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Speaker:
Carlo Pagano
Affiliation:
University of Leiden/MPIM
Date:
Thu, 20/09/2018 - 16:30 - 17:30
Location:
MPIM Lecture Hall 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 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.
(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.
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