Elyes BOUGHATTAS

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Reading group on the Fibration Method


Aim This reading group is dedicated to the fibration method over number fields. We shall go from the historical case of Hasse-Minkowski's theorem to the more elaborated conjecture of Harpaz and Wittenberg. Some notes are expected to arise afterwards.


Schedule

ID Date Topic Speaker Room
1 30/01/2024 Background and motivation Happy 8W 1.32
2 06/02/2024 Embryo case of the fibration method and Hasse-Minkowski theorem Harry 8W 1.32
3 13/02/2024 Spectral sequences, Brauer group and Harari's formal lemma Austin and Julie 6E 2.1
4 20/02/2024 The fibration method modulo Schinzel Jiazhi EB 0.15
4.5 27/02/2024 The fibration method modulo Schinzel Elyes and Jiazhi 8W 1.34
5 05/03/2024 Fibration method with a rational section Sam 8W 1.32
6 12/03/2024 Approximation properties on connected linear algebraic groups Abdulmuhsin 8W 1.34
7 19/03/2024 HW conjecture and the fibration method (part 1) Tudor EB 0.15
8 26/03/2024 HW conjecture and the fibration method (part 2) Dan 6W 1.1

Suggested plan

This plan may be subject to alteration. Some extra sessions may appear if any tool is needed to be discussed more thoroughly.

Session 1: Background and motivation
  1. Main approximation properties:
    1. Weak approximation (WA), weak weak approximation (WWA), Hasse principle (HP)
    2. Brauer-Manin pairing, (BMWA) and (BMHP)
    3. Iskovskikh's nonexample to the Hasse principle: \(y^2+z^2=(3-x^2)(x^2-2)\)
    4. Skorobogatov's nonexample to (BMHP): \((x^2+1)y^2=(x^2+2)z^2=3(t^4-54t^2-117t-243)\)
    5. State Colliot-Thélène's conjecture on (BMWA) for rationally connected varieties, and discuss rational connectedness
  2. What should the fibration method be?
    1. State the fibration method and discuss rational connectedness of the generic fibre
    2. State known case 1: all fibres are split
    3. State known case 2: all fibres are Severi-Brauer varieties, under Schinzel's hypothesis
    4. State known case 3: there exists a rational section
  3. Arithmetico-geometric tools needed:
    1. Implicit functions theorem
    2. Any of the approximation properties listed above are birational invariants
    3. Lang-Weil bounds for split varieties
    4. Lang-Nishimura theorem (?)
  4. State Harpaz-Wittenberg (HW) conjecture and state the aim: HW conjecture \(\Longrightarrow\) fibration method
Session 2: Embryo case of the fibration method and Hasse-Minkowski theorem

  1. Prove the fibration method when all fibres are split (see Proposition 3.15 of [W18])
  2. State and prove Hasse-Minkowski theorem (see [Chapter IV, \(\S3.2\) of [S73]])
    1. May assume the case \(n=3\), which is due to Legendre.
    2. Prove that \(n=3 \Rightarrow n=4\) by using and proving Proposition \(3.17\) of [W18]. Stress the use of Dirichlet theorem.
    3. Prove the \(n\geq5\) case by following \(iv)\) of the proof of Theorem \(8\) in \(\S3.2\) of [S73]
  3. Comment on the proof by saying that it uses a fibration method, and suggest that a generalisation might be possible by using Schinzel's hypothesis (H) (families of Severi-Brauer varieties over \(\mathbf{P}^1\))
Session 3: Spectral sequences, Brauer group and Harari's formal lemma

  1. Introduction on spectral sequences
    1. May follow Appendix \(A.7\) in Harari's book "Galois cohomology and class field theory"
    2. Then focus on Hochschild-Serre spectral sequence (Theorem \(1.44\) in ibidem)
  2. Prove that if \(X/k\) is rationally connected, then \(\mathrm{Br}(X)/\mathrm{Br}(k)\) is finite, using Hochschild-Serre spectral sequence (follow \((ii)\) of Remarks \(2.4\) in [W18])
  3. State Harari's formal lemma (see Theorem \(13.4.3\) in [CTS21])
Session 4 and 4.5: The fibration method modulo Schinzel

  1. State Schinzel's hypothesis à la Serre and relate it to Schinzel's hypothesis (H) (\(\S4\) of [CTSD94])
  2. Define corestrictions, cup-products and local invariants of cyclic algebras
  3. Prove fibration method for families of Severi-Brauer varieties, conditionally under Schinzel (prove (\(a\)) in Theorem \(1.1\) and the implication (\(a\))\(\Rightarrow\)(\(e\)))
Session 5: Fibration method with a rational section

  1. Fibration method when the generic fibre has a rational point [Théorème 3 of [H07]]
Session 6: Approximation properties on connected linear algebraic groups

  1. Introduction to linear algebraic groups (examples, examples!)
  2. Proof of (BMWA) for connected linear algebraic groups (Théorème 5.3.1 in [H94])
Session 7: HW conjecture and the fibration method (part 1)
  1. State Harpaz-Wittenberg (HW) conjecture (Conjecture 14.2.12 in [CTS21])
    1. When stating conjecture HW, start by the conclusion
    2. Then give a geometric interpretation of it, the same way we did for Schinzel's hypothesis, by drawing suitable models over \(\mathrm{Spec}(\mathscr{O}_{k,S})\)
  2. State the Fibration Method that will be proved modulo HW in both this session and the next one, that is Theorem 14.2.14 in [CTS21]
  3. Proof of "Conjecture HW \(\Rightarrow\) Fibration method" when there is no Brauer-Manin obstruction in the fibres (Theorem 14.2.18 in [CTS21])
Session 8: HW conjecture and the fibration method (part 2)
  1. First prove Theorem 14.2.21 in [CTS21]
  2. Then combine Theorem 14.2.18 — proved in previous session — and Theorem 14.2.21 in [CTS21] to give a sketch of proof of "Conjecture HW \(\Rightarrow\) Fibration method" (follow p.365 of [CTS21])

References