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

“该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。“首席研究员（PI）将推进Shimura曲线和四元数代数的算法理论，并应用于算术Fuchsian群上的Hilbert模形式和自守形式的计算。 在继续与Matthew Greenberg的联合工作中，PI将通过Shimura曲线的（1度）上同调来推广计算Hilbert模形式的Hecke模的算法范围，并将在完全真实的域上构建模块椭圆曲线的穷举表。 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基本算法问题。经典的未解决问题通常是丰富和统一的数学结构的起源。 亚历山大的丢番图（Diophantus of Alexandria）最早在两千年前就开始用整数求解代数方程。 今天，数学家们认识到，几何性质往往支配算术对象的行为。此外，计算工具提供了一种测试几何的手段，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。

"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.