A decomposition theorem on differential polynomials of theta functions of high level

Quadratic forms and their theta series - infinitesimal aspects

We study the theta map which assigns to a real quadratic form its theta series. We introduce two invariants reflecting whether the differential of the theta map vanishes or is degenerate. We provide examples of lattices where this differential is zero. These invariants turn out to be modular forms for integral lattices. We illustrate this in the rank two case.

The spectral decomposition of $|θ|^2$

Let $θ$ be an elementary theta function, such as the classical Jacobi theta function. We establish a spectral decomposition and surprisingly strong asymptotic formulas for $\langle |θ|^2, φ\rangle$ as $φ$ traverses a sequence of Hecke-translates of a nice enough fixed function. The subtlety is that typically $|θ|^2 \notin L^2$. Applications to the subconvexity, quantum variance and $4$-norm problems are indicated.

库德拉 - 米尔森西塔函数的勒夫谢茨分解

Lefschetz decompositions of Kudla-Millson theta functions

研究库德拉 - 米尔森西塔函数，证明其勒夫谢茨分解即模分解，表明库德拉 - 米尔森提升的像在X本原上同调中，推导X低阶上同调维数公式，成果覆盖紧致超凯勒流形模空间。

Hecke型虚假θ函数的通用公式

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。

参数模群与theta级数

Paramodular groups and theta series

本文研究参数模群的Hecke代数与边界分量，定义参数模theta级数，证明在平方自由水平和足够大权下生成尖形式空间，并给出Garrett分解的新几何证明。

A theory of theta functions to the quintic base

Properties of four quintic theta functions are developed in parallel with those of the classical Jacobi null theta functions. The quintic theta functions are shown to satisfy analogues of Jacobi's quartic theta function identity and counterparts of Jacobi's Principles of Duplication, Dimidiation and Change of Sign Formulas. The resulting library of quintic transformation formulas is used to describe series multisections for modular forms in terms of simple matrix operations. These efforts culminate in a formal technique for deducing congruences modulo powers of five for a variety of combinator…

三元二次格陪集的 Theta 级数

Theta series of ternary quadratic lattice cosets

本文证明三元格陪集 theta 级数的三部分自然分解分别对应亏格、spinor 亏格和类不变量，验证相关猜想，并推广 $p$-邻元以构造类代表元算法。

通过θ提升获得GL_n上的Whittaker函数

Whittaker functions on ${\rm GL}_n$ via theta lifting

本文通过θ提升方法推导了GL_n(C)上Whittaker函数的新显式公式，并计算了相关的Asai局部积分。

关于雅可比簇上黎曼θ-函数的若干微分方程

Some Differential Equations for the Riemann $θ$-Function on Jacobians

本文利用受Arakelov理论启发的Fay公式变体，证明了黎曼曲面雅可比簇上黎曼θ函数满足的若干微分方程。

群 $\mathrm{GSp}_4$ 上的 theta 算子

A theta operator for the group $\mathrm{GSp}_4$

本文利用修订的 Howe 方法，在 Siegel 三维簇上构造了 $\mathrm{GSp}_4$ 的 $p$-进模形式层上的 theta 算子。

Theta operators on Siegel modular forms and Galois representations

We describe the effect of the differential operators defined by Boecherer-Nagaoka, Flander-Ghitza and Yamauchi on the Galois representations (conjecturally) attached to Siegel modular eigenforms.

Higher pullbacks of modular forms on orthogonal groups

We apply differential operators to modular forms on orthogonal groups $\mathrm{O}(2, \ell)$ to construct infinite families of modular forms on special cycles. These operators generalize the quasi-pullback. The subspaces of theta lifts are preserved; in particular, the higher pullbacks of the lift of a (lattice-index) Jacobi form $ϕ$ are theta lifts of partial development coefficients of $ϕ$. For certain lattices of signature (2, 2) and (2, 3), for which there are interpretations as Hilbert-Siegel modular forms, we observe that the higher pullbacks coincide with differential operators introduce…

