L-functions over number fields and function fields

l -函数是解析数论的核心，因为它们编码了几个重要对象的性质，比如质数的分布，它们是只能被自己和1整除的正整数，如2,3,5,7,11,13,17…2300年前，欧几里得证明了素数是无穷多个，200年前，高斯提出了一个素数p到x的渐近公式。高斯还推测渐近是非常精确的，误差项非常小。渐近性在1896年由Hadamard和de la Vallee-Poussin证明，并称为“素数定理”。但证明渐近拟合和高斯猜想一样好，仍然是一个开放的问题，被称为“黎曼假设”，是克莱千年问题之一（奖金100万美元……）黎曼假设相当于黎曼ζ函数的零点位置的知识，它是第一个“l函数”。从那时起，l -函数的概念在许多方向上得到了推广，这些l -函数的解析性质（如它们的零点的位置）与数论中许多最深刻的问题有关，这些问题已解决或未解决。在过去的几十年里，Katz和Sarnak的开创性工作已经出现，这是非常富有成效的研究“l -函数族”，它是l -函数具有一些共同特征的集合，因为l -函数族的统计为单个l -函数提供了有价值的信息。对l函数的理解是我研究项目的核心。在过去的几年里，我一直专注于特殊的族，如l函数的“三次扭转”族。关于“二次扭转”的理论已经得到了很好的理解，但是在文献中关于三次扭转族的研究很少，特别是与关于二次扭转族的大量文献相比。我还研究了与“椭圆曲线”相关的l函数，它再次提供了关于椭圆曲线的非常重要的信息，例如通过Birch和Swinnerton-Dyer猜想，Clay千禧年问题的另一个。这些l-函数也与Sato-Tate猜想有关，该猜想在2010年由Taylor证明，表明与椭圆曲线相关的某些l-函数在其定域的某些部分是定义良好的（或解析的）且非零。我们对l -函数知识的提高对我们对算术对象结构的理解产生了深远的影响，如素数、椭圆曲线和许多其他对象，对l -函数的理解是我研究计划的核心。

L-functions lie in the heart of analytic number theory, because they encode the properties of several important objects, such as distribution of prime numbers, which are positive integers only divisible by themselves and 1, as 2,3,5,7,11,13,17... It was proven by Euclid 2300 years ago that they are infinitely many primes, and an asymptotic formula for the number of primes p up to x was conjectured by Gauss 200 years ago. Gauss also conjectured that the asymptotic is very precise, with a very small error term. The asymptotic was proven by Hadamard and de la Vallee-Poussin in 1896, and called "The Prime Number Theorem". But proving that the fit to the asymptotic is as good as Gauss conjectured is still an open problem, which is called "The Riemann Hypothesis", and is one of the Clay Millenium Problem (with a prize of 1 million dollars...) The Riemann Hypothesis is equivalent to the knowledge of  the location of the zeroes of the Riemann zeta function, which is the first "L-function".  Since then, the concept of L-functions was generalized in many directions, and the analytic properties of those L-functions (as the location of their zeroes) are related to many of the deepest questions, solved or unsolved, in number theory. In the last decades, it has emerged for the seminal work of Katz and Sarnak that is is very fruitful to study "families of L-functions", which are sets of L-functions sharing some common features, because statistics for the family of L-function provide valuable information for individual L-functions. The understanding of L-functions is at the core of my research program. I have focused in the last years on special families, as the family of "cubic twists" of L-functions. The theory is well understood for "quadratic twists", but there are very few works on families of cubic twists in the literature, especially compared to the abundance of literature on families of quadratic twists. I also study L-functions attached to "elliptic curves", which again provide very important information about the elliptic curves, for example through the Birch and Swinnerton-Dyer conjecture, another of the Clay Millenium Problem. Those L-functions are also related to the Sato-Tate conjecture, which was proven in 2010 by Taylor by showing that certain L-functions associated to elliptic curves are well-defined (or analytic) and non-zero in some part of their domain. Improvement of our knowledge of L-functions have profound consequences to our understanding of the structure of arithmetic objects, as primes, elliptic curves, and many others, and the understanding of L-functions is at the core of my research program.