This term (summer 2024) the algebraic number theory study group will be on the work of Dimitrov, Gao, Habegger and Kühne. The talks will be at 2:30 in K0.18 in the King's building (Strand campus). Note that if you are interested in attending and external to King's then you will need to contact the seminar organisers Aled Walker and Vaidehee Thatte to ensure that you are able to access the building.
Talk 1 (01/05): Introduction (Netan)
In this talk we will give a(n) (a)historical introduction to the Mordell conjecture, it's proofs and its refinements. In particular we will explain the challenges in extending Vojta's proof to obtain a uniform bound in the form proved by Dimitrov, Gao and Habegger. We will then discuss Bogomolov's conjecture and give a vague sketch of some of the recent methods introduced for controlling 'small points' on curves.
Talk 2 (08/05): Heights (Lazar)
This talk will give an introduction to Weil heights over number fields and Neron--Tate heights of abelian varieties. Properties of heights in families, in particular the Silverman--Tate theorem, will also be discussed.
Talk 3 (15/05): Vojta's Theorem (Zerui)
This talk will discuss the Bombieri--Vojta proof of the Mordell conjecture, using gap principles for points of large height.
Talk 4 (29/05): Equidistribution and Bogomolov's conjecture (Chris)
This talk will give some background on Arakelov theory, Manin--Mumford, and Bogomolov's conjecture, and then explain the role of equidistribution of small points in the proof of Bogomolov's conjecture for a curve.
Talk 5 (12/06): Moduli of abelian varieties (Yicheng)
This talk will provide some foundational material on abelian schemes and moduli of abelian varieties over the complex numbers.
Talk 6 (19/06): The Betti map and degeneracy (Alex)
Talk 7 (26/06): The new gap principle (Peter)
References:
Uniformity in Mordell--Lang for curves, Dimitrov, Gao and Habegger.
A consequence of the relative Bogomolov conjecture, Dimitrov, Gao and Habegger.
Equidistribution in Families of Abelian Varieties and Uniformity, Kühne.
Recent developments of the Uniform Mordell-Lang Conjecture , Gao.
Generic rank of Betti map and Unlikely Intersections, Gao.
Heights in families of abelian varieties and the Geometric Bogomolov Conjecture, Gao and Habegger.
Equidistribution of small points on abelian varieties, Zhang.
Small points and Arakelov theory, Zhang.
Diophantine Geometry: An Introduction, Hindry and Silverman.
Heights in Diophantine Geometry, Bombieri and Gubler.