Modular forms and arithmetic geometry

模形式和算术几何

• 批准号: RGPIN-2017-06579
• 负责人: Kudla, Stephen
• 金额: $ 3.13万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=688669
• 关键词: Modular; forms; arithmetic; geometry

Imagine a crystal as a lattice of atoms arranged in space according to a regular repeating pattern. Theta functions, the mathematical objects which encode the locations of the lattice points (atoms) as one moves outward from a fixed point, have many remarkable properties and have been studied since the 19th century. Recently, much more complicated theta functions have been introduced, for example in the study of lattice points inside or outside the light cone Minkowski space. Still more exotic ones occur in the study of arithmetic geometry -- the theory of whole number solutions to polynomial equations. Theta functions are special cases of a wider class of functions, modular forms and automorphic forms. The goal of this research is to develop new types of theta functions and automorphic forms and to investigate their applications in number theory and their connections to physics, in particular, to the counting functions which arise in string theory.

把晶体想象成一个由原子组成的晶格，在空间中按照规则的重复模式排列。Theta函数是一种数学对象，当一个人从一个固定点向外移动时，它对晶格点（原子）的位置进行编码，它有许多显著的特性，自19世纪以来一直被研究。最近，更复杂的函数被引入，例如在光锥闵可夫斯基空间内外的点阵点的研究。在等差几何的研究中还有更奇特的——多项式方程的整数解理论。函数是一类更广泛的函数的特殊情况，模形式和自同构形式。本研究的目标是开发新型的函数和自同构形式，并研究它们在数论中的应用及其与物理学的联系，特别是与弦理论中出现的计数函数的联系。