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1000个判别式小于B的数域。一个自然的问题是，我们是否可以计算这些数域。赖特在1989年证明这是可能的阿贝尔扩展，和其他具体情况下，这是可能的证明达文波特，海尔布龙和Bhargava。本课题的目的是证明有限交换伽罗瓦群的导体满足一定的限制条件，即有界和平方和的数域是可以计数的，为此我定义了计数函数并研究了相应的Dirichlet级数.为了做到这一点，我将首先应用类场论，以便能够研究在adeles上的问题，然后使用调和分析来研究狄利克雷级数，现在在adeles上。然后，我将使用泊松求和公式将狄利克雷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分析性质来验证原始狄利克雷级数的分析性质。最后，我将使用傅立叶变换来推导所需的渐近公式。虽然我将计数的导体，一个可能的扩展，这将是计数的判别式代替。我使用的文本是文件“数字段与规定的规范”由Loughran，弗雷和牛顿，论文的朗道和塞尔在寻找渐近的数字表示的两个平方和的书“先进的分析数论：L函数”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.