Seminar: Diophantine Geometry (Fall 2023)
Topic: Heights in Diophantine Geometry
This seminar is intended as the first of a twosemester sequal on diophantine geometry: In this semester, we would survey various major theorems and conjectures in diophantine geometry: the strong MordelWeil theorem, Faltings's theorem, Vojta's conjecture, abcconjecture, and, if time allows, Zhang's theorem on small points. To provide a satisfactory introduction to the theorems above. we will survey various basic machineary needed:

Height: Weil height, NéronTate height, Faltings' height

Roth's theorem

Abelian varieties

Nevanlinna theory
Once we have the basic frameworks in handy, we shall be able to offer a relatively direct proof on the theorems mentioned above.
Logistics (To be continuously updated thorughout the semester)

When: Mondays, 6:007:15 PM ET

Where: Math 528

Organizer: Xiaorun Wu

References:

Main References

Enrico Bombieri and Walter Gubler Heightsin Diophantine Geometry, Cambridge University Press

Marc Hindry and Joseph H. Silvennan Diophantine Geometry: An Introduction, Springer GTM 201

References to modules: (to be updated throughout semester)

[Bha] Bhagav Bhatt  The HodgeTate decomposition via perfectoid spaces, link
Plan:
Module 1 (34 Weeks): Weil Height, Roth's theorem
Heights in projective & affine space
Heights of polynomials
Local Heights
Global Heights
Weil Heights & explicit bonds
Siegel's Lemma
Proof of Roth's Theorem
Metrized line bundles and local heights
Module 2 (23 Weeks): Abelian varieties, NéronTate height
Group varieties, elliptic curve, picard variety
Theorem of square and dual abelian variety
Theroem of cube
Isogeny multiplication by n
NéronTate height
Néron Symbol
Hilbert's irreducibility theorem
Module 3 (2 Weeks): MordellWeil Theorem
Weak MordellWeil for elliptic curve
Weak MordellWeil for abelian varieties
Kummer theory; Galois cohomology
Strong MordellWeil
Module 4 (3 Weeks): Falting's Theorem, abcconjecture
Vojta divisor
Mumford's method for upper bound
Vojta's divisor of small height
Proof of Falting's theorem
Belyi's theorem
abcconjecture
Module 5 (2 Weeks): Nevanlinna theory, Vojta's conjectures
Nevanlinna theory
AhlforsShimizu characteristic
Holomorphic curves in Nevanlinna theorem
Vojta's dictionary & conjectures
abctheorem for function fields
Tentative Syllabus:
See here
Schedule

Week 0 (09/07)

Xiaorun Wu

Logistics & Introduction

We had a short session today to go through the general logistics, as well as updated our syllabus with some new topcis proposed. For a detailed memo, please see below. The actual seminar begins next week (see the description for next week).

Week 1 (09/11)

Xiaorun Wu

Overview of Seminar & Heights in Projective Space, Local/Global Heights, Weil Heights

We will first give the overview and the structure of the seminar. Then we will start with defining Heights in projective space, for which we walk through Kronecker's theorem, Dirichlet's unit theorem, and Liouville's inequality. Next, we will define heights of a polynomial, for which we will prove Jensen's formula and Northcott's theorem, whhich would allow us a crude bound of the height of a polynomial. We next formally introduce Weil's height, starting with local & global heights.

notes here

Week 2 (09/18)

Xiaorun Wu

Siegel's lemma & Roth's theorem over the rationals (Introduction)

Continue from last time, we will define heights of a polynomial, for which we will prove Jensen's formula and Northcott's theorem, which would allow us a crude bound of the height of a polynomial. We next formally introduce Weil's height, starting with local & global heights. Next, we will introduce Siegel's lemma, which will be used to prove Roth's theorem. If time allows, we will briefly mention Roth's theorem over number fields.

Week 3 (09/25)

Xiaorun Wu

Roth's theorem over the rationals; Roth's theorem

We will continue from last time, and then introduce Roth's theorem for number fields. If time allows, we will give Lang's general formulation of Roth's theorem. To do this, we will introduce Roth's lemma, from which we will introduce the Wronskian criterion. Finally, we will give the complete proof of Roth's theorem.

Week 4 (10/01)

Xiaorun Wu

Metrized line bundles and local heights, Abelian Varieties
