Nikos Bagis
In this article we give evaluations of certain series of hyperbolic functions using Jacobi elliptic functions theory. We also define some new functions that enable us to give characterization of not solvable class of series.
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Generalized Addition Theorems for the Jacobi Elliptic Functions
该研究推广了雅可比椭圆函数的经典加法定理，得到多个含不同项数的加法定式及行列式恒等式，为相关领域提供了新的数学工具。
In 1989, Ming Luo \cite{L2} showed that the Fibonacci number $U_n$ is Triangular if and only if $n=\pm1,2,4,8,10$. For this, he established a Jacobi Symbol Criterion. Moreover, he observed that this problem is equivalent to finding all integer points on two elliptic curves. In this paper, we prove a Jacobi Symbol Criterion for more general families of binary recurrences. In addition, applying the criterion and elementary methods, we determine all integer points on the elliptic curves $y^2=5x^2(x+3)^2+4(-1)^n$.
We show that Hida's families of $p$-adic elliptic modular forms generalize to $p$-adic families of Jacobi forms. We also construct $p$-adic versions of theta lifts from elliptic modular forms to Jacobi forms. Our results extend to Jacobi forms previous works by Hida and Stevens on the related case of half-integral weight modular forms.
Elliptic genera, real algebraic varieties and quasi-Jacobi forms
第一部分综述了L.Borisov和作者在文献中得到的椭圆类的推进公式及其应用。在剩余的部分中，我们讨论了环的准Jacobi形式，它允许的功能，这是椭圆属的殆复流形和扩展Ochanine椭圆属某些奇异的真正的代数簇的特点。
Non-vanishing of Jacobi Poincaré series
证明了在适当的条件下，整数权和矩阵指数的指数型Jacobi Poincaré级数不恒为零。对于经典Jacobi型，我们构造了由“前”几个Poincaré级数组成的基，并给出了与权相关和无关的条件，从而保证了经典Jacobi Poincaré级数的非零性.证明了某些Kloosterman型和的相等性。同时，得到了当奇素数整除指数时Jacobi Poincaré级数不为零的一个结果。
Two-dimensional series evaluations via the elliptic functions of Ramanujan and Jacobi
本文第一次以封闭形式给出了某些二重级数的值，这些级数是满格和的。椭圆函数、奇异模、类不变量和Rogers-Ramanujan连分数在我们的评估中起着核心作用
General Berndt-Type Integrals and Series Associated with Jacobi Elliptic Functions
本文利用围道积分、双曲和及矩阵方法，研究分母具有任意正次数的一般Berndt型积分，并将其表示为含Γ^4(1/4)和π^{-1}的有理系数多项式，同时提出若干猜想。
A Jacobi theta series and its transformation laws
我们考虑了Jacobi θ级数的推广，并证明了每一个这样的函数是一个拟Jacobi形式。在一定条件下，我们建立了这些函数关于Jacobi群的变换律，并证明了这些函数是Jacobi型。在建立这些结果，我们构造其他功能，也是雅可比形式。这些结果是由顶点算子代数理论中的应用所激发的。
Formal series of Jacobi forms
本文证明了在一般参数模水平下，满足对合条件的标量 Jacobi 形式形式级数必然收敛，并构成参数模 Fricke 本征形式在标准 1-尖点处的 Fourier-Jacobi 展开。
In this text we generalize the classical Jacobi Eisenstein series as they were discussed by Eichler and Zagier to arbitrary lattices. We use two different methods to derive the general Fourier expansion. The last two sections give formulas in the case where the lattice is an even unimodular lattice or $L$ is of the type $NA_{1}$.
A decomposition of the Fourier-Jacobi coefficients of Klingen Eisenstein series
本文研究了Siegel模形式空间的Klingen分解与Jacobi形式空间的Dulinski类似分解之间的关系。
We construct harmonic weak Maass forms that map to cusp forms of weight $k\geq 2$ with rational coefficients under the $ξ$-operator. This generalizes work of the first author, Griffin, Ono, and Rolen, who constructed distinguished preimages under this differential operator of weight $2$ newforms associated to rational elliptic curves using the classical Weierstrass theory of elliptic functions. We extend this theory and construct a vector-valued Jacobi--Weierstrass $ζ$-function which is a generalization of the classical Weierstrass $ζ$-function.
Vector valued formal Fourier-Jacobi series
H. Aoki证明了辛群Sp_2（Z）的任何对称形式的Fourier-Jacobi级数都是全纯Siegel模形式的Fourier-Jacobi展开式。结合青木定理和向量值模形式的事实，证明了向量值对称形式Fourier-Jacobi级数的一个类似结果。最近M. Raum使用不同的方法。作为一个应用，借助W. Zhang，得到了与正交群相关的Shimura簇上余维为2的特殊圈的模性结果.
