Jay Jorgenson, Cormac O'Sullivan
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.
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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 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.
The classical Kronecker limit formula describes the constant term in the Laurent expansion at the first order pole of the non-holomorphic Eisenstein series associated to the cusp at infinity of the modular group. Recently, the meromorphic continuation and Kronecker limit type formulas were investigated for non-holomorphic Eisenstein series associated to hyperbolic and elliptic elements of a Fuchsian group of the first kind by Jorgenson, Kramer and the first named author. In the present work, we realize averaged versions of all three types of Eisenstein series for $Γ_0(N)$ as regularized theta …
Consider an elliptic curve defined over an imaginary quadratic field $K$ with good reduction at the primes above $p\geq 5$ and has complex multiplication by the full ring of integers $\mathcal{O}_K$ of $K$. In this paper, we construct $p$-adic analogues of the Eisenstein-Kronecker series for such elliptic curve as Coleman functions on the elliptic curve. We then prove $p$-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.
Let $Γ\subset\mathrm{PSL}_{2}(\mathbb{R})$ be a Fuchsian subgroup of the first kind acting by fractional linear transformations on the upper half-plane $\mathbb{H}$, and let $M=Γ\backslash\mathbb{H}$ be the associated finite volume hyperbolic Riemann surface. Associated to any cusp of $M$, there is the classically studied non-holomorphic (parabolic) Eisenstein series. In 1979, Kudla and Millson studied non-holomorphic (hyperbolic) Eisenstein series associated to any closed geodesic on $Γ\backslash\mathbb{H}$. In 2004, Jorgenson and Kramer introduced so-called elliptic Eisenstein series associa…
$p$-adic properties of Eisenstein-Kronecker cocycles over imaginary quadratic fields and $p$-adic interpolation
本文证明了 Eisenstein-Kronecker 上循环的整性与同余性质，并构造 p-adic 测度插值虚二次域上 Hecke L-函数的临界值。
In this paper, we give a proof of the classical Kronecker limit formulas using the distribution relation of the Eisenstein-Kronecker series. Using a similar idea, we then prove $p$-adic analogues of the Kronecker limit formulas for the $p$-adic Eisenstein-Kronecker functions defined in our previous paper.
An Eisenstein--Kronecker Modular Symbol and Beilinson--Kato Distributions
本文构造了一个Eisenstein-Kronecker模符号，其特化恢复Borisov-Gunnells符号，并通过归一化将其分布等同于Colmez的Beilinson-Kato分布实现，从而建立了循环Manin关系并生成相关线性关系。
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.
A classical construction of Katz gives a purely algebraic construction of Eisenstein--Kronecker series using the Gauß--Manin connection on the universal elliptic curve. This approach gives a systematic way to study algebraic and $p$-adic properties of real-analytic Eisenstein series. In the first part of this paper we provide an alternative algebraic construction of Eisenstein--Kronecker series via the Poincaré bundle. Building on this, we give in the second part a new conceptional construction of Katz' two-variable $p$-adic Eisenstein measure through $p$-adic theta functions of the Poincaré b…
Eisenstein-Kronecker classes on Hilbert moduli spaces and a new construction of Katz's $p$-adic measure
将Kings-Sprang的Eisenstein-Kronecker类推广到Hilbert模空间上的万有交换概形，并与Katz的Eisenstein级数比较，构造插值这些类的$p$-进测度，恢复Katz的$p$-进Eisenstein测度。
Kronecker's first limit formula gives the polar and constant terms of the Laurent series expansion of the Eisenstein series for SL(2,Z) at s=1. In this article, we generalize the formula to certain maximal parabolic Eisenstein series associated to SL(n,Z) for n greater than or equal to 2. We also show how the generalized formula can be used to give the polar and constant terms of the Dedekind zeta function of any number field at s=1.
Higher Schwarzians of Elliptic Double Covers and Eisenstein--Kronecker Functions
本文通过射影核计算椭圆双覆盖的高阶 Aharonov 不变量，将其分解为 Eisenstein 项与 Eisenstein–Kronecker 函数，并揭示其与 Hasse 不变量的联系及约化性质。
Let $X$ be a smooth, compact, projective Kähler variety and $D$ be a divisor of a holomorphic form $F$, and assume that $D$ is smooth up to codimension two. Let $ω$ be a Kähler form on $X$ and $K_{X}$ the corresponding heat kernel which is associated to the Laplacian that acts on the space of smooth functions on $X$. Using various integral transforms of $K_{X}$, we will construct a meromorphic function in a complex variable $s$ whose special value at $s=0$ is the log-norm of $F$ with respect to $μ$. In the case when $X$ is the quotient of a symmetric space, then the function we construct is a …
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.
