Published on *Max Planck Institute for Mathematics* (https://www.mpim-bonn.mpg.de)

Posted in

- Talk [1]

Speaker:

Carlo Pagano
Affiliation:

University of Leiden/MPIM
Date:

Thu, 2018-09-20 16:30 - 17:30 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?

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,

by the distribution of certain

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.

(and for more refine invariants of it) has been introduced in a joint work with E. Sofos in 2017 in the case of

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

**Links:**

[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39

[2] https://www.mpim-bonn.mpg.de/node/3444

[3] https://www.mpim-bonn.mpg.de/node/7671