Automorphic Forms, and their links to Arithmetic and Geometry

本专题将研究自守形式对几何和数论的影响。项目的第一部分是与K. Paranjape，将卡-丘流形的有理数的维数m的某些基本全纯尖形式的重量m+1与合理的系数。第二部分的目标是与J. Cogdell一起完善GL（n）的全能匡威定理，以便对相关的L函数要求更少，从而增加在更有趣的情况下的潜在适用性。第三部分，与N. Dunfield，将使用CM自守形式来构造一个算术型的紧致双曲3-流形M（n）的显式塔，使得Thurston单位球中的球面数随着n变大而无穷大，实际上是指数地无穷大。第四个项目是与D. Prasad，将使用全球的方法来解决一个猜想，当自对偶不可约表示的乘法群的除法代数D在一个局部域留下不变的对称，或交替，双线性形式，这将有潜在的应用到根数。该项目的主旨是理解一些对称性的表现在数学和超越。可以说，自然界中所有有趣的事物都有某种对称性，尽管并非总是如此。数学家和物理学家经常从观察或计算中得到离散的数集合，然后建立生成函数，而迫切需要知道这些函数是否具有隐藏的对称性，比如隐藏变量的反演不变性。当这种对称性出现时，它们通常描述自守函数的音调，自守函数是连续的、美丽的实体，如圆盘上的波形。他们的离散色调，实验和理论上，令人兴奋的数量，如长度的测地线，素数，和同余解决方案的多项式方程。开发它们是一项值得努力的奋进，还有许多金矿尚未被发现。

This project will investigate the impact of automorphic forms on geometry and number theory. The first part of the project, joint with K. Paranjape, will associate Calabai-Yau manifolds over the rationals of dimension m to certain basic holomorphic cusp forms of weight m+1 with rational coefficients. The goal of the second part, joint with J. Cogdell, is to refine the omnipotent converse theorem for GL(n) so as to require less on the associated L-functions, thereby increasing the potential applicability in more interesting situations. The third part, joint with N. Dunfield, will use CM automorphic forms to construct an explicit tower of compact hyperbolic 3-manifolds M(n) of arithmetic type such that the numbered of fibred faces in the Thurston unit ball goes to infinity, in fact exponentially, as n becomes large. The fourth project, joint with D. Prasad, will use global methods to settle a conjecture on when self dual irreducible representations of the multiplicative group of a division algebra D over a local field leave invariant a symmetric, or alternating, bilinear form, and this will have potential applications to the root number.The main thrust of the project is to comprehend some of the manifestations of symmetry in Mathematics and beyond. One could say that everything interesting in the natural world has some aspect of symmetry, though not always visibly so. Often mathematicians and physicists build generating functions outof discrete collections of numbers arising from observations or calculations, and it is a pressing problem to know if these functions admit hidden symmetries, like the invariance under inversion of a hidden variable. When such symmetries arise, they are often describing the tones of automorphic functions, which are continuous, pulchritudinous entities such as the waveforms on a disk. Their discrete tones are linked, experimentally and theoretically, to exciting quantities such as lengths of geodesics, primes, and congruence solutions of polynomial equations. Exploiting them is a worthy endeavor, and there are many gold mines yet to be discovered.