L-functions over number fields and function fields

数域和函数域上的 L 函数

• 批准号: RGPIN-2019-05536
• 负责人: David, Chantal
• 金额: $ 2.33万
• 依托单位: Concordia University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758944
• 关键词: functions; over; number; fields; function

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.

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.