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
Session 1: Background and motivation
- Main approximation properties:
- Weak approximation (WA), weak weak approximation (WWA), Hasse principle (HP)
- Brauer-Manin pairing, (BMWA) and (BMHP)
- Iskovskikh's nonexample to the Hasse principle: \(y^2+z^2=(3-x^2)(x^2-2)\)
- Skorobogatov's nonexample to (BMHP): \((x^2+1)y^2=(x^2+2)z^2=3(t^4-54t^2-117t-243)\)
- State Colliot-Thélène's conjecture on (BMWA) for rationally connected varieties, and discuss rational connectedness
- What should the fibration method be?
- State the fibration method and discuss rational connectedness of the generic fibre
- State known case 1: all fibres are split
- State known case 2: all fibres are Severi-Brauer varieties, under Schinzel's hypothesis
- State known case 3: there exists a rational section
- Arithmetico-geometric tools needed:
- Implicit functions theorem
- Any of the approximation properties listed above are birational invariants
- Lang-Weil bounds for split varieties
- Lang-Nishimura theorem (?)
- State Harpaz-Wittenberg (HW) conjecture and state the aim: HW conjecture \(\Longrightarrow\) fibration method
- Prove the fibration method when all fibres are split (see Proposition 3.15 of [W18])
- State and prove Hasse-Minkowski theorem (see [Chapter IV, \(\S3.2\) of [S73]])
- 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 \(\mathbf{P}^1\))
- Introduction on spectral sequences
- May follow Appendix \(A.7\) in Harari's book "Galois cohomology and class field theory"
- Then focus on Hochschild-Serre spectral sequence (Theorem \(1.44\) in ibidem)
- 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])
- State Harari's formal lemma (see Theorem \(13.4.3\) in [CTS21])
- State Schinzel's hypothesis à la Serre and relate it to Schinzel's hypothesis (H) (\(\S4\) of [CTSD94])
- Define corestrictions, cup-products and local invariants of cyclic algebras
- Prove fibration method for families of Severi-Brauer varieties, conditionally under Schinzel (prove (\(a\)) in Theorem \(1.1\) and the implication (\(a\))\(\Rightarrow\)(\(e\)))
- Fibration method when the generic fibre has a rational point [Théorème 3 of [H07]]
- Introduction to linear algebraic groups (examples, examples!)
- Proof of (BMWA) for connected linear algebraic groups (Théorème 5.3.1 in [H94])
- State Harpaz-Wittenberg (HW) conjecture (Conjecture 14.2.12 in [CTS21])
- When stating conjecture HW, start by the conclusion
- Then give a geometric interpretation of it, the same way we did for Schinzel's hypothesis, by drawing suitable models over \(\mathrm{Spec}(\mathscr{O}_{k,S})\)
- 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]
- Proof of "Conjecture HW \(\Rightarrow\) Fibration method" when there is no Brauer-Manin obstruction in the fibres (Theorem 14.2.18 in [CTS21])
References
- [CTS21] J.-L. Colliot-Thélène, A. N. Skorobogatov, The Brauer-Grothendieck group, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 71. Cham: Springer. xv, 453 p. (2021). PDF of a preliminary version
- [CTSSD98] J.-L. Colliot-Thélène, A. N. Skorobogatov, P. Swinnerton-Dyer, Rational points and zero cycles on fibred varieties: Schinzel's hypothesis and Salberger's device. J. Reine Angew. Math. 495, 1-28 (1998). PDF version
- [CTSD94] J.-L. Colliot-Thélène, P. Swinnerton-Dyer, Hasse principle and weak approximation for pencils of Severi-Brauer and similar varieties., J. Reine Angew. Math. 453, 49-112 (1994).
- [H94] D. Harari, Méthode des fibrations et obstruction de Manin, Duke Math. J. 75, No. 1, 221-260 (1994). PDF version
- [H07] D. Harari, Quelques propriétés d’approximation reliées à la cohomologie galoisienne d’un groupe algébrique fini, Bull. Soc. Math. Fr. 135, No. 4, 549-564 (2007). PDF version
- [P17] B. Poonen, Rational points on varieties, Graduate Studies in Mathematics 186. Providence, RI: American Mathematical Society (AMS). xv, 337 p. (2017). PDF version
- [S73] J.-P. Serre, A course in arithmetic, Graduate Texts in Mathematics. 7. New York-Heidelberg-Berlin: Springer-Verlag. viii, 115 p. DM 21.10 (1973). PDF version
- [W18] O. Wittenberg, Rational points and zero-cycles on rationally connected varieties over number fields, Algebraic geometry: Salt Lake City 2015, 597–635. Proc. Sympos. Pure Math., 97.2 American Mathematical Society, Providence, RI (2018). PDF version