Reel kuadratik sayı cisimlerinin Picard grubu hakkında

Computing the class group of a number field has always been one of the main problems of number theory. For this purpose, several algorithms have been introduced so far, and it turns out that some of the computations underlying these algorithms can be viewed as ones within the Arakelov class group (or Picard group) of the number field in question. In this thesis, we first construct the oriented Arakelov class group of a number field K, $\widetilde{\pic^{0}(K)}$, that is analogous to the Picard group of an algebraic curve using basic notions attached to an oriented Arakelov divisor, e.g. degree, pairs consisting of fractional K-ideals and units in $K \otimes_{\Q} \R$. We then observe that $\widetilde{\pic^{0}(K)}$ is an extension of the narrow ideal class group of K, which will be denoted by $\cl^+(K)$, by a real Lie group. Afterwards, we focus on primitive integral indefinite binary quadratic forms of fixed discriminant and define an action of $\gl_2(\Z)$ on the set consisting of these forms. We will notice that the integrality, primitivity, and discriminant are invariants of this action. To combine these two main notions, we restrict our attention to real quadratic number fields. We show that there is a natural bijection between $\cl^+(K)$ and the set of equivalence classes of integral binary quadratic forms with fixed discriminant when K is a real quadratic field and then express $\widetilde{\pic^{0}(K)}$ in the language of indefinite binary quadratic forms.