In this paper, using p-adic integration with values in spaces of modular forms, we construct the p-adic analogue of Weil's elliptic functions according to Eisenstein in the book "Elliptic functions according to Eisenstein and and Kronecker". This construction extends Serre's p-adic family of Eisenstein series in "Formes modulaires et fonctions zêta p-adiques". We show that the power series expansion of Weil's elliptic functions also exists in the p-adic case.
Eisenstein series are real analytic functions which play a central role in spectral theory of the hyperbolic Laplacian. Kronecker limit formulas determine their connection to modular forms. The main result of this work is Theorem 7.2 in which a Kronecker limit formula for a family of non-congruence subgroups associated with the Fermat curves is presented. As an application we can determine the scattering constants for the Fermat curves in Theorem 8.1.
Non-vanishing of Cubic Dirichlet L-Functions over the Eisenstein Field
本文针对艾森斯坦域上的本原三次狄利克雷特征族，建立了L函数第一矩的渐近公式和第二矩的上界，无条件证明了该族中无穷多个L函数在中心点不消没。
We study the properties of Eisenstein-Kronecker numbers, which are related to special values of Hecke $L$-function of imaginary quadratic fields. We prove that the generating function of these numbers is a reduced (normalized or canonical in some literature) theta function associated to the Poincare bundle of an elliptic curve. We introduce general methods to study the algebraic and $p$-adic properties of reduced theta functions for CM abelian varieties. As a corollary, when the prime $p$ is ordinary, we give a new construction of the two-variable $p$-adic measure interpolating special values …
Mollified Moments of Cubic Dirichlet L-Functions over the Eisenstein Field
在广义黎曼假设下，证明艾森斯坦域上三次狄利克雷 L-函数在中心点有正密度不消失，通过计算第一加扰矩并估计高阶矩上界，不消失比例显式但极小。
Eisenstein-Kronecker classes, integrality of critical values of Hecke $L$-functions and $p$-adic interpolation
本文证明任意全虚数域上代数Hecke特征标的临界$L$-值除以周期为代数整数，并构造$p$-进测度插值普通情形下的临界值，推广了Damerell、Shimura和Katz的结果。
Several-variable Kronecker limit formula over global function fields
本文为全局函数域上多变量 Eisenstein 级数建立了 Kronecker 极限公式，利用 Berkovich 解析结构对 Heegner 闭链进行参数化，并应用于解释 Dedekind-Weil zeta 函数和 Dirichlet $L$-函数的 Kronecker 项。
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 …
We prove that the generating function of overpartition $M2$-rank differences is, up to coefficient signs, a component of the vector-valued mock Eisenstein series attached to a certain quadratic form. We use this to compute analogs of the class number relations for $M2$-rank differences. As applications we split the Kronecker-Hurwitz relation into its "even" and "odd" parts and calculate sums over Hurwitz class numbers of the form $\sum_{r \in \mathbb{Z}} H(n - 2r^2)$.
Analytic Properties of an Orthogonal Fourier-Jacobi Dirichlet Series
本文研究了正交傅里叶-雅可比狄利克雷级数的解析性质，通过积分表示和theta对应关系，推导了其在复数域上的解析延拓及E_8晶格中的函数方程。
A first Kronecker limit formula for Kleinian groups
本文为Kleinian群证明了首个Kronecker极限公式，推广Goldstein方法，利用Eisenstein级数Fourier展开构造Dedekind eta函数类似物，并研究其性质。
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…
The problem of quantum unique ergodicity (QUE) of weight 1/2 Eisenstein series for Γ_0(4) leads to the study of certain double Dirichlet series involving GL2 automorphic forms and Dirichlet characters. We study the analytic properties of this family of double Dirichlet series (analytic continuation, convexity estimate) and prove that a subconvex estimate implies the QUE result.
Eisenstein Series on Metaplectic Covers and Multiple Dirichlet Series
本文计算了整体元辛群上由环面诱导的爱森斯坦级数的首个惠特克系数，并将其与对偶根系统的外尔群多重狄利克雷级数建立联系，从而证实了 Brubaker-Bump-Friedberg 猜想。
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 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 introduce the formal double Eisenstein space $\mathcal{E}_k$, which is a generalization of the formal double zeta space $\mathcal{D}_k$ of Gangl-Kaneko-Zagier, and prove analogues of the sum formula and parity result for formal double Eisenstein series. We show that $\mathbb Q$-linear maps $\mathcal{E}_k\rightarrow A$, for some $\mathbb Q$-algebra $A$, can be constructed from formal Laurent series (with coefficients in $A$) that satisfy the Fay identity. As the prototypical example, we define the Kronecker realization $ρ^{\mathfrak{K}}: \mathcal{E}_k\rightarrow \mathbb Q[[q]]$, which lifts …
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.
