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Serre's conjecture II

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Unsolved problem in mathematics
Must the Galois cohomology set of a simply connected, semisimple algebraic group over a perfect field of cohomological dimension at most 2 always equal zero?
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In mathematics, specifically number theory, Serre's conjecture II is the statement that if G is a simply connected, semisimple algebraic group over a perfect field F of cohomological dimension at most 2, then the Galois cohomology set H1(F, G) is zero.[1][2] It was proposed by Jean-Pierre Serre in 1962, as an higher-dimension equivalent of his conjecture I (which was later proven).[3]

A converse of the conjecture holds: if the field F is perfect and if the cohomology set H1(F, G) is zero for every semisimple, simply connected algebraic group G, then the p-cohomological dimension of F is at most 2 for every prime p.[4] The conjecture has been proven for all groups over all perfect fields; however, it remains open for anisotropic E6, E7 and E8 groups and trialitarian D4 group over imperfect fields.

Known cases

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The conjecture holds in the case where F is a local field (such as p-adic field), a global field with no real embeddings (such as Q(√−1)) or a totally imaginary number field.[5][6] This is a special case of the Kneser–Harder–Chernousov Hasse principle for algebraic groups over global fields. (Note that such fields do indeed have cohomological dimension at most 2.[2]) The conjecture holds when F is finitely generated over complex numbers and has transcendence degree at most 2. The conjecture also holds for l-special fields, complete valued fields and function fields.[7]

The conjecture is known to hold for certain groups G. For special linear groups, it is a consequence of the Merkurjev–Suslin theorem.[8] Building on this result, the conjecture holds if G is a classical group (type A, B, C or D with no triality); or a group of type F4 and G2 on a perfect field.[9] The conjecture also holds for classical groups over imperfect fields.[10] Over perfect fields, the conjecture also holds if G is an isotropic or quasi-split exceptional group (except for E8);[11][12] or a pseudo-reductive group.[13]

References

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  1. ↑ Serre, Jean-Pierre (1962). "Cohomologie galoisienne des groupes algébriques linéaires" [Galois cohomology of linear algebraic groups]. Colloque sur la théorie des groupes algébriques, Bruxelles: 53–68. OCLC 1708914.
  2. 1 2 Serre, Jean-Pierre (1994) [1964]. Cohomologie galoisienne. Lecture Notes in Mathematics. Vol. 5 (5th ed.). Springer Berlin, Heidelberg. doi:10.1007/BFb0108758. ISBN 978-3-540-58002-7.
  3. ↑ Izquierdo, Diego; Lucchini Arteche, Giancarlo (2025-11-01). "Transfer principles for Galois cohomology and Serre's conjecture II". Advances in Mathematics. 480 110532. arXiv:2308.00903. doi:10.1016/j.aim.2025.110532. ISSN 0001-8708.
  4. ↑ Serre, Jean-Pierre (1995). "Cohomologie galoisienne : progrès et problèmes". Astérisque. 227: 229–247. MR 1321649. Zbl 0837.12003.
  5. ↑ Kneser, Martin (1965-02-01). "Galois-Kohomologie halbeinfacher algebraischer Gruppen überp-adischen Körpern. I". Mathematische Zeitschrift (in German). 88 (1): 40–47. doi:10.1007/BF01112691.
  6. ↑ Kneser, Martin (1965-06-01). "Galois-Kohomologie halbeinfacher algebraischer Gruppen über p-adischen Körpern. II". Mathematische Zeitschrift (in German). 89 (3): 250–272. doi:10.1007/BF02116869.
  7. ↑ de Jong, A. J.; He, Xuhua; Starr, Jason Michael (2008). "Families of rationally simply connected varieties over surfaces and torsors for semisimple groups". arXiv:0809.5224 [math.AG].
  8. ↑ Merkurjev, A. S.; Suslin, A. A. (1983). "K-cohomology of Severi-Brauer varieties and the norm-residue homomorphism". Math. USSR Izvestiya. 21 (2): 307–340. Bibcode:1983IzMat..21..307M. doi:10.1070/im1983v021n02abeh001793.
  9. ↑ Bayer-Fluckiger, Eva; Parimala, Raman (1995). "Galois cohomology of the classical groups over fields of cohomological dimension ≤ 2". Inventiones Mathematicae. 122 (1): 195–229. Bibcode:1995InMat.122..195B. doi:10.1007/BF01231443. S2CID 124673233.
  10. ↑ Berhuy, Grégory; Frings, Christoph; Tignol, Jean-Pierre (2007-11-01). "Galois cohomology of the classical groups over imperfect fields". Journal of Pure and Applied Algebra. 211 (2): 307–341. doi:10.1016/j.jpaa.2007.01.001. S2CID 122036284.
  11. ↑ Gille, Philippe (2001). "Cohomologie galoisienne des groupes quasi-déployés sur des corps de dimension cohomologique ≤ 2" [Galois cohomology of quasi-split groups over fields of cohomological dimension ≤ 2]. Compositio Mathematica (in French). 125 (3): 283–325. doi:10.1023/A:1002473132282. S2CID 124765999.
  12. ↑ Gille, Philippe (2019). Groupes algébriques semi-simples en dimension cohomologique ≤2. Lecture Notes in Mathematics. Vol. 2238. doi:10.1007/978-3-030-17272-5. ISBN 978-3-030-17271-8.
  13. ↑ Nguyen, Mac Nam Trung (2026-03-09), Serre conjecture II for pseudo-reductive groups, arXiv, arXiv:2603.08061, doi:10.48550/arXiv.2603.08061, S2CID 286377008, retrieved 2026-08-18
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