Groupe de Travail

RATIONAL POINTS ON CURVES OVER FINITE FIELDS (2022-2023)

REFERENCE: Rational Points on Curves over Finite Fields by Jean-Pierre Serre.

Thursday 14h.

  • 29/9/2022, Samuele Anni, Organisation

  • 13/10/2022, Leonardo Colò, 2.1-2.3, Weil's bound. Notes available at the following link.

  • 20/10/2022, Zoé Yvon, 2.4-2.7, Jacobians

  • 3/11/2022, Pierre Tchamitchian, 3.1, Honda-Tate

  • 24/11/2022, Pierre Tchamitchian, 3.2-3.4, Honda-Tate

  • 24/11/2022, David Kohel, 3.5, Polarisations

  • 1/12/2022, Ali Issa, 3.8-3.11, Tensor products & hermitian forms

  • 8/12/2022, 10h, David Kohel, 3.6-3.7, Polarisations

  • à venir (mars), Samuele Anni, 4.1-4.4, Genus 3

  • à venir (mars), David Kohel, 5.1-5.8, Ihara 1

  • à venir (mars), David Kohel, 5.9-5.11, Ihara 2

  • à venir (mars), Yves Aubry, 6.1-6.6, Oesterlé

  • à venir (mars), Yves Aubry, 7.1-7.3, q=2

CORPS LOCAUX (2020-2021)

REFERENCE: Corps Locaux by Jean-Pierre Serre (CL) and Algebraïsche Zahlentheorie by Neukirch (AZ).

Goals: proof of the local Kronecker-Weber theorem and the statement of local class field theory

  • 22/10/2020, Samuele Anni, Organisation

  • 5/11/2020 - 10h30, Bastien Pacifico, Discrete Valuation Rings - (CL) I.1-I.3 and (AZ) II.3

  • 12/11/2020 - 10h, Zoé Yvon, Extensions - (CL) I.4-I.8; (AZ) II.7 and II.9

  • 18/11/2020 - 16h, Leonardo Colò, Completion and topology - (CL) II.1-II.3; (AZ) II.4 and II.8

  • 3/12/2020 - 14h, Bastien Pacifico, Structure of complete Discrete Valuation Rings - (CL) II.4-II.5

  • 17/12/2020 - 14h, David Kohel, Witt vectors - (CL) II.6

  • 7/1/2021 - 10h, Zoé Yvon, Discriminant and different - (CL) III.1-III.4; (AZ) II.4 and II.8

  • 14/1/2021 - 14h, Samuele Anni, Higher ramification groups I - (CL) IV.1-IV.4 and (AZ) II.10

  • 21/1/2021 - 10h, Samuele Anni, Higher ramification groups II - (CL) IV.1-IV.4 and (AZ) II.10

  • 28/1/2021 - 10h, David Kohel, Norm I- (CL) IV.2, V.1-V.3 and (AZ) II.10

  • 18/2/2021 - 16h, Samuele Anni, Norm II- (CL) IV.2, V.1-V.3 and (AZ) II.10

  • 25/2/2021 - 10h, Leonardo Colò, The Hasse-Arf theorem I (CL) V.4 - V.7 and (AZ) V.6

  • 11/3/2021 - 16h, Leonardo Colò, The Hasse-Arf theorem II (CL) V.4 - V.7 and (AZ) V.6

  • 18/3/2021 - 10h, Bastien Pacifico, Kummer theory (CL) X.3 and (AZ) IV.3

  • 25/3/2021 - 10h, Samuele Anni, Group cohomology and Galois cohomology

  • 1/4/2021 - 10h, David Kohel, Group cohomology and Galois cohomology

  • 15/4/2021 - 16h, Zoé Yvon, Local Kronecker-Weber (AZ) V.1

  • 27/5/2021 - 14h, Samuele Anni, Local class field theory (CL) XIII and (AZ) V.1

Algèbres de quaternions (2019-2020)

REFERENCE: Quaternion algebras by John Voight

  • 10/10/2019, Samuele Anni, Beginnings (Chapter 2), Exercises: 4, 7, 9 and 10.

  • 24/10/2019, Elena Berardini, Involutions(Chapter 3), Exercises: 1, 5, 10, 14 and theorem 3.5.1.

  • 7/11/2019, Bastien Pacifico, Quadratic Forms (Chapter 4), Exercises: 5, 8, 11 and 13.

  • 14/11/2019, Leonardo Colò, Ternary Forms (Chapter 5), Exercises: 5, 8, 14 and 18.

  • 21/11/2019, David Kohel, Characteristic 2 (Chapter 6), Exercises: 8, 11, 13, example 6.3.3 and 6.3.8.

  • 28/11/2019, Samuele Anni, Simple Algebras (Chapter 7), Exercises: 1, 4, 5, 8, 10, 24, 25 and 28.

  • 12/12/2019, Alejandro Giangreco, Simple Algebras & Involutions (Chapter 8), Exercises: 1,5 and 7.

  • 16/1/2020, Samuele Anni, Summary of Part I Algebra.

  • 23/1/2020, Alejandro Giangreco, Orders (Chapter 10), Exercises: 3, 5, Lemmas 10.2.9 and 10.3.2.

  • 30/1/2020, Alejandro Giangreco, Hurwitz order (Chapter 11), Exercises: 1 and 8.

  • 6/2/2020, Leonardo Colò, Local fields and ternary quadratic forms (Chapter 12), Exercises: 1, 6 and 12.

  • 13/2/2020, Bastien Pacifico, Quaternion algebras over local fields (Chapter 13), Exercises: 1, 4, Lemma 13.3.2 and Lemma 13.3.4.

  • 20/2/2020, Alejandro Giangreco, Quaternion algebras over Q (Chapter 14), Exercises: 3, 9 and 10.