Distribution problems for L-functions and number fields

这个研究项目属于数论领域，数论是数学中处理整数性质和整数系数方程解的部分。这样的方程叫做丢番图方程。椭圆曲线是丢番图方程理论的基本研究对象之一。许多数论问题可以在称为数域的复数子集中进行更广泛的研究。在过去的十年中，我从解析的角度研究了与椭圆曲线和数域相关的几个数论问题，并使用了解析数论中的基本工具l -函数。我计划在未来五年内进一步研究这些问题。我的目标是将解析数论的一些基本结果，如最初对黎曼ζ函数和狄利克雷l函数所陈述的，推广到自同构l函数的一般情况；证明l函数族新的值分布定理，并探讨这些结果在数域不变量分布（如Euler-Kronecker常数）中的应用；研究用椭圆曲线的分域及其相关的l函数表述的几何设置中出现的分布问题。这项研究的结果将丰富我们在数论几个关键领域的知识，并将有助于为该领域的进一步研究铺平道路。此外，该研究项目为学生和博士后提供了良好的培训机会。

This research program is in the field of number theory, which is the part of mathematics that deals with properties of integers and solutions of equations with integer coefficients. Such equations are called Diophantine equations. Elliptic curves are one of the fundamental objects of study in the theory of Diophantine equations. Many number theoretical problems can be studied more generally in certain subsets of complex numbers called number fields. Over the past decade, I have studied several number theoretical problems related to elliptic curves and number fields from an analytic point of view and by employing L-functions which are fundamental tools in analytic number theory. I plan to further my research in such problems over the next five years. I aim to extend some of the fundamental results of analytic number theory, as originally stated for the Riemann zeta function and Dirichlet L-functions, to the general setting of automorphic L-functions; to prove new value distribution theorems for certain families of L-functions and explore the applications of such results in the distribution of invariants of number fields such as Euler-Kronecker constants; and to investigate distribution problems arising in geometric settings formulated in terms of division fields of elliptic curves and their associated L-functions. The outcome of this research will enrich our knowledge in several key areas of number theory and will help to pave the way for further studies in the field. In addition, this research program provides excellent training opportunities for students and postdoctoral fellows.