PD GDL 2021

Goal: The aim of this seminar is to understand a new proof of the Mordell conjecture due to Lawrence and Venkatesh [LV], published in 2020.
Schedule: Every Friday afternoon, starting on April 16 2021.
Zoom link (ask Nicola for the password)
Oroganizers: Adrian I. and Nicola M.

PROGRAM


  1. Introduction. Outlines of the proof of Faltings and Lawrence-Venkatesh.
    Speaker: Adrian I.
    Schedule: 16/04, at 14:00, Room 2BC60


  1. The Shafarevich conjecture and the Kodaira-Parshin family of curves. Reduction of the Mordell conjecture to the Shafarevich conjecture.
    References: [\S2; Dar], [\SS 2-3; Szp]
    Speaker: Daniele T.
    Schedule: 23/04, at 16:00, Room 1AD100


  1. Proof of Faltings' theorem of the finiteness of the l-adic Galois representations with fixed Serre weights and bounded ramification.
    References: [\SS 2.5&2.7; Dar], [Del]
    Speakers: Eduardo W. and Shilung W.
    Schedule: 30/04, at 16:00, Room 1AD100


  1. The Tate conjecture for Abelian varieties over finite fileds (J. Tate) and number fields (G. Faltings).
    Speaker: Yukihide N.
    Schedule: 07/05, at 14:00, Room 2BC60


  1. Algebraic de Rham cohomology
    Speaker: Ernesto M.
    Schedule: 14/05, at 14:00


  1. Gauss-Manin connection for schemes over the complex numbers, versus over p-adics. Period mappings and Galois representations.
    Speaker: Luca M. and Ernesto M.
    Schedule: 21/05, at 14:00


  1. Introduction to p-adic Hodge theory. The S-unit equation ([LV] \S 4.1, \S 4.2).
    Speaker: Adrian I. and Nicola M.
    Schedule: 04/06, at 14:00


  1. Monodromy of Kodaira-Parshin families. ([LV], \S 8.)
    Speaker: Remke K. 11/06 at 14:00

REFERENCES

  • [LV] B. Lawrence and A. Venkatesh.
    Diophantine problems and p-adic period mappings. arXiv
    This is the main reference

  • [Noo] M.P. Noordman.
    Siegel's theorem via the Lawrence-Venkatesh method. arXiv

  • [Dar] H. Darmon.
    Rational points on curves. PDF
    Expository notes on Diophantine equations

  • [Del] P. Deligne.
    Representation l-adiques. Numdam
    Part of Asterisque 127

  • [Poo] B. Poonen.
    A p-adic approach to rational points on curves. arXiv
    A 12 pages gentle introduction to [LV]. Good for a first overview. See also the youtube lectures linked below

  • [Szp] L. Szpiro.
    La conjecture de Mordell (d'après G. Faltings). Numdam
    Overview of the Faltings proof

LINKS