Counting number fields with finite Abelian Galois group of bounded conductor that can be described as the sum of two squares.

Hermite-Minkowski定理指出，对于任何正的常数B，只有有限多个判别式小于B的数域。一个自然的问题是，我们是否能计算这些数域。1989年，Wright证明了这对于Abel扩张是可能的，Davenport、海尔布伦和Bhargava也证明了这是可能的。这个项目的目的是证明人们可以计数有限交换伽罗华群的数域，其导体满足一定的限制，即它是有界的，并且是两个平方的和.为此，我将定义一个计数函数并研究相关的Dirichlet级数.为了做到这一点，我将首先应用类场理论，以便能够研究Adeles上的问题，然后使用调和分析来研究Dirichlet级数，现在Adeles上。然后，我将使用泊松求和公式将狄里克莱级数写成傅立叶变换，因为它们更容易显式计算，并且我可以使用傅立叶变换的解析性质来验证原始狄里克莱级数的解析性质。最后，我将使用傅立叶变换来推导所需的渐近公式。虽然我将使用导体计数，但可能的扩展是通过判别式计数。我使用的文本是Loughran，Frei和牛顿的《具有规定范数的数域》，Landau和Serre关于寻找可由两个平方和表示的数的渐近的论文，以及Moreno的《高级解析数论：L函数》一书，作为调和分析的参考。

The Hermite-Minkowski theorem states that for any positive constant B, there are only finitely many number fields of discriminant less than B. A natural question which then arises is whether we can count these number fields. In 1989 Wright proved this was possible for Abelian extensions, and other specific cases where this is possible were proved by Davenport, Heilbronn and Bhargava. The aim of this project is to prove that one can count the number fields with finite abelian Galois group whose conductor satisfies certain restrictions, namely that it is bounded and the sum of two squares.In order to do so I will define a counting function and study the associated Dirichlet series. In order to do this I will first apply class field theory, in order to be able to study the problem over the adeles, and then use harmonic analysis to study the Dirichlet series, now over the adeles. I will then use a Poisson summation formula to write the Dirichlet series in terms of the Fourier transforms as they are easier to explicitly calculate, and I can use the analytic properties of the Fourier transforms to verify the the analytic properties of the original Dirichlet series. Finally, I will use the Fourier transforms to derive the required asymptotic formula.Although I will be counting by conductor, a possible extension of this would be to count by discriminant instead.The texts I am using are the papers "Number fields with prescribed norms" By Loughran, Frei and Newton, the paper of Landau and that of Serre on finding the asymptotic of numbers representable by the sum of two squares and the book "Advanced analytic number theory: L functions" by Moreno as a reference on harmonic analysis.