Automorphic Forms, and their links to Arithmetic and Geometry

这个项目将研究自同构形式对几何和数论的影响。该项目的第一部分是与K. Paranjape合作，将维度m的有理数上的Calabai-Yau流形与权值m+1的具有有理数系数的某些基本全纯顶点形式联系起来。第二部分的目标是与J. Cogdell共同完善GL(n)的全能逆定理，从而减少对相关l函数的要求，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联合，将使用CM自同构形式构造一个算术型的紧双曲3-流形M(n)的显式塔，使得Thurston单位球中的纤维面数量趋于无穷大，实际上是指数地，当n变大时。第四个项目，与D. Prasad合作，将使用全局方法来解决一个猜想，即在局部域上除法代数D的乘法群的自对偶不可约表示何时保持对称或交替双线性形式不变，这将有潜在的应用于根数。该项目的主要目的是理解对称性在数学及其他领域的一些表现形式。人们可以说，自然界中所有有趣的事物都有某种对称性，尽管并不总是明显如此。数学家和物理学家经常从观察或计算中产生的离散数字集合中构建生成函数，并且知道这些函数是否承认隐藏的对称性是一个紧迫的问题，例如隐变量的反转下的不变性。当这种对称性出现时，它们通常描述自同构函数的音调，这些自同构函数是连续的、有脉动的实体，如磁盘上的波形。它们的离散音调在实验和理论上都与令人兴奋的量有关，如测地线的长度、素数和多项式方程的同余解。开采它们是一项值得的努力，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.