UCL Discovery

## On the 2-part of class groups and Diophantine equations

Chan, Yik Tung; (2020) On the 2-part of class groups and Diophantine equations. Doctoral thesis (Ph.D), UCL (University College London).

## Abstract

This thesis contains several pieces of work related to the 2-part of class groups and Diophantine equations. We first give an overview of some techniques known in computing the 2-part of the class groups of quadratic number fields, including the use of the Rédei symbol and Rédei reciprocity in the study of the 8-rank of the class groups of quadratic fields. We review the construction of governing fields for the 8-rank by Corsman and extend a proof of Smith on the distribution of the 8-rank for imaginary quadratic fields, to real quadratic fields, conditional on the general Riemann hypothesis. In joint work with Peter Koymans, Djordjo Milovic, and Carlo Pagano, we improve a previous lower bound by Fouvry and Klüners, on the density of the solvability of the negative Pell equation over the set of squarefree positive integers with no prime factors congruent to 3 mod 4. We show how Rédei reciprocity allows us to apply techniques introduced by Smith to obtain this improvement. In joint work with Djordjo Milovic, using Kuroda's formula, we study the average behaviour of the unit group index in certain families of totally real biquadratic fields Q(√p,√d) parametrised by the prime p. In joint work with Christine McMeekin and Djordjo Milovic, we study certain cyclic totally real number fields K, in which we attach a quadratic symbol spin(a,σ) to each odd prime ideal a and each non-trivial σ in Gal(K/Q). We prove a formula for the density of primes ideals p such that spin(p,σ) = 1 for all non-trivial σ in Gal(K/Q). Finally, we study integral points on the quadratic twists E_D:y²=x³-D²x of the congruent number curve. We show that the number of non-torsion integral points on E_D is << (3.8)^{\rank E_D(Q)} and its average is bounded above by 2. We deduce that the system of simultaneous Pell equations aX²-bY²=d, bY²-cZ²=d for pairwise coprime positive integers a,b,c,d, has at most << (3.6)^{ω(abcd)} integer solutions.

Type: Thesis (Doctoral) Ph.D On the 2-part of class groups and Diophantine equations UCL (University College London) An open access version is available from UCL Discovery English Copyright © The Author 2020. Original content in this thesis is licensed under the terms of the Creative Commons Attribution 4.0 International (CC BY 4.0) Licence (https://creativecommons.org/licenses/by/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. Class group, Diophantine equation, elliptic curve, number theory UCLUCL > Provost and Vice Provost OfficesUCL > Provost and Vice Provost Offices > UCL BEAMSUCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences https://discovery.ucl.ac.uk/id/eprint/10107372