In this article we show analytic properties of certain Rankin-Selberg type Dirichlet series for holomorphic Jacobi cusp forms of integral weight and of half-integral weight. The numerators of these Dirichlet series are the inner products of Fourier-Jacobi coefficients of two Jacobi cusp forms. The denominators and the range of summation of these Dirichlet series are like the ones of the Koecher-Maass series. The meromorphic continuations and functional equations of these Dirichlet series are obtained. Moreover, an identity between the Petersson norms of Jacobi forms with respect to linear isom…
Isomorphism between Jacobi forms of index $D_{2n+1}$ and elliptic modular forms of level $2$
本文研究了$D_{2n+1}$指数雅可比形式与水平为2的椭圆模形式之间的同构关系，并给出了雅可比-艾森斯坦级数的傅里叶系数公式，以及从Zagier-Eisenstein级数构造半正定模形式的方法。
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.
Berndt-Type Integrals and Series Associated with Ramanujan and Jacobi Elliptic Functions
本文以闭式求解含双曲与三角函数的Berndt型无穷积分，通过Gamma函数特殊值及椭圆函数展开求双曲和，并给出显式公式与猜想。
H-Harmonic Maass-Jacobi Forms of Degree 1: The Analytic Theory of Some Indefinite Theta Series
在以前的工作中，Zwegers（和Zagier）定义的一元$\widehatμ$ -函数和他的签名格$(r\!+\!1,1)$上的不定θ级数都是海森堡调和的Maaesthet-Jacobi形式。本文将Heisenberg调和性的概念推广到任意多个椭圆变量的Maaubrill-Jacobi形式，得到了符号为$(r\!+\!s,s)$的非退化格的“乘积型”的不定θ级数.因此，我们得到一个干净的推广$\widehatμ$这些负定格。通过对Heisenberg调和Maaesthet-Jacobi型扭点的限制，得到了在Zagier和Zwegers意义下的高阶调和弱Maaesthet-Jacobi型。特别是，我们解释了一些模块化完成，所谓的退化不定θ系列的背景下，更高的深度混合模拟模形式。本文发展的Heisenberg调和Maaesthet-Jacobi型的结构理论也解释了Zwegers二元$\widehatμ$ -函数奇异分裂为亚纯Jacobi型和一元Maaesthet-Jacobi型之和的现象。
We study Taylor expansions of Jacobi forms of lattice index. As the main result, we give an embedding from certain space of such forms, whether scalar-valued or vector-valued, integral-weight or half-integral-weight, of any level, with any character, into a product of finitely many spaces of modular forms. As an application, we investigate linear relations among Jacobi theta series of lattice index. Many linear relations among the second powers of such theta series associated with the $D_4$ lattice and $A_3$ lattice are obtained, along with relations among the third powers of series associated…
A Serre derivative for even weight Jacobi Forms
利用变形或扭曲的Eisenstein级数，构造了偶权Jacobi形式上的Jacobi-Serre导数，推广了经典的模形式上的Serre导数.作为应用，我们得到了指数$1$ Eisenstein级数$E_{4,1}, E_{6,1}$和一个新定义的$E_{2,1}$的Ramanujan方程.最后，我们将变形的爱森斯坦级数直接与经典的第一雅可比θ函数联系起来。
Berndt-Type Integrals of Order Three and Series Associated with Jacobi Elliptic Functions
本文利用Jacobi椭圆函数的Fourier级数和Maclaurin级数展开及围道积分法，通过Gamma函数特殊值建立了六类双曲和与两族三阶Berndt型积分的显式求值，并给出了相关推论与示例。
On Sun's Conjectures for Truncated Jacobi-Symbol Determinants via Supersingular Elliptic Curves
该研究针对孙智伟提出的截断雅可比符号行列式，通过统一方法结合超奇异椭圆曲线，证明了孙的多个相关猜想及强化形式，揭示了这类恒等式的通用机制。
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。
Finding the complement of an elliptic curve inside a Jacobian
本文针对亏格2曲线X到椭圆曲线E的非平凡映射，给出一个简单算法，用于计算X的雅可比簇分解中互补的椭圆曲线因子。
Pairings of harmonic Maass-Jacobi forms involving special values of partial $L$-functions
本文利用Jacobi型的Eichler-Shimura上同调理论，研究了调和Maass-Jacobi型与Jacobi积分之间的关系.作为应用，我们研究了由斜全纯Jacobi尖点形式的部分$L$ -函数的特殊值所定义的两个Jacobi积分之间的配对.我们得到了这种配对与斜全纯Jacobi尖点形式的Petersson内积之间的联系。这个结果可以看作椭圆模形式的Haberland公式对Jacobi形式的类似。
