Negative Pell, Higher Redei reciprocity, Higher genus theory and Arakelov ray class groups

In this talk we shall consider precise instances of the following question. Let S be a finite set of rational places, how does the unit group of a random real quadratic field typically embed in the various completions above S? An instance of this question, when S consists of the infinite place, is given by Stevenhagen's conjecture. I will explain a joint work with P. Koymans, where we make progress on this conjecture by means of Smith's method. I will explain our two main innovations. The first one is a generalization of Redei's reciprocity to governing expansions. The second one is a generalization of Gauss's genus theory, with the language of governing expansions, yielding sharp upper bounds for the 2-torsion of class groups of multi-quadratic fields. Finally I will explain the concept of Arakelov ray class groups, a construction introduced in a joint work with A. Bartel. With this notion we can make partial progress on the motivating question, in the case where S consists of non-archimedean places.