Seminar: Diophantine Geometry (Fall 2023)

Topic: Heights in Diophantine Geometry

This seminar is intended as the first of a two-semester sequal on diophantine geometry: In this semester, we would survey various major theorems and conjectures in diophantine geometry: the strong Mordel-Weil theorem, Faltings's theorem, Vojta's conjecture, abc-conjecture, 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, Néron-Tate height, Faltings' height
• Roth's theorem
• Abelian varieties
• Nevanlinna theory

Logistics (To be continuously updated thorughout the semester)

• When: Mondays, 6:00-7:15 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, Springer-Verlag, New York–Heidelberg–Berlin 1980, 11–45.

   

Plan:

Module 1 (3-4 Weeks): Weil Height, Roth's theorem

Module 2 (2-3 Weeks): Abelian varieties, Néron-Tate height

Module 3 (2 Weeks): Mordell-Weil Theorem

Module 4 (3 Weeks): Falting's Theorem, abc-conjecture

Module 5 (2 Weeks): Nevanlinna theory, Vojta's conjectures

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 Mordell-Weil. After that, we will be convering some basics of Abelian Varieties, one of the key ingredients for the proof of general Mordell-Weil theorem.

notes here

Week 5 (10/09)

Xiaorun Wu

Abelian Varieties, Néron-Tate Heights

This week, we will continue our discussion on abelian varieties. After finishing up isogeny multiplication by n, we will formally introduce Néron-Tate 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 Mordell-Weil.

notes here

Week 6 (10/16)

Xiaorun Wu

Néron-Tate Heights (Cont'd)

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

notes here and also see Bianca's notes, chapter 1-4 here

Week 7 (10/23)

Xiaorun Wu

Weak Mordell-Weil Theorem

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

notes here

Week 8 (10/30)

Xiaorun Wu

Weak Mordell-Weil Theorem (Cont'd)

We will continue on the proof of weak Mordell-Weil: 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 Mordell-Weil Theorem.

notes here

Week 9 (11/09)

Xiaorun Wu

Strong Mordell-Weil Theorem

The main objective of this paper is to extend the result on strong Mordell-Weil 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 Mordell-Weil 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 Mordell-Weil Theorem for general abelian varieties, then we give a generalized version of Fermat's descent theorem, which will allow us to prove strong Mordell-Weil Theorem.

notes here

Week 10 (11/13)

Xiaorun Wu

Strong Mordell-Weil Theorem (Cont'd)

We will continue on last week and finish the proof of storng Mordell-Weil 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 Mordell-Weil 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 Néron-Tate 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