = 0 modulo p when p>= 13 is prime. This follows from a more general theorem on the non-existence of congruences in (E_2^r)(E_4^s)(E_6^t) where r is non-negative and r,s,t are integers.
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…
In the case of Siegel modular forms of degree $n$, we prove that, for almost all prime ideals $\frak{p}$ in any ring of algebraic integers, mod $\frak{p}^m$ cusp forms are congruent to true cusp forms of the same weight. As an application of this property, we give congruences for the Klingen-Eisenstein series and cusp forms, which can be regarded as a generalization of Ramanujan's congruence. We will conclude by giving numerical examples.
There are classical congruences between the class number of an imaginary quadratic field and a Bernoulli number or an Euler number. Under the BSD conjecture, Onishi obtained an elliptic generalization of these congruences, which gives congruences between the order of the Tate-Shafarevich group of certain elliptic curves with CM by the Gauss integers ring and Mordell-Weil rank 0, and a coefficient of power series expansion of an elliptic function associated to Gauss integers ring. In this paper, we provide Onishi's type congruences for the Eisenstein integers case.
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。
In this paper we present several finite families of congruences between cusp forms and Eisenstein series of higher weights at powers of prime ideals. We formulate a conjecture which describes properties of the prime ideals and their relation to the weights. We check the validity of the conjecture on several numerical examples.
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.
A note on the spaces of Eisenstein series on general congruence subgroups
本文提出利用主同余子群的Hecke艾森斯坦级数理论研究一般同余子群上谱艾森斯坦级数的新方法，得到相关空间的基与系数性质，并简化证明了Eichler-Shimura同构定理。
Eisenstein congruences in tame families and the class group of a metabelian extension
本文利用tame族中Eisenstein同余存在性的新结果，解释了一个特定metabelian数域中素数分裂行为的计算观察，延续了Ribet开创的范式。
Eisenstein cohomology and congruences for the ratios of Rankin--Selberg $L$-functions
本文利用改进的埃森斯坦上同调理论，证明了Rankin-Selberg L函数临界值比值与对象之间的同余关系。
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。
Hilbert modular Eisenstein congruences of local origin
本文研究了希尔伯特模形式与欧拉特征级数之间的欧拉特征 congruences，探讨了其模数来源及新形式满足的条件。
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-…
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。
We investigate level $p$ Eisenstein congruences for GSp$_4$, generalisations of level $1$ congruences predicted by Harder. By studying the associated Galois and automorphic representations we see conditions that guarantee the existence of a paramodular form satisfying the congruence. This provides theoretical justification for computational evidence found in the author's previous paper.
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 $E$.
We study congruences between cuspidal modular forms and Eisenstein series at levels which are square-free integers and for equal even weights. This generalizes our previous results from Naskręcki [17] for prime levels and provides further evidence for the sharp bounds obtained under restrictive ramification conditions. We prove an upper bound on the exponent in the general square-free situation and also discuss the existence of the congruences when the coefficients belong to the rational numbers and weight equals 2.
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, …
Cuspidal subgroups associated with non-rational Eisenstein maximal ideals
本文推广了Ramanujan型Eisenstein同余式到一般水平N，证明了在非平方自由水平下非有理Eisenstein极大理想对应的Ribet猜想推广成立。
Eisenstein congruences among Euler systems
本文研究 Garrett–Rankin–Selberg 型 Euler 系统间的 Eisenstein 同余，借助 Bloch–Kato 猜想、Perrin-Riou 形式体系及 $p$-进 $L$-函数比较，统一处理三类上同调类。
We use the Jacquet-Langlands correspondence to generalize well-known congruence results of Mazur on Fourier coefficients and L-values of elliptic modular forms for prime level in weight 2 both to nonsquare level and to Hilbert modular forms.
Modularity theorems for Eisenstein congruences in prime-power level
针对满足 $N \equiv 1 \bmod p$ 的素数 $p, N \geq 5$，证明了级 $N$ 和 $N^2$ 下权为2的 $p$ 进赫克代数艾森斯坦局部化的模性定理，关联通用伪形变环的对应商环。
We work out instances of a general conjecture on congruences between Hecke eigenvalues of induced and cuspidal automorphic representations of a reductive group, modulo divisors of certain critical L-values, in the case that the group is a split orthogonal group. We provide some numerical evidence in the case that the group is SO(4,3) and the L-function is the spinor L-function of a genus 2, vector-valued, Siegel cusp form. We also consider the case that the group is SO(4,4) and the L-function is a triple product L-function.
