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)

[WEI] A. Weil, L’arithmétique sur les courbes algébriques, Acta Math. 52 (1929), 281–315. Also OEuvres Scientifiques – Collected Papers. Vol. I, Corrected Second Printing, SpringerVerlag, New York–Heidelberg–Berlin 1980, 11–45.
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.

notes here

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.

notes here

Week 4 (10/02)

Xiaorun Wu

Metrized line bundles and local heights, Abelian Varieties

This week, we will finish up on the proof of Roth's theorem. After that, we will introduce metrized line bundle, which will be later used to prove MordellWeil. After that, we will be convering some basics of Abelian Varieties, one of the key ingredients for the proof of general MordellWeil theorem.

notes here

Week 5 (10/09)

Xiaorun Wu

Abelian Varieties, NéronTate Heights

This week, we will continue our discussion on abelian varieties. After finishing up isogeny multiplication by n, we will formally introduce NéronTate height and Néron symbol. If time allows, we will be talking about Hilbert's irreducibility theorem, which would be another important ingredient for the proof of strong MordellWeil.

notes here

Week 6 (10/16)

Xiaorun Wu

NéronTate Heights (Cont'd)

This week, we will be continuing on our discussion of NéronTate height. After that, we will be proving Hilbert's irreducibility theorem. This would set up for the proof of MordellWeil: if time allows, we will provide a brief overview of the proof of MordellWeil, as well as introducing the weak MordellWeil on elliptic curves.

notes here and also see Bianca's notes, chapter 14 here

Week 7 (10/23)

Xiaorun Wu

Weak MordellWeil Theorem

This week, we will formally start on the proof of MordellWeil theorem on ellitpic curves. We will introduce Fermat's descent, which would be a sequence of lemmas needed for the proof of weak MordellWeil. After completing the proof of Fermat's descent, we will start on the proof of MordellWeil.

notes here

Week 8 (10/30)

Xiaorun Wu

Weak MordellWeil Theorem (Cont'd)

We will continue on the proof of weak MordellWeil: putting everything we have discussed in the previous section together, using main results from Fermat's descent, we will finish up the proof of weak MordellWeil Theorem.

notes here

Week 9 (11/09)

Xiaorun Wu

Strong MordellWeil Theorem

The main objective of this paper is to extend the result on strong MordellWeil theorem for Elliptic Curves to Abelian Varieties, namely the finite generation of the group of rational points of an abelian variety defined over a number field. In this week, we will introduce additional tools that would allow us to extend this result to general MordellWeil Theorem. As in the case of Elliptic Curve, we will split into two steps: in the first step, we will outline the proof for weak MordellWeil Theorem for general abelian varieties, then we give a generalized version of Fermat's descent theorem, which will allow us to prove strong MordellWeil Theorem.

notes here

Week 10 (11/13)

Xiaorun Wu

Strong MordellWeil Theorem (Cont'd)

We will continue on last week and finish the proof of storng MordellWeil theorem. In particular, finishing up the discussion of generalized version of Fermat's descent theorem, and put all the ingredients we already have for the strong MordellWeil Theorem.

notes here (same as last week)

Week 11 (11/20)

Xiaorun Wu

Falting's theorem, introduction

This week, we will provide a brief historical overview of Falting's theorem. We will introduce Vojta's divisors, as well as Mumford's method and an upper bound of NéronTate height. We will recall results from Roth's lemma, which will

notes here

Week 12 (11/27)

Xiaorun Wu

Falting's theorem (cont'd)

Using the tools from last week, we will give a complete proof of Falting's theorem: Let C be an irreducible projective smooth curve of genus \(g \ge 2\), defined over a number field K, with a point P0 defined over K. The proof of Faltings’s Theorem would be a direct consequence of the following theorem, which is known as the Vojta’s theorem.

notes here

Week 13 (12/04)

Xiaorun Wu

Nevanlinna theory: an overview, Vojta's conjecture

We will give a quick overview on Nevanlinna theory, which I feel important as it bridges the concepts of complex analysis with Arithmetic Geometry. We will survey some of the important result, which would set up for a discussion of Vojta's conjecture & abc conjecture.

notes here