We introduce a method that is based on Fourier series expansions related to Jacobi elliptic functions and that we apply to determine new identities for evaluating hyperbolic infinite sums in terms of the complete elliptic integrals $K$ and $E$. We apply our method to determine generalizations of a family of $\text{sech}$-sums given by Ramanujan and generalizations of a family of $\text{csch}$-sums given by Zucker. Our method has the advantage of producing evaluations for hyperbolic sums with sign functions that have not previously appeared in the literature on hyperbolic sums. We apply our met…
Fourier transform pairs and Eisenstein-type series related to Jacobi elliptic functions
本文通过梅林变换计算了Jacobi椭圆函数相关的傅里叶变换，并得到了十六个Eisenstein型级数及其解析延拓、函数方程和显式表达式。
Modular forms of orthogonal type and Jacobi theta-series
本文考虑任意正定格L的半整指标的Jacobi形式（Eichler和Zagier书中的经典Jacobi形式对应于格A_1=<2>）。我们用Jacobi θ-级数给出了根系奇异权和临界权的Jacobi形式的许多例子。我们描述了Jacobi形式的半整指数的Jacobi提升。在某些情况下，给出了新的反射模形式的加法提升构造。
We prove modularity of formal series of Jacobi forms that satisfy a natural symmetry condition. They are formal analogues of Fourier-Jacobi expansions of Siegel modular forms. From our result and a theorem of Wei Zhang, we deduce Kudla's conjecture on the modularity of generating series of special cycles of arbitrary codimension and for all orthogonal Shimura varieties.
Using ordinary and exponential generating functions, we explore the reversion of power series defined by $2$nd order recurrences. We express the reversions in terms of Jacobi and Thron continued fractions. We find relations with Eulerian expressions using a transformation of continued fractions.
We give a simple and a more explicit proof of a mod $4$ congruence for a series involving the little $q$-Jacobi polynomials which arose in a recent study of a certain restricted overpartition function.
Maass-Jacobi Poincaré series and Mathieu Moonshine
Mathieu Moonshine将权为零、指数为1的弱Jacobi形式附加到Mathieu的最大散在单群的每个共轭类上。我们介绍了一个修改这个任务，从而弱Jacobi形式取代半全纯的马斯-Jacobi形式的重量和指数为2。本文证明了一类权为1的Maass-Jacobi Poincaré级数的收敛性，并以此刻画了最大Mathieu群的半全纯Maass-Jacobi形式.
Correction to "Kudla's Modularity Conjecture and Formal Fourier-Jacobi Series"
本研究针对论文《Kudla模性猜想与形式Fourier-Jacobi级数》，更正了其中引理4.4的错误及该引理在定理4.5中的应用问题，完善了相关数论研究的论证基础。
On Mordell-Weil groups of Jacobians over function fields
研究了域$K=k(t)$上阿贝尔簇的算法，其中$k$是任意域.主要结果是将$K$上某些Jacobian的Mordell-Weil群与$k$上其它Jacobian的同态联系起来。我们的方法也产生完全显式点的椭圆曲线上的无界秩在$\Fpbar(t)$和一个新的建设的椭圆曲线适度高秩在$\C(t)$。
Mock Jacobi forms in basic hypergeometric series
我们证明了一些$q$ -级数，如泛仿θ函数，是θ项与权为1/2的仿Jacobi形式的线性和，当它们被限制在扭点上并乘以$q$的适当幂时，它们成为实解析模形式的全纯部分.证明了某些$q$ -级数的线性和是权为1/2的弱全纯模形式，这是由于模Jacobi形式的湮灭或模Jacobi形式的完备化.作为应用，我们得到了划分的秩与曲柄之间的关系。
A calculation of $L$-series in terms of Jacobi sums
让我们考虑定义在有限域上的函数域的循环扩张。对于一个字符（非平凡）的这种扩展，我们计算，作为一个线性组合的产品的Jacobi和，系数的多项式给出的Dirichtlet $L$ -系列。在最后一节中，我们展示了这种计算的应用。
