The period-index problem in WC-groups IV: a local transition theorem

Clark, Pete L.

Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 3, pp. 583-606.

Soit $K$ un corps de valuation discrète complet avec corps résiduel parfait $k$. En supposant des bornes supérieures pour la relation entre l’indice et la période pour des groupes de Weil-Châtelet sur $k$, nous déduisons des bornes supérieures correspondantes pour la relation entre l’indice et la période pour des groupes de Weil-Châtelet sur $K$. À une constante dépendant seulement de la dimension d’un torseur près, nous retrouvons des théorèmes de Lichtenbaum et Milne dans un contexte “sans dualité”. Nos techniques utilisent les modèles LLR des torseurs sous des variétés abeliennes avec bonne réduction et une généralisation de l’obstruction période-indice à la cohomologie plate. Dans un appendice, nous considérons des sujets apparentés relevant de l’arithmétique du corps.

Let $K$ be a complete discretely valued field with perfect residue field $k$. Assuming upper bounds on the relation between period and index for WC-groups over $k$, we deduce corresponding upper bounds on the relation between period and index for WC-groups over $K$. Up to a constant depending only on the dimension of the torsor, we recover theorems of Lichtenbaum and Milne in a “duality free” context. Our techniques include the use of LLR models of torsors under abelian varieties with good reduction and a generalization of the period-index obstruction map to flat cohomology. In an appendix, we consider some related issues of a field-arithmetic nature.

[BoXa] S. Bosch and X. Xarles, Component groups of Néron models via rigid uniformization. Math. Ann. 306 (1996), 459–486. | MR | Zbl

[BLR] S. Bosch, W. Lütkebohmert and M. Raynaud, Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete 21, Springer-Verlag, 1990. | MR | Zbl

[CG] J.-P. Serre, Cohomologie Galoisienne. Lecture Notes in Mathematics 5, 5th revised edition, Springer-Verlag, 1994. | MR | Zbl

[Ch] C. Chevalley, Démonstration d’une hypothèse de M. Artin. Abh. Math. Sem. Univ. Hamburg 11 (1936), 73–75. | Zbl

[CL] J.-P. Serre, Corps Locaux. Hermann, Paris, 1962. | MR

[ClSh] P.L. Clark and S. Sharif, Period, index and potential Sha. Algebra and Number Theory 4 (2010), No. 2, 151–174. | MR | Zbl

[ClXa] P.L. Clark and X. Xarles, Local bounds for torsion points on abelian varieties. Canad. J. Math. 60 (2008), no. 3, 532–555. | MR

[deJ] A.J. de Jong, The period-index problem for the Brauer group of an algebraic surface. Duke Math. J. 123 (2004), no. 1, 71–94. | MR | Zbl

[FA] M.D. Fried and M. Jarden, Field arithmetic. Third edition. Revised by Jarden. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas, 11. Springer-Verlag, Berlin, 2008. | MR

[Ge] L. Gerritzen, Periode und Index eines prinzipal-homogenen Raumes über gewissen abelschen Varietäten. Manuscripta Math. 8 (1973), 131–142. | MR | Zbl

[GeJa] W.-D. Geyer and M. Jarden, Non-PAC fields whose Henselian closures are separably closed. Math. Research Letters 8 (2001), 509–519. | MR | Zbl

[GLL] O. Gabber, Q. Liu and D. Lorenzini, Moving Lemmas and the Index of Algebraic Varieties. 2009 preprint.

[Gr67] M.J. Greenberg, Rational points in Henselian discrete valuation rings. Publ. Math. IHES 31 (1967), 59–64. | Numdam | MR

[FMV] M.J. Greenberg, Lectures on forms in many variables. W. A. Benjamin, Inc., New York-Amsterdam 1969. | MR | Zbl

[GiSz] P. Gille and T. Szamuely, Central Simple Algebras and Galois Cohomology. Cambridge Studies in Advanced Mathematics 101, Cambridge University Press, 2006. | MR | Zbl

[Ha] T. Harase, On the index-period problem for algebraic curves and abelian varieties. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20 (1973), 13–20. | MR | Zbl

