CAREER: Explicit Methods in Arithmetic Geometry

研究了上同调自同构形式的算法和同余半算术群参数化曲线的算法。这一建议将抽象理论和实际计算联系起来，将研究与工具和具体项目的开发联系起来，这些将整合到本科和研究生教育中。在第一个提议的活动中，PI将证明存在一种算法来计算算术群作为Hecke模块的上同调。PI将在计算机代数系统中鲁棒地实现所提出的算法，收集和分析数据，并根据发现的现象推测和证明结果。在第二项活动中，PI将研究Galois Belyi盖的算术应用，这些应用是由三角形群模嵌入到四元数单元群中产生的。PI将研究模块化问题，椭圆曲线的参数化问题，以及由此构造产生的特殊点。拟议的研究将允许明确调查朗兰兹对应-自同构形式，代数群和伽罗瓦表示之间的深层联系-在经典和新颖的设置，探索令人兴奋的新领域。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。PI进一步提出了许多在本科和研究生阶段的具体项目，所提出的算法工具将整合到教学、培训和学习中。PI将继续拓展本科生，指导本科生和研究生，以及旨在向学生听众传达高水平数学的说说性写作和广泛的讲座。最后，PI将为从佛蒙特州数学倡议（VMI）毕业的教师领导设计一门关于密码学数学的课程。

The PI proposes to research algorithms for cohomological automorphic forms and the arithmetic of curves parametrized by congruence semi-arithmetic groups. This proposal interrelates abstract theory and practical computation, linking research with the development of tools and concrete projects which will be integrated into undergraduate and graduate education. In the first proposed activity, the PI will prove that there exists an algorithm to compute the cohomology of an arithmetic group as a Hecke module. The PI will implement the proposed algorithm robustly in a computer algebra system, collect and analyze data, and conjecture and prove results based on the discovered phenomena. In the second activity, the PI will study arithmetic applications of Galois Belyi covers that arise from a modular embedding of a triangle group into a quaternionic unit group. The PI will investigate questions of modularity, parametrization of elliptic curves, and special points arising from this construction. The proposed research will allow explicit investigation of the Langlands correspondence--the deep connection between automorphic forms, algebraic groups, and Galois representations--in both classical and novel settings, exploring exciting new ground.Classical unsolved problems often serve as the genesis for the formulation of a rich and unified 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. The PI further proposes many specific projects at the undergraduate and graduate level, and the proposed algorithmic tools will be integrated into teaching, training, and learning. The PI will continue undergraduate outreach, mentoring of undergraduate and graduate students, and expository writing and extensive lectures aimed at communicating high level mathematics to a student audience. Finally, the PI will design a course on the mathematics of cryptography for teacher leaders who have graduated from the Vermont Mathematics Initiative (VMI)