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。
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.
This brief article gives an alternative interpretation, based on a theorem of Berkovich, of the Eisenstein classes in the cohomology of Shimura varieties, used in forthcoming work of the author with K. W. Lan, R. Taylor, and J. Thorne.
Lambda-invariants of Mazur--Tate elements attached to Ramanujan's tau function and congruences with Eisenstein series
该研究针对权12模形式$\Delta$，证明其与特定艾森斯坦级数模$p$（$p\in\{3,5,7\}$）同余，并推导了相关Mazur–Tate元的岩泽不变量公式，验证了此前的数值数据。
Congruences with Eisenstein series and mu-invariants
本文研究带剩余可约伽罗瓦表示的Hida族中μ不变量的变化，证明其下界通常可用p进ζ函数表示且无界，在特定条件下证明下界为等式及p进L函数的性质，并证明相关Selmer群的类似结论，确立了这类模形式的主猜想。
Let F be the imaginary quadratic field of discriminant -3 and OF its ring of integers. Let Gamma be the arithmetic group GL_3 (OF), and for any ideal n subset OF let Gamma_0 (n) be the congruence subgroup of level n consisting of matrices with bottom row (0,0,*) bmod n. In this paper we compute the cohomology spaces H^{nu - 1} (Gamma_0 (n); C) as a Hecke module for various levels n, where nu is the virtual cohomological dimension of Gamma. This represents the first attempt at such computations for GL_3 over an imaginary quadratic field, and complements work of Grunewald--Helling--Mennicke and …
In his paper, 'On torsion in the cohomology of locally symmetric varieties', Peter Scholze has introduced a new, purely topological method to construct the cohomology classes on arithmetic quotients of symmetric spaces of rational reductive groups originating from the cohomology of the similar quotients of Levi subgroups of maximal parabolic subgroups. We extend this construction beyond the cases he considers, and, in the complex case, to the cohomology of local systems.
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.
Two results on Eisenstein part of homology and cohomology groups of Bianchi modular groups
本文明确构造了 Bianchi 模群同调群的 Eisenstein 环，并研究上同调部分，进而得到尖点上同调群的渐近维数公式。
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.
In this article, several cohomology spaces associated to the arithmetic groups $\mathrm{SL}_3(\mathbb{Z})$ and $\mathrm{GL}_3(\mathbb{Z})$ with coefficients in any highest weight representation $\mathcal{M}_λ$ have been computed, where $λ$ denotes their highest weight. Consequently, we obtain detailed information of their Eisenstein cohomology with coefficients in $\mathcal{M}_λ$. When $\mathcal{M}_λ$ is not self dual, the Eisenstein cohomology coincides with the cohomology of the underlying arithmetic group with coefficients in $\mathcal{M}_λ$. In particular, for such a large class of represe…
In this work we prove the so-called "torsion congruences" between abelian $p$-adic $L$-functions that are related to automorphic representations of definite unitary groups. These congruences play a central role in the non-commutative Iwasawa theory as it became clear in the works of Kakde, Ritter and Weiss on the non-abelian Main Conjecture for the Tate motive. We tackle these congruences for a general definite unitary group of $n$ variables and we obtain more explicit results in the special cases of $n=1$ and $n=2$. In both of these cases we also explain their implications for some particular…
Equivariant primitives of Eisenstein series for congruence subgroups
本文证明主同余子群的爱森斯坦级数的等变原函数恰为非全纯爱森斯坦级数，给出推广全模群结果的闭公式，并研究亏格零模曲线上权为2的情形，将其表示为以Hauptmodul有理函数为自变量的单值对数。
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 …
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.
Rationality results are proved for the ratios of critical values of Rankin-Selberg L-functions of GL(n) x GL(n') over a totally imaginary field F, by studying rank-one Eisenstein cohomology for the group GL(N)/F, where N = n+n', generalizing the methods and results of previous work with Guenter Harder where the base field was totally real. In contrast to the totally real situation, the internal structure of the totally imaginary base field has a delicate effect on the rationality results.
