首页/论文/2004.04716

arXiv 2004.047162020-04math.NTmath.CAmath.CV

Modular forms, deformation of punctured spheres, and extensions of symmetric tensor representations

Gabriele Bogo

备注：: 15 pages. The result on differential equations has been improved. A new section on vector-valued modular forms and extension of symmetric tensor representations is included

英文摘要：

Let~$X=\Po/Γ$ be an~$n$-punctured sphere, $n>3$. We introduce and study~$n-3$ deformation operators on the space of modular forms~$M_*(Γ)$ based on the classical theory of uniformizing differential equations and accessory parameters. When restricting to modular functions, we recover a construction in Teichmüller theory related to the deformation of the complex structure of~$X$. We describe the deformation operators in terms of derivations with respect to Eichler integrals of weight-four cusp forms, and in terms of vector-valued modular forms attached to extensions of symmetric tensor representations.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Accessory Parameters for Four-Punctured Spheres

We study the accessory parameter problem for four-punctured spheres from the point of view of modular forms. The value of the accessory parameter giving the uniformization is characterized as the unique zero of a system of equations. This gives an effective method to compute the uniformizing differential equation. As an application, we compute numerically and study the local expansion of the real-analytic function associating to a four-punctured sphere the value of its uniformizing parameter, and make some observations on its coefficients.

2022-03-28math.CAmath.CVmath.NT

Hyperbolic Metric, Punctured Riemann Sphere and Modular Functions

We derive a precise asymptotic expansion of the complete Kähler-Einstein metric on the punctured Riemann sphere with three or more omitting points. By using Schwarzian derivative, we prove that the coefficients of the expansion are polynomials on the two parameters which are uniquely determined by the omitting points. Futhermore, we use the modular form and Schwarzian derivative to explicitly determine the coefficients in the expansion of the complete Kähler-Einstein metric for punctured Riemann sphere with $3, 4, 6$ or $12$ omitting points.

2020-04-17math.DGmath.NT

Hypergeometric series, modular linear differential equations, and vector-valued modular forms

We survey the theory of vector-valued modular forms and their connections with modular differential equations and Fuchsian equations over the three-punctured sphere. We present a number of numerical examples showing how the theory in dimensions 2 and 3 leads naturally to close connections between modular forms and hypergeometric series.

2015-03-18math.NT

Lauricella超几何函数、穿孔Riemann球面的幂单基本群及其motivic余作用

Lauricella hypergeometric functions, unipotent fundamental groups of the punctured Riemann sphere, and their motivic coactions

本文提出一种与超几何积分相关的motives理论，证明散射振幅背景下Gauss超几何函数上的motivic Galois群作用猜想，并展示Lauricella超几何函数的局部与全局作用的兼容性。

2024-08-24hep-phmath-phmath.AG

最优球填充的模形式不等式的代数证明

Algebraic proof of modular form inequalities for optimal sphere packings

本文为8维和24维最优球填充的Viazovska及Cohn-Kumar-Miller-Radchenko-Viazovska模形式不等式提供了代数证明。

2025-08-13math.NT

Dual linear programming bounds for sphere packing via modular forms

We obtain new restrictions on the linear programming bound for sphere packing, by optimizing over spaces of modular forms to produce feasible points in the dual linear program. In contrast to the situation in dimensions 8 and 24, where the linear programming bound is sharp, we show that it comes nowhere near the best packing densities known in dimensions 12, 16, 20, 28, and 32. More generally, we provide a systematic technique for proving separations of this sort.

2021-04-19math.MGmath.NT

更多相关论文收起更多论文

Seifert纤维化同调3-球的同调块的模变换

Modular transformations of homological blocks for Seifert fibered homology $3$-spheres

本文针对任意Seifert纤维化整同调3-球，给出了同调块的显式模变换公式，并基于此推导出Witten-Reshetikhin-Turaev不变量的渐近展开公式，为Andersen版本的Witten渐近猜想提供了新证明。

2023-07-27hep-thmath-phmath.GT

Explicit Chabauty-Kim theory for the thrice punctured line in depth two

Let $X= \mathbb{P}^1 \setminus \{0,1,\infty\}$, and let $S$ denote a finite set of prime numbers. In an article of 2005, Minhyong Kim gave a new proof of Siegel's theorem for $X$: the set $X(\mathbb{Z}[S^{-1}])$ of $S$-integral points of $X$ is finite. The proof relies on a `nonabelian' version of the classical Chabauty method. At its heart is a modular interpretation of unipotent $p$-adic Hodge theory, given by a tower of morphisms $h_n$ between certain $\mathbb{Q}_p$-varieties. We set out to obtain a better understanding of $h_2$. Its mysterious piece is a polynomial in $2|S|$ variables. Our…

2014-03-18math.AGmath.NT

On Viazovska's modular form inequalities

Viazovska proved that the $E_8$ lattice sphere packing is the densest sphere packing in 8 dimensions. Her proof relies on two inequalities between functions defined in terms of modular and quasimodular forms. We give a direct proof of these inequalities that does not rely on computer calculations.

2023-03-23math.MGmath.NT

A note on Schwartz functions and modular forms

We generalize the recent work of Viazovska by constructing infinite families of Schwartz functions, suitable for Cohn-Elkies style linear programming bounds, using quasi-modular and modular forms. In particular for dimensions $d \equiv 0 \pmod{8}$ we give the constructions that lead to the best sphere packing upper bounds via modular forms. In dimension $8$ and $24$ these exactly match the functions constructed by Viazovska and Cohn, Kumar, Miller, Radchenko, and Viazovska which resolved the sphere packing problem in those dimensions.

2019-05-08math.NT

四 punctured 球体特征品种的强近似

Strong Approximation for the Relative Character Variety of the Four-Times Punctured Sphere

研究Markoff型方程在有限域上的解轨道，分析纯映射类群作用下的传输性质，分类退化参数并探讨其对大轨道计数的影响。

2026-07-30math.DSmath.GRmath.NT

Automorphic forms for triangle groups

For triangle groups, the (quasi-)automorphic forms are known just as explicitly as for the modular group SL$(2,\bbZ)$. We collect these expressions here, and then interpret them using the Halphen differential equation. We study the arithmetic properties of their Fourier coefficients at cusps and Taylor coefficients at elliptic fixed-points --- in both cases integrality is related to the arithmeticity of the triangle group. As an application of our formulas, we provide an explicit modular interpretation of periods of 14 families of Calabi-Yau threefolds over the thrice-punctured sphere.

2013-07-16hep-thmath.GRmath.NT