Automorphic Forms and L-Functions

自同构型和L函数是一类特殊的数学函数，可用于研究数论中的一系列广泛的问题。例如，Riemann Zeta函数是L函数的最简单的例子，它已被证明在研究素数的分布时是不可或缺的。其他类型的L函数对于了解某些多项式方程是否有解，或者更广泛地说，有多少解是至关重要的。这些问题往往与L函数在某一特定点的值有多大有关。本课题是关于研究L函数的值的新工具的发展。研究人员将研究超出凸性范围的L函数的矩，并推导出L函数中心值的新界。一个相关的主题是开发研究新的L函数族的基本工具，包括任意级别新形式的Petersson公式。这些工具对于发展研究三元二次型表示问题的解析方法是必要的。此外，研究人员还将研究自同构形式的均匀分布的不同方面，例如受限量子唯一遍历(QUE)以及在更高等级设置下的QUE。最后，研究者将研究各种类型的自同构函数的零点。所使用的方法将是解析数论的技术，如求和公式、指数和和积分，以及自同构形式的谱理论。

Automorphic forms and L-functions are special kinds of mathematical functions that are useful for studying a broad array of questions in number theory. For instance, the Riemann zeta function, which has proven to be indispensable in studying the distribution of the prime numbers, is the simplest example of an L-function. Other types of L-functions are crucial for understanding whether certain polynomial equations have solutions, or more generally, how many solutions there are. Often these questions are related to how large the value of an L-function is at a special point. This project concerns the development of new tools for studying the values of L-functions.The investigator will study moments of L-functions that are beyond the convexity range, and deduce new bounds for central values of L-functions. A related theme is to develop foundational tools to study new families of L-functions, including a Petersson formula for newforms of arbitrary level. Such tools are necessary to advance the analytic method for studying the representation problem for ternary quadratic forms. In addition, the investigator will study different aspects of the equidistribution of automorphic forms, such as restricted quantum unique ergodicity (QUE) as well as QUE in higher rank settings. Finally, the investigator will study the zeros of automorphic functions of various types. The methods employed will be techniques from analytic number theory such as summation formulas, exponential sums and integrals, and the spectral theory of automorphic forms.