Euler-Kronecker constants of modular forms: beyond Dirichlet $L$-series
本文构造涉及 ${\mathfrak S}_3$、${\mathfrak S}_4$ 扩张所对应 Artin $L$-级数的 Euler–Kronecker 常数例子，并据此解决 2023 年关于模形式同余的问题。
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.
First, we recount a history of how certain methods using natural self-adjoint operators have, thus far, failed to prove the Riemann Hypothesis. In Section 2, we set the analytical context necessary to have genuine proofs in later sections, rather than attractive heuristics. In Section 3, we recall the utility of designed pseudo-Laplacians by reproving meromorphic continuation of certain Eisenstein series and proving a spacing result for zeros of $ζ_k(s)$ for $k$ a complex quadratic field with negative determinant. In Section 4, we construct an automorphic Hamiltonian which has purely discrete …
The specializations of the motivic elliptic polylog are called motivic Eisenstein classes. For applications to special values of L-Functions, it is important to compute the realizations of these classes. In this paper, we prove that the syntomic realization of the motivic Eisenstein classes, restricted to the ordinary locus of the modular curve, may be expressed using p-adic Eisenstein-Kronecker series. These p-adic modular forms are defined using the two-variable p-adic measure with values in p-adic modular forms constructed by Katz.
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 compute the Fourier coefficients of a minimal parabolic Eisenstein distribution on the double cover of SL$(3)$ over $\mathbb{Q}$. Two key aspects of the paper are an explicit formula for the constant term, and formulas for the Fourier coefficients at the ramified place $p=2$. Additionally, the unramified non-degenerate Fourier coefficients of this Eisenstein distribution fit into the combinatorial description provided by Brubaker-Bump-Friedberg-Hoffstein in `Weyl group multiple Dirichlet series. III. Eisenstein series and twisted unstable $A_r$'.
We consider, for even $s$, the secant Dirichlet series $ψ_s (τ) = \sum_{n = 1}^{\infty} \frac{\sec (πn τ)}{n^s}$, recently introduced and studied by Lal\'ın, Rodrigue and Rogers. In particular, we show, as conjectured and partially proven by Lal\'ın, Rodrigue and Rogers, that the values $ψ_{2 m} ( \sqrt{r})$, with $r > 0$ rational, are rational multiples of $π^{2 m}$. We then put the properties of the secant Dirichlet series into context by showing that they are Eichler integrals of odd weight Eisenstein series of level $4$. This leads us to consider Eichler integrals of general Eisenstein ser…
A multiple Dirichlet series in two variables is constructed as a Mellin transform of a higher order Eisenstein series. It is shown to extend to a meromorphic function and satisfy two independent functional equations.
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.
The space of toroidal automorphic forms was introduced by Zagier in the 1970s: a GL_2-automorphic form is toroidal if it has vanishing constant Fourier coefficients along all embedded non-split tori. The interest in this space stems (amongst others) from the fact that an Eisenstein series of weight s is toroidal for a given torus precisely if s is a non-trivial zero of the zeta function of the quadratic field corresponding to the torus. In this paper, we study the structure of the space of toroidal automorphic forms for an arbitrary number field F. We prove that it decomposes into a space span…
Eisenstein series part of the primitive representations for even rank quadratic forms
本文研究了二次型本原与非本原表示数的关系，提出处理偶秩二次型本原表示数公式中带二次狄利克雷特征的艾森斯坦级数部分的定理，并给出特殊情况公式示例。
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 consider certain double series of Eisenstein type involving hyperbolic-sine functions. We define certain generalized Hurwitz numbers, in terms of which we evaluate those double series. Our main results can be regarded as a certain generalization of well-known results of Hurwitz, Herglotz, Katayama and so on. Our results also include recent formulas of the third-named author which are double analogues of the formulas of Cauchy, Mellin, Ramanujan, Berndt and so on, about certain Dirichlet series involving hyperbolic functions. As an application, we give some evaluation formulas for $q$-zeta f…
