Statistics of Diophantine problems and the Cohen-Lenstra heuristics

We shall consider the following three questions:

 

(1) What is the number of positive square-free integers D going up to a large number X for which the equation a^2-Db^2=-1 has
a solution in a and b integers? 

 

(2) For how many positive integers D up to a large number X, the elliptic curve

 Dy^2=x^3-x has rank at least 2? 

 

(3) If c denotes an integer, usually how large is the unit group of a randomly chosen real quadratic field when reduced modulo c?"

 

These are fundamental Diophantine problems sharing a common feature: as I shall explain in this talk they are all secretly governed
by the distribution of certain ray class groups or Selmer groups respectively. A natural probabilistic model for the class group of random quadratic fields has been introduced by Cohen and Lenstra in 1983 (respectively for Selmer groups by Bhargava--Kane--Lenstra--Poonen--Rains in 2013), and I will briefly prove the equivalence with a natural model arising from large random matrices. Dramatic progress
on the distribution of the two part of class groups and Selmer groups has been made by Alexander Smith in a recent breakthrough,

which has lead in particular to a satisfactory answer to Question 2 above. I will mention some joint work in progress with Peter Koymans, towards an extension of this result for the p-part of class groups in the family of degree p cyclic extensions of the rationals. I will explain some joint work in progress with S.Chan--D.Milovic--P.Koymans on Question 1. I will spend some time showing that Question 3 is governed by the joint distribution of ray class group of modulus c and the ordinary class group. A probabilistic model for such a pair
(and for more refine invariants of it) has been introduced in a joint work with E. Sofos in 2017 in the case of imaginary quadratic fields, where we also established the joint distribution of the 4-rank of the two groups: I will explain how our result and a theorem of I.Varma
on the average 3-torsion, fit in the general set of conjectures we have introduced. I will conclude explaining joint work in progress
with A. Bartel on the extension of this to real quadratic fields in the direction of question 3.