[HKS] D. Harbater, J. Hartmann and D. Krashen, Applications of patching to quadratic forms and central simple algebras. Invent. Math. 178 (2009), no. 2, 231–263. | MR

[La52] S. Lang, On Quasi Algebraic Closure. Annals of Math. 55 (1952), 373–390. | MR | Zbl

[La56] S. Lang, Algebraic groups over finite fields. Amer. J. Math. 78 (1956), 555–563. | MR | Zbl

[LaTa] S. Lang and J. Tate, Principal homogeneous spaces over abelian varieties. Amer. J. Math. 80 (1958), 659–684. | MR | Zbl

[LiuGa] Q. Liu, following O. Gabber, Separable index of smooth algebraic varieties, 2009 preprint.

[Li68] S. Lichtenbaum, The period-index problem for elliptic curves. Amer. J. Math. 90 (1968), 1209–1223. | MR | Zbl

[Li70] S. Lichtenbaum, Duality theorems for curves over $p$-adic fields. Invent. Math. 7 (1969), 120–136. | MR | Zbl

[Lie] M. Lieblich, Twisted sheaves and the period-index problem. Compos. Math. 144 (2008), no. 1, 1–31. | MR | Zbl

[LLR] D. Lorenzini, Q. Liu and M. Raynaud, Néron models, Lie algebras, and reduction of curves of genus one. Invent. math. 157 (2004), 455–518. | MR | Zbl

[MeSu] A. S. Merkur’ev and A.A. Suslin, $K$-cohomology of Severi-Brauer varieties and the norm residue homomorphism. Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 1011–1046, 1135–1136. | MR | Zbl

[Mil] J. Milne, Étale cohomology. Princeton Mathematical Series, 33. Princeton University Press, Princeton, N.J., 1980. | MR | Zbl

[Mum] D. Mumford, Abelian varieties. Tata Institute of Fundamental Research Studies in Mathematics, No. 5. Bombay; Oxford University Press, London 1970. | MR | Zbl

[O’N] C.H. O’Neil, The period-index obstruction for elliptic curves. J. Number Theory 95 (2002), 329–339. | MR | Zbl

[PoSt] B. Poonen and M. Stoll, The Cassels-Tate pairing on polarized abelian varieties. Ann. of. Math. (2) 150 (1999), 1109–1149. | MR | Zbl

[Sa] D.J. Saltman, Division algebras over $p$-adic curves. (English summary) J. Ramanujan Math. Soc. 12 (1997), no. 1, 25–47. | MR | Zbl

[Sc] F. K. Schmidt, Die Theorie der Klassenkörper über einem Körper algebraischer Funktionen in einer Unbestimmten und mit endlichem Koeffizientenbereich. Sitz.-Ber. phys. med. Soz. 62 (1931), 267–284. | Zbl

[SH] I.R. Shafarevich, Principal homogeneous spaces defined over a function field. (Russian) Trudy Mat. Inst. Steklov. 64 (1961), 316–346. | MR | Zbl

[St] R. Steinberg, Cohomologie galoisienne des groupes algébriques linéaires. Colloques de Bruxelles, 1962, 53–67. | MR

[Te] O. Teichmüller, p-Algebren. Deutsche Math. 1 (1936), 362–368. | Zbl

[Ts] C. Tsen, Divisionsalgebren über Funktionenkörper. Nachr. Ges. Wiss. Göttingen (1933), 335. | Zbl

[WCI] P.L. Clark, Period-index problems in WC-groups I: elliptic curves. J. Number Theory 114 (2005), 193–208. | MR | Zbl

[WCII] P.L. Clark, Period-index problems in WC-groups II: abelian varieties. Submitted.

[WCIII] P.L. Clark, Period-index problems in WC-groups III: biconic curves. Preprint.

[WCV] P.L. Clark, Period-index problems in WC-groups V: Cartier symbols. In preparation.

[Za] Ju. G. Zarhin, Noncommutative cohomology and Mumford groups. (Russian) Mat. Zametki 15 (1974), 415–419. | MR | Zbl