The relation between Eisenstein series and the $J$-homomorphism is an important topic in chromatic homotopy theory at height $1$. Both sides are related to the special values of the Riemann $ζ$-function. Number theorists have studied the twistings of the Riemann $ζ$-functions and Eisenstein series by Dirichlet characters. Motivated by the Dirichlet equivariance of these Eisenstein series, we introduce the Dirichlet $J$-spectra in this paper. The homotopy groups of the Dirichlet $J$-spectra are related to the special values of the Dirichlet $L$-functions. Moreover, we find Brown-Comenetz duals …
We give explicit expressions for the Fourier coefficients of Eisenstein series twisted by Dirichlet characters and modular symbols on $Γ_0(N)$ in the case where $N$ is prime and equal to the conductor of the Dirichlet character. We obtain these expressions by computing the spectral decomposition of automorphic functions closely related to these Eisenstein series. As an application, we then evaluate certain sums of modular symbols in a way which parallels past work of Goldfeld, O'Sullivan, Petridis, and Risager. In one case we find less cancellation in this sum than would be predicted by the co…
Elliptic modular graph functions and forms (eMGFs) are defined for arbitrary graphs as natural generalizations of modular graph functions and forms obtained by including the character of an Abelian group in their Kronecker--Eisenstein series. The simplest examples of eMGFs are given by the Green function for a massless scalar field on the torus and the Zagier single-valued elliptic polylogarithms. More complicated eMGFs are produced by the non-separating degeneration of a higher genus surface to a genus one surface with punctures. eMGFs may equivalently be represented by multiple integrals ove…
Algebraic Eisenstein classes for $\mathrm{GL}_n$ with applications to the Bianchi case
本文为 Kings-Sprang 构造的代数 Eisenstein-Kronecker 类给出显式 current，在 Bianchi 情形下提供 Eisenstein 上同调类的纯代数构造，并将尖点常数项用部分 Hecke L-值表达。
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。
Summation formulas for Hurwitz class numbers and other mock modular coefficients
本文证明了中等增长拟模形式前 n 个系数的加权和公式，并将其应用于赫尔维茨类数与负半整数权爱森斯坦级数系数，得到二次狄利克雷 L-值形式的求和结果。
We introduce a new algorithm, which is faster and requires less computing resources than the ones previously known, to compute the Euler-Kronecker constants $\mathfrak{G}_q$ for the prime cyclotomic fields $\mathbb{Q}(ζ_q)$, where $q$ is an odd prime and $ζ_q$ is a primitive $q$-root of unity. With such a new algorithm we evaluated $\mathfrak{G}_q$ and $\mathfrak{G}_q^+$, where $\mathfrak{G}_q^+$ is the Euler-Kronecker constant of the maximal real subfield of $\mathbb{Q}(ζ_q)$, for some very large primes $q$ thus obtaining two new negative values of $\mathfrak{G}_q$: $\mathfrak{G}_{9109334831}…
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…
Explicit formula for the $(\text{GL}_2, \text{GL}_2)$ theta lift via Bruhat decomposition
本文通过 Bruhat 分解和 Kronecker-Eisenstein 级数构造流，给出了无平滑化的类型 II $(\ ext{GL}_2, \ ext{GL}_2)$ theta 提升的显式公式，并推广到 CM 域情形。
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…
Averages of quadratic twists of long Dirichlet polynomials
本文在Lindelöf假设下计算了由Kronecker符号扭曲的长Dirichlet多项式的平均值，验证了其与[CFKRS]配方中的0-和1-交换项一致。
The aim of this paper is to define some new number-theoretic functions including necklaces polynomials and the numbers of special words such as Lyndon words. By using Dirichlet convolution formula with well-known number-theoretic functions, we derive some new identities and relations associated with Dirichlet series, Lambert series, and also the family of zeta functions including the Riemann zeta functions and polylogarithm functions. By using analytic (meromorphic) continuation of zeta functions, we also derive identities and formulas including Bernoulli numbers and Apostol-Bernoulli numbers.…
Eisenstein degeneration of Beilinson--Kato classes and circular units
本文研究穿过 Eisenstein 级数临界 $p$-稳定化的 Beilinson–Kato Euler 系统，在多个 $p$-进 $L$-函数消失时建立其与分圆单位的显式分解关系。
Algebraicity of Spin $L$-functions for $\mathrm{GSp}_6$
本研究证明了$\u200b\u200b\u200bmathrm{GSp}_6$上全纯西格尔特征形式对应的尖点自守表示的Spin $L$-函数在对称中心右侧临界点的代数性，通过关联艾森斯坦级数性质的策略，突破了群无已知模问题、关联模型非唯一的难点。
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