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:

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)

Plan:

Module 1 (3-4 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 (2-3 Weeks): Abelian varieties, Néron-Tate height

  • Group varieties, elliptic curve, picard variety
  • Theorem of square and dual abelian variety
  • Theroem of cube
  • Isogeny multiplication by n
  • Néron-Tate height
  • Néron Symbol
  • Hilbert's irreducibility theorem

    • Module 3 (2 Weeks): Mordell-Weil Theorem

  • Weak Mordell-Weil for elliptic curve
  • Weak Mordell-Weil for abelian varieties
  • Kummer theory; Galois cohomology
  • Strong Mordell-Weil

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

  • Vojta divisor
  • Mumford's method for upper bound
  • Vojta's divisor of small height
  • Proof of Falting's theorem
  • Belyi's theorem
  • abc-conjecture

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

  • Nevanlinna theory
  • Ahlfors-Shimizu characteristic
  • Holomorphic curves in Nevanlinna theorem
  • Vojta's dictionary & conjectures
  • abc-theorem 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