Andrews and Merca investigated a truncated version of Euler's pentagonal number theorem and showed that the coefficients of the truncated series are nonnegative. They also considered the truncated series arising from Jacobi's triple product identity, and they that its coefficients are nonnegative. This conjecture was posed by Guo and Zeng independently and confirmed by Mao and Yee using different approaches. In this paper, we provide a new combinatorial proof of their nonnegativity result related to Euler's pentagonal number theorem. Meanwhile, we find an analogous result for a truncated serie…
On the Number of Distinct Legendre, Jacobi and Hessian Curves
给出了有限域上Legendre、Jacobi、Hessian和广义Hessian曲线族中直到同构的不同椭圆曲线个数的显式公式。
Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions
在本文中，我们得到了许多无限族的显式精确公式涉及平方或三角数，其中两个推广雅可比的4和8平方的身份，分别为4n^2或4 n（n+1）平方，而不使用尖点形式。事实上，我们同样推广到无限家庭的所有雅可比的明确规定的程度2，4，6，8兰伯特级数展开的经典θ函数。此外，我们还将Jacobi关于2个正方形、2个三角形、6个正方形、6个三角形的特殊分析分别推广到12个正方形、12个三角形、20个正方形、20个三角形。这些结果，取决于新的扩展的权力的各种产品的经典θ函数，出现在设置的雅可比椭圆函数，相关的连分数，经常C-分数，汉克尔或Turanian决定，傅里叶级数，兰伯特级数，包含/排除，拉普拉斯展开式公式的决定，和舒尔功能。这些无穷族恒等式的舒尔函数形式类似于麦克唐纳的η-函数恒等式。此外，麦克唐纳工作中出现的幂4 n（n+1）、2n^2+n、2n^2-n也出现在我们分析的适当位置。我们的一般方法的一个特殊情况产生的两个Kac-Wakimoto证明的身份涉及表示一个正整数由4n^2或4 n（n+1）三角数的和，分别。
Modularity of formal Fourier--Jacobi series from a cohomological point of view
本文通过建立 $\mathcal{A}_g$ 紧化上线丛的上同调消失结果，分别在复数域和整数环上刻画了亏格 2 及带级结构的形式 Fourier–Jacobi 级数的模性。
We study generating functions of certain shapes of planar polygons arising from homological mirror symmetry of elliptic curves. We express these generating functions in terms of rational functions of the Jacobi theta function and Zwegers' mock theta function and determine their (mock) Jacobi properties. We also analyze their special values and singularities, which are of geometric interest as well.
Asymptotic behavior of partial and false theta functions arising from Jacobi forms and regularized characters
我们证明了几个渐近结果的部分和假θ函数所产生的Jacobi形式，作为模变量$τ$趋于$0$沿虚轴，和椭圆变量$z$是不受限制的复平面。我们观察到，这些功能表现出斯托克斯现象-这些功能的渐近行为急剧不同的椭圆变量$z$位于复平面内。我们应用这些结果研究了共形域论中的$(1,p)$ -单重顶点算子代数的正则特征和量子维数的渐近展开.特别是，这恢复和扩展了几个已知的结果有关的正则化量子尺寸，这是一个主要的动机来源。
Theta Series Associated with the Weil-Schroedinger Representation
Andre Weil发现的Weil表示在研究θ级数变换性质中起着重要作用。本文定义了Jacobi群的Weil-Schroedinger表示，并证明了与Weil-Schroedinger表示相关的θ级数是关于Jacobi模群的适当算术子群的Jacobi形式.
Intermediate Jacobian and Some Arithmetic Properties of Kummer-surface-type CY 3-folds
在这篇文章中，我们研究了Kummer曲面型CY三重$\hat{T/G}$的算法方面，其特征在于只有孤立奇点的三环面轨道$T/G$的临界分辨率。直到同构为止，只有两个这样的空间$\hat{T/G}$具有$|G|=3, 7$，并且两个$T$都具有CM椭圆曲线的三重积结构。这些$\hat{T/G}$的（Griffiths）中间雅可比行列式被明确地标识为$T$结构中出现的相应椭圆曲线。我们进一步给出了$\hat{T/G}$的$\QZ$ -结构，并验证了它们的模块性.
We use the method of tiling to give elementary combinatorial proofs of some celebrated $q$-series identities, such as Jacobi triple product identity, Rogers-Ramanujan identities, and some identities of Rogers. We give a tiling proof of the q-binomial theorem and a tiling interpretation of the q-binomial coefficients. A new generalized $ k $-product $q$-series identity is also obtained by employing the `tiling-method', wherein the generating function of the set of all possible tilings of a rectangular board is computed in two different ways to obtain the desired $q$-series identity. Several new…
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