Modular varieties, arithmetic and geometry

该项目将研究自同构形式研究中产生的模变体以及某些Calabi-Yau流形中的算术和几何编码。该项目的第一部分将研究证明整个四元数Shimura曲面的Tate猜想的可能性，如果成功，将导致在所有经典类型的Shimura变体上的这个问题上的进展。第二个项目是与K. Paranjape合作，旨在将具有m维有理上对合的Calabi-Yau变种与权值m+1和有理系数的某些基本全纯顶点形式联系起来。这里的重点将放在三维的情况下，以理解二次曲线和函数积。PI还将继续他对某些其他主题的研究，包括与N. Dunfield关于算术型双曲3流形的圆振动的工作，以及与P. Michel关于l值的精确平均值的工作。当人们遇到的许多问题表现出某些对称性时，例如在镜面反射下的周期性或不变性，它们就可以用数学方法加以说明。这在破译密码、研究晶体和宝石等方面都很重要。这个项目的主要目的是理解对称的一些表现形式，特别是当它们的存在不明显的时候。数学家、物理学家和其他人通常从可能来自实验的离散数字集合开始，然后形成它们的生成函数，并询问它们是否编码隐藏的对称性。当这种和谐的安排出现在自然界时，它们经常描述自同构函数的音调，这些自同构函数是连续的实体，就像磁盘上的波形一样。它们的离散频率与曲线长度、素数和多项式方程的同余解等令人兴奋的结构有关。该项目的一个特别重点是确定伽罗瓦对称的存在何时意味着环境空间中存在特殊的弯曲子空间，这些子空间反过来与某些zeta函数的极点相关。最终目的是通过几何和分析更好地理解数字形成的普遍性和力量。

This project will investigate the arithmetic and geometry encoded in the modular varieties arising from the study of automorphic forms, as well as in certain Calabi-Yau manifolds. The first part of the project will investigate the possibility of proving the Tate conjecture for the entire class of quaternionic Shimura surfaces, which if successful, will lead to progress on this question for divisors on all Shimura varieties of classical type. The second project, joint with K. Paranjape, will aim to associate Calabi-Yau varieties with involution over the rationals of dimension m to certain basic holomorphic cusp forms of weight m+1 and rational coefficients. The emphasis here will be on the 3-dimensional case, with a view to understanding quadratic twists and functorial products. The PI will also continue his investigation into certain other topics, including the works with N. Dunfield on the circle fibrations of hyperbolic 3-manifolds of arithmetic type, and with P. Michel concerning the exact averages of L-values.Many problems one encounters become amenable to elucidation by the mathematical method when they exhibit some symmetry such as periodicity or invariance under mirror reflection. This is important in the cracking of codes and in the study of crystals and precious stones, for example. The main thrust of this project is to comprehend some of the manifestations of symmetry, especially when their presence is not evident. Mathematicians, Physicists, and others often start with discrete collections of numbers, possibly from experimentation, then form their generating functions, and ask if they encode hidden symmetries. When such harmonious arrangements arise in nature, they frequently describe the tones of automorphic functions, which are continuous entities like the waveforms on a disk. Their discrete frequencies are linked to exciting constructs like lengths of curves, prime numbers, and congruence solutions of polynomial equations. A particular focus of the project is to determine when the presence of Galois symmetries implies the existence of special curved subspaces of the ambient space, relating in turn to the poles of certain zeta functions. The ultimate aim is to understand the ubiquity and power of number formations better through geometry and analysis.