关于奇自旋A₁^(1)-字符串函数、交叉自旋恒等式以及类似假theta猜想的恒等式

On odd-spin $A_{1}^{(1)}$-string functions, cross-spin identities, and mock theta conjecture-like identities

本文研究了奇自旋A₁^(1)-字符串函数的极有限分解，并发现新的类似假theta猜想的恒等式。

有限域上对偶数上曲线的theta提升的类似物

Analog of theta-lifting for a curve over dual numbers over a finite field

本文研究了有限域对偶数环上曲线的自守函数框架下的theta提升实例，证明其可由自守函数空间的轨道分解理解，且所有强尖点函数均可通过适当双重覆盖的theta提升构造。

Superconnections, theta series, and period domains

We use superconnections to define and study some natural differential forms on period domains $\mathbb{D}$ that parametrize polarized Hodge structures of given type on a rational quadratic vector space $V$. These forms depend on a choice of vectors $v_1,\ldots,v_r \in V$ and have a Gaussian shape that peaks on the locus where $v_1,\ldots,v_r$ become Hodge classes. We show that they can be rescaled so that one can form theta series by summing over a lattice $L^r \subset V^r$. These series define differential forms on arithmetic quotients $Γ\backslash \mathbb{D}$. We compute their cohomology cla…

2级艾森斯坦级数的Ramanujan-Shen微分方程

Ramanujan-Shen's differential equations for Eisenstein series of level 2

受经典艾森斯坦级数微分方程研究启发，从第二类Jacobi theta函数推导2级艾森斯坦级数的新微分方程，为已有相关微分方程组提供新刻画，并得到Ramanujan tau函数的数论结果。

关于广义亚费米子理论的弦函数、mock theta函数与false theta函数，II

On string functions of the generalized parafermionic theories, mock theta functions, and false theta functions, II

本文通过发展容许弦函数的拟周期概念并计算Zagier--Zwegers极有限分解，扩展了仿射Kac--Moody代数$A_1^{(1)}$的结果，得到了用Ramanujan的mock theta函数表示的$1/3$、$2/3$和$1/5$层弦函数的新恒等式族。

Negative index Jacobi forms and quantum modular forms

In this paper, we consider the Fourier coefficients of a special class of meromorphic Jaocbi forms of negative index. Much recent work has been done on such coefficients in the case of Jacobi forms of positive index, but almost nothing is known for Jacobi forms of negative index. In this paper we show, from two different perspectives, that their Fourier coefficients have a simple decomposition in terms of partial theta functions. The first perspective uses the language of Lie super algebras, and the second applies the theory of elliptic functions. In particular, we find a new infinite family o…

Differential Birkhoff decomposition and the renormalization of multiple zeta values

In the Hopf algebra approach of Connes and Kreimer on renormalization of quantum field theory, the renormalization process is views as a special case of the Algebraic Birkhoff Decomposition. We give a differential algebra variation of this decomposition and apply this to the study of multiple zeta values.

超弦散射振幅的模形式特征：广义Eisenstein级数与Theta提升

Modular Features of Superstring Scattering Amplitudes: Generalised Eisenstein Series and Theta Lifts

本文研究超弦散射振幅中的广义Eisenstein级数，证明积分关联函数大N展开系数与两圈模图函数可通过theta提升统一描述，并表示为L值抵消的广义Eisenstein级数的有理组合。

On the transcendence degree of the differential field generated by Siegel modular forms

It is a classical fact that the elliptic modular functions satisfies an algebraic differential equation of order 3, and none of lower order. We show how this generalizes to Siegel modular functions of arbitrary degree. The key idea is that the partial differential equations they satisfy are governed by Gauss--Manin connections, whose monodromy groups are well-known. Modular theta functions provide a concrete interpretation of our result, and we study their differential properties in detail in the case of degree 2.

部分Theta函数的首根的组合学

The Combinatorics of the Leading Root of the Partial Theta Function

本文通过有根树与堆叠多联骨牌或Ferrers图的组合解释，给出了部分Theta函数首根系数的显式组合意义，并推导了其渐近公式。