Rationality of the periods of Eisenstein series
本文推广Zagier–Gangl的观察，证明一般同余子群上谱Eisenstein级数在Eichler–Shimura同构下的像定义于分圆数域，并推广相关不变量、初等给出Maass Eisenstein级数的Fourier展开。
On the residual Eisenstein cohomology of unitary groups
本文研究数域二次扩张上酉群的剩余艾森斯坦上同调，识别艾森斯坦级数的上同调相关极点并证明残数留存，以F=Q(∛2)上秩3酉群为例显式构造非零类，方法可推广。
We give a formula for the Eisenstein cohomology of local systems on the partial compactification of the moduli of principally polarized abelian varieties given by rank 1 degenerations. For genus 2 we give a formula for the full Eisenstein cohomology.
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.
Congruences of Eisenstein series of level $Γ_1(N)$ via Dieudonné theory of formal groups
本文基于Katz的代数几何解释，通过将黎曼-希尔伯特对应重新表述为高度1形式$A$-模的Dieudonné理论，给出了水平$Γ_1(N)$艾森斯坦级数同余的新解释，并推广至高度大于1的形式群。
Classical periods of Eisenstein series and Bernoulli polynomials in the equivariant cohomology of a torus
本文通过取值于currents的群上链为环面的$\ ext{GL}_2$-等变多重对数类提供显式代表元，并基于权为$2$的Eisenstein级数构造，几何地推导出Dedekind-Rademacher同态及相应周期的经典公式。
For a totally real field $F$, a finite extension $\mathbf{F}$ of $\mathbf{F}_p$ and a Galois character $χ: G_F \to \mathbf{F}^{\times}$ unramified away from a finite set of places $Σ\supset \{\mathfrak{p} \mid p\}$ consider the Bloch-Kato Selmer group $H:=H^1_Σ(F, χ^{-1})$. In an earlier paper of the authors it was proved that the number $d$ of isomorphism classes of (non-semisimple, reducible) residual representations $\overlineρ$ giving rise to lines in $H$ which are modular by some $ρ_f$ (also unramified outside $Σ$) satisfies $d \geq n:= \dim_{\mathbf{F}} H$. This was proved under the assu…
On rationality of certain Eisenstein cohomology
研究与退化主系列表示相关的艾森斯坦上同调，通过特定方法证明了某些合理性结果。
In this article, Eisenstein cohomology of the arithmetic group $G_2(\mathbb{Z})$ with coefficients in any finite dimensional highest weight irreducible representation has been determined. We accomplish this by studying the cohomology of the boundary of the Borel-Serre compactification.
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…
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…
The paper investigates a significant part of the automorphic, in fact of the so-called Eisenstein cohomology of split odd orthogonal groups over Q. The main result provides a description of residual and regular Eisenstein cohomology classes for maximal parabolic Q-subgroups in case of generic cohomological cuspidal automorphic representations of their Levi subgroups. That is, such identifying necessary conditions on these latter representations as well as on the complex parameters in order for the associated Eisenstein series to possibly yield non-trivial classes in the automorphic cohomology.
Kurokawa-Mizumoto congruence and differential operators on automorphic forms
该研究给出与Klingen-Eisenstein级数相关的向量值黑川-水本同余成立的充分条件，并借助表示论对自守形式上的微分算子作出新的诠释。
We set up a general framework to compute the exact multiplicity with which certain automorphic representations appear in both the cuspidal and Eisenstein cohomology of locally symmetric spaces. We apply this machinery to Eisenstein series on $GSp_4$ and split $G_2$. In the case of $G_2$, we will also obtain new information about the archimedean components of certain CAP representations using Arthur's conjectures.
Geodesic clustering of zeros of Eisenstein series for congruence groups
研究了同余群中Eisentein级数零点的测地聚类特性，揭示了零点收敛行为及代数零点分布。
The paper deals with establishing bounds for Eisenstein series on congruence quotients of the upper half plane, with control of both the spectral parameter and the level. The key observation in this work is that we exploit better the structure of the amplifier by just supporting on primes for the Eisenstein series, which can use both the analytic method as Young did to get a lower bound for the amplifier and the geometric method as Harcos--Templier did to obtain a more efficient treatment for the counting problem.
We develop two applications of the Kronecker's limit formula associated to elliptic Eisenstein series: A factorization theorem for holomorphic modular forms, and a proof of Weil's reciprocity law. Several examples of the general factorization results are computed, specifically for certain moonshine groups, congruence subgroups, and, more generally, non-compact subgroups with one cusp. In particular, we explicitly compute the Kronecker limit function associated to certain elliptic points for a few small level moonshine groups.
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