Quaternion algebras, Shimura curves, and modular forms: Algorithms and arithmetic

四元数代数、Shimura 曲线和模形式：算法和算术

• 批准号: 0901971
• 负责人: John Voight
• 金额: $ 7.48万
• 依托单位: University of Vermont & State Agricultural College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2011-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901971&HistoricalAwards=false
• 关键词: Quaternion; algebras; Shimura; curves; modular

"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)."The principal investigator (PI) will advance the algorithmic theory of Shimura curves and quaternion algebras, with applications for the computation of Hilbert modular forms and automorphic forms on arithmetic Fuchsian groups. In continuation of joint work with Matthew Greenberg, the PI will generalize the scope of an algorithm to compute the Hecke module of Hilbert modular forms via the (degree 1) cohomology of a Shimura curve and will build exhaustive tables of modular elliptic curves over totally real fields. The PI will at the same time investigate the underlying algorithmic problems for quaternion algebras from both the perspective of computational complexity as well as practical implementation.Classical unsolved problems often serve as the genesis for the formulation of a rich and uni fied mathematical fabric. Diophantus of Alexandria first sought solutions to algebraic equations in integers almost two thousand years ago. Today, mathematicians recognize that geometric properties often govern the behavior of arithmetic objects.Furthermore, computational tools provide a means to test conjectures and can sometimes furnish partial solutions; at the same time, theoretical advances fuel dramatic improvements in computation. The theory, design, and implementation of algorithms in arithmetic geometry is a burgeoning area, and there are many exciting applications of these methods to diverse fields.

“这项奖励是根据2009年美国复苏和再投资法案（公法111-5）资助的。”主要研究者（PI）将推进Shimura曲线和四元数代数的算法理论，并将其应用于计算算术Fuchsian群上的Hilbert模形式和自同构形式。在与Matthew Greenberg的合作中，PI将推广一种算法的范围，该算法通过Shimura曲线的（1度）上同调来计算Hilbert模形式的Hecke模，并将在全实数域上建立模椭圆曲线的穷尽表。PI将同时从计算复杂性和实际实现的角度研究四元数代数的潜在算法问题。经典的未解决的问题往往是形成丰富而统一的数学结构的起源。大约两千年前，亚历山大的丢番图第一次寻求整数代数方程的解。今天，数学家们认识到几何性质常常支配着算术对象的行为。此外，计算工具提供了一种检验猜想的方法，有时可以提供部分解；与此同时，理论的进步推动了计算的巨大进步。算术几何中算法的理论、设计和实现是一个新兴的领域，这些方法在不同的领域有许多令人兴奋的应用。