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，如素数的分布，素数是只能被自己整除的正整数，1表示为2，3，5，7，11，13，17…2300年前，欧几里得证明了它们是无穷多个素数，200年前，高斯猜想出了素数p到x的渐近公式。高斯还猜测，渐近是非常精确的，误差项非常小。1896年Hadamard和de la Vallee-Poussin证明了这个渐近性，并称之为“素数定理”。但是，证明与渐近的拟合性与高斯猜想一样好仍然是一个悬而未决的问题，这被称为“黎曼假说”，是Clay千禧问题之一(奖金为100万美元……)黎曼假设相当于关于黎曼Zeta函数的零点位置的知识，这是第一个“L函数”。从那时起，L函数的概念被从多个方向推广，而那些L函数的解析性质(作为它们的零点位置)与数论中许多最深的问题有关，无论是已解的还是未解的。在过去的几十年里，卡茨和萨纳克的开创性工作--“L函数族”的研究取得了丰硕的成果，“L函数族”是具有某些共同特征的L函数族，因为L函数族的统计数据为单个L函数提供了宝贵的信息。对L功能的理解是我研究项目的核心。在过去的几年里，我把重点放在特殊的家庭上，就像L的“立方扭曲”家庭--功能。这一理论为“二次扭曲”所熟知，但文献中关于三次扭曲家族的著作很少，尤其是与大量关于二次扭曲家族的文献相比。我还研究了附加在“椭圆曲线”上的L函数，它们同样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 Millum问题的另一个猜想。这些L函数也与佐藤泰猜想有关。2010年，Taylor证明了与椭圆曲线相关的某些L函数是定义良好的(或解析的)函数，并且在它们的定义域的某些部分非零。我们对L函数的了解的提高对我们理解算术对象的结构有着深远的影响，如素数、椭圆曲线等，而对L函数的理解是我研究项目的核心。

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.