Carlo Pagano: On the distribution of (ray) class groups of random quadratic fields and related problems
Let \(l\) be a prime number. I shall explain a recent joint work with P. Koymans where, conditionally on the Generalized Riemann Hypothesis, we establish a conjecture of F. Gerth on the statistical behavior of the \(l\)-Sylow of the class group of a cyclic degree \(l\) extension of \(\mathbb{Q}\). Our work builds on a recent breakthrough of A. Smith that covered the case \(l=2\) and the quadratic fields to be imaginary. Altogether this settles a set of conjectures advanced by F. Gerth and directly inspired on the Cohen--Lenstra heuristics. In this talk I shall also give a brief introduction to these heuristics. I shall explain how using large random matrices over suitable DVR's one can explain intuitively the main term in our asymptotic result and its difference with the main term in Smith's theorem. These heuristics models have recently been generalized in several directions. I shall explain a recent joint work with E. Sofos where, using homological algebra, we extended these heuristics to a statistical model for ray class groups of imaginary quadratic number fields and we could prove some special cases of our conjectures; time allowing I shall explain some of the additional algebraic challenges that we had to overcome to arrive at formulating an analogous of Gerth's conjecture.
