Duality in abelian varieties and formal groups over local fields

N. M. Glazunov

Čebyševskij sbornik, Tome 19 (2018) no. 1, pp. 44-56.

Résumé

The article is dedicated to the memory of Oleg Nikolaevich Vvedenskii (1937–1981). O.N. Vvedenskii was a student of the academician I.R. Shafarevich. O.N. Vvedenskii's research and the results obtained are related to duality in elliptic curves and with the corresponding Galois cohomology over local fields, with Shafarevich-Tate pairing and with other pairings, with local and quasi-local of class fields theory of elliptic curves, with the theory of Abelian varieties of dimension greater than 1, with the theory of commutative formal groups over local fields. The paper presents both the results obtained by O.N. Vvedenskii, and new selected results, developing research in the directions of the fundamental groups of schemes, the principal homogeneous spaces (torsors), and duality. The first part of the article presented here is an introduction both to the results obtained by O.N. Vvedenskii in the direction of duality of Abelian varieties and formal groups, and in new selected results, developing research in the directions of the fundamental groups of schemes, the principal homogeneous spaces (torsors), and duality. The Introduction gives preliminary information and presents the content of the article. In the first section we give a brief survey of selected results on the theory of algebraic, quasialgebraic and proalgebraic groups and group schemes. Further, in Section 2 we present selected results on fundamental groups of algebraic varieties, on fundamental groups of schemes, and in Section 3 — selected results on principal homogeneous spaces (torsors), developing research by O.N. Vvedenskii and other authors. In Section 4 we give information on duality, and in Section 5 the paper presents the results by O.N. Vvedenskii on the arithmetic theory of formal groups and their development. The results of this section, represented over local and quasi-local fields K, over their rings of integers, and over their residue fields k, are connected (1) with the formal structure of Abelian varieties, (2) with commutative formal groups, (3) with corresponding homomorphisms. In the article, algebraic varieties, Abelian schemes, and commutative formal group schemes are defined, as a rule, over local and quasi-local fields, over their rings of integers, and over their residue fields. But these objects are also briey considered over global fields, since O.N. was interested in the subject of algebraic varieties over global fields and he carried out corresponding studies. It is assumed that the characteristic of the residue fields is more than 3, unless otherwise specified. I am grateful to V.N. Chubarikov for offering to publish the article in Chebyshevskii Sbornik. Special thanks to N.M. Dobrovolsky for help and support in the process of preparing the article for publication.

Détail

Keywords: duality; Abelian variety; local field; Picard group; formal group; group scheme; fundamental group; torsor; global field; proalgebraic group; group of universal norms.

Bibliographie

[1] Shafarevich I. R., Sochineniya, ch. 2, v. 3, Fizmatlit, M., 1996, 637 pp.

[2] Shafarevich I. R., Osnovy algebraicheskoj geometrii, V 2 t., Nauka, M., 1988 | MR

[3] Vvedenskij O. N., “Dvojstvennost' v ehllipticheskih krivyh nad lokal'nym polem. II”, Izv. AN SSSR. Ser.: Matematika, 30:4 (1966), 891—922

[4] Vvedenskij O. N., “O lokal'nyh “polyah klassov” ehllipticheskih krivyh”, Izv. AN SSSR. Ser.: Matematika, 37:1 (1973), 20—88 | MR

[5] Vvedenskij O. N., “O “universal'nyh normah” formal'nyh grupp, opredelennyh nad kol'com lokal'nogo polya”, Izv. AN SSSR. Ser.: Matematika, 37:4 (1973), 737—751 | MR

[6] Vvedenskij O. N., “O kvazi-lokal'nyh “polyah klassov” ehllipticheskih krivyh. I”, Izv. AN SSSR. Ser.: Matematika, 40:5 (1976), 969—992 | MR

[7] Vvedenskij O. N., “O sparivaniyah v ehllipticheskih krivyh nad global'nymi polyami”, Izv. AN SSSR. Ser.: Matematika, 42:2 (1978), 237—260 | MR

[8] Vvedenskij O. N., “Ehffekt Artina v abelevyh mnogoobraziyah. II”, Izv. AN SSSR. Ser.: Matematika, 45:1 (1981), 23—46 | MR

[9] Miln Dzh., Ehtal'nye kogomologii, Mir, M., 1983, 393 pp. | MR

[10] Serre J.-P., “Groupes proalgebriques”, Publications mathematiques IHES, 7, 1960, 65 pp. | MR | Zbl

[11] Tate J., “Duality theorems in Galois cohomology over number fields”, Proceedings of Int. Congress of Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1962, 288—295 | MR

[12] Shatz S. S., “Cohomology of artinian group schemes over local fields”, Ann. of Math., 79:2 (1964), 411—449 | DOI | MR | Zbl

[13] Grothendieck A., Artin M., Verdier J. L., Théorie des Topos et cohomologie étale des schémas (SGA4), Lecture Notes in Math., 269, 270, 305, Springer-Verlag, Berlin–N.Y., 1972 | DOI | MR

[14] Lichtenbaum S., “The Weil-etale topology for number rings”, Ann. of Math., 170:2 (2009), 657—683 | DOI | MR | Zbl

[15] Bosch S., Liu Q., “Rational points of the group of components of a Néron model”, Manuscripta math., 98 (1999), 275—293 | DOI | MR | Zbl

[16] Morin B., “The Weil-étale fundamental group of a number field. II”, Sel. Math., New Ser., 17:1 (2011), 67—137 | DOI | MR | Zbl

[17] Saavedra R., Catégories Tannakiennes, Lecture Notes in Math., 265, Springer-Verlag, Berlin–N.Y., 1972, 418 pp. | MR

[18] Nori M., “On the representations of the fundamental group”, Compos. Math., 33:1 (1976), 29—41 | MR | Zbl

[19] Broshi M., “$G$-torsors over a Dedekind scheme”, J. Pure Appl. Algebra, 217:1 (2013), 11—19 | DOI | MR | Zbl

[20] Conrad B., “Reductive group schemes”, Autour des schémas en groupes, École d'Été “Schémas en groupes”, I, Société Mathématique de France (SMF), Paris, 2014, 42—43 ; 93—444 | MR | Zbl

[21] Biswas I., Dos Santos João Pedro P., “Abelianization of the $F$-divided fundamental group scheme”, Proc. Indian Acad. Sci., Math. Sci., 127:2 (2017), 281—287 | DOI | MR | Zbl

[22] Tziolas N., “Quotients of schemes by $\alpha_p$ or $\mu_p$ actions in characteristic $p > 0$”, Manuscr. Math., 152:1–2 (2017), 247—279 | DOI | MR | Zbl