# Arakelov ray class groups, the Cohen--Lenstra heuristics, Higher genus theory and Higher Redei reciprocity

I will explain, partly ongoing, work about the distribution of ray class groups and unit groups of real quadratic fields and more general families. I will explain work in progress, jointly with A. Bartel, revolving around the notion of Arakelov ray class group, a compact abelian group that encodes at the same time a ray class group of given conductor c and the reduction of the unit group modulo c. I will explain how we generalize the classical Cohen--Lenstra heuristics to the setting of Arakelov ray class groups, extending to the real quadratic case previous work done jointly with E. Sofos in the imaginary case. Furthermore

controlling mixed moments of a certain Arakelov cohomology map, one can partly control the distribution modulo 4-th powers of the unit group modulo c, one of the first results in this new direction. In the second part of my talk I will explain how this work relates with recent dramatic progress on the Cohen--Lenstra heuristic, due to the breakthrough of A. Smith on the

Cohen--Lenstra and the Goldfeld's conjecture. I will overview some of the ideas of his work, and explain recent joint work with P. Koymans addressing Gerth's conjecture, under GRH. Finally I will explain the connection between Smith's method and the first part of the talk and how, in a recent work with P. Koymans, this connection brought us to a result of independent

interest. Namely a generalization of the classical Gauss genus theory about the two torsion of class groups and a generalization of a classical reciprocity law of Redei, from the case of quadratic (resp. bi-quadratic) number fields to the case of multi-quadratic ones.

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