Igor Rostislavovich Shafarevich and His Mathematical Heritage

S. O. Gorchinskiy ; Vik. S. Kulikov ; A. N. Parshin ; V. L. Popov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebra, number theory, and algebraic geometry, Tome 307 (2019), pp. 9-31.

Résumé

A brief biography of Igor Rostislavovich Shafarevich and an overview of all his main scientific publications are presented.

Bibliographie

[1] V. A. Abrashkin, “Group schemes of period $p$ over the ring of Witt vectors”, Sov. Math., Dokl., 32 (1985), 310–315 | MR | Zbl

[2] V. A. Abrashkin, “Honda systems of group schemes of period $p$”, Math. USSR, Izv., 30:3 (1988), 419–453 | DOI | MR | Zbl | Zbl

[3] V. A. Abrashkin, “Galois moduli of period $p$ group schemes over a ring of Witt vectors”, Math. USSR, Izv., 31:1 (1988), 1–46 | DOI | MR | Zbl

[4] S. Ju. Arakelov, “Families of algebraic curves with fixed degeneracies”, Math. USSR, Izv., 5:6 (1971), 1277–1302 | DOI | MR | Zbl

[5] S. Ju. Arakelov, “Intersection theory of divisors on an arithmetic surface”, Math. USSR, Izv., 8:6 (1974), 1167–1180 | DOI | MR

[6] Arakelov S.J., “Theory of intersections on an arithmetic surface”, Proc. Int. Congr. Math. (Vancouver, 1974), v. 1, Can. Math. Congr., Montreal, 1975, 405–408 | MR

[7] Artin M., “Supersingular K3 surfaces”, Ann. sci. Éc. norm. supér. Sér. 4, 7:4 (1974), 543–567 | MR | Zbl

[8] I. K. Babenko, “Problems of growth and rationality in algebra and topology”, Russ. Math. Surv., 41:2 (1986), 117–175 | DOI | MR | Zbl

[9] Baker A., “Linear forms in the logarithms of algebraic numbers”, Mathematika, 13:2 (1966), 204–216 | DOI | MR | Zbl

[10] Bedulev E., Viehweg E., “On the Shafarevich conjecture for surfaces of general type over function fields”, Invent. math., 139:3 (2000), 603–615 | DOI | MR | Zbl

[11] Bégueri L., Dualité sur un corps local à corps résiduel algébriquement clos, Mém. Soc. math. France, 4, Gauthier-Villars, Paris, 1980 | MR

[12] G. V. Belyĭ, “On Galois extensions of a maximal cyclotomic field”, Math. USSR, Izv., 14:2 (1980), 247–256 | DOI | MR | Zbl | Zbl

[13] Bertapelle A., “Local flat duality of abelian varieties”, Manuscr. math., 111:2 (2003), 141–161 | DOI | MR | Zbl

[14] Bester M., “Local flat duality of abelian varieties”, Math. Ann., 235:2 (1978), 149–174 | DOI | MR | Zbl

[15] Birch B.J., “Diophantine analysis and modular functions”, Algebraic geometry: Papers presented at the Bombay Colloquium (1968), Oxford Univ. Press, London, 1969, 35–42 | MR

[16] Block R.E., Wilson R.L., “Classification of the restricted simple Lie algebras”, J. Algebra, 114:1 (1988), 115–259 | DOI | MR | Zbl

[17] F. A. Bogomolov, “Classification of surfaces of class $\mathrm {VII}_0$ with $b_2=0$”, Math. USSR, Izv., 10:2 (1976), 255–269 | DOI | MR

[18] F. A. Bogomolov, “Holomorphic tensors and vector bundles on projective varieties”, Math. USSR, Izv., 13:3 (1979), 499–555 | DOI | MR | Zbl

[19] Z. I. Borevich and I. R. Shafarevich, Number Theory, Nauka, Moscow, 1964 | MR | Zbl | Zbl

[20] Academic, New York, 1966 | MR | Zbl | Zbl

[21] Brauer R., “Über die Konstruktion der Schiefkörper, die von endlichem Rang in bezug auf ein gegebenes Zentrum sind”, J. reine angew. Math., 168 (1932), 44–64 | MR

[22] Brückner H., Explizites Reziprozitätsgesetz und Anwendungen, Heft 2, Vorles. Fachbereich Math. Univ. Essen, Univ. Essen, Essen, 1979 | MR

[23] Burns D., \textup {Jr.}, Rapoport M., “On the Torelli problem for kählerian K-3 surfaces”, Ann. sci. Éc. norm. supér. Sér. 4, 8:2 (1975), 235–273 | MR | Zbl

[24] Châtelet F., “Méthode galoisienne et courbes de genre un”, Ann. Univ. Lyon. Sect. A, 9 (1946), 40–49 | MR

[25] V. I. Danilov, “Nonsimplicity of the group of unimodular automorphisms of the affine plane”, Math. Notes, 15:2 (1974), 165–167 | DOI | MR | Zbl | Zbl

[26] Deligne P., “La conjecture de Weil pour les surfaces K3”, Invent. math., 15 (1972), 206–226 | DOI | MR | Zbl

[27] B. N. Delone and D. K. Faddeev, “Investigations in the geometry of the Galois theory”, Mat. Sb., 15:2 (1944), 243–284

[28] S. P. Demushkin, A. I. Kostrikin, S. P. Novikov, A. N. Parshin, L. S. Pontryagin, A. N. Tyurin, and D. K. Faddeev, “Igor' Rostislavovich Shafarevich (on his sixtieth birthday)”, Russ. Math. Surv., 39:1 (1984), 189–200 | DOI | MR | Zbl | Zbl

[29] S. P. Demuškin and I. R. Šafarevič, “The imbedding problem for local fields”, Am. Math. Soc. Transl., Ser. 2,, 27 (1963), 267–288 | MR | MR | Zbl | Zbl

[30] S. P. Demuškin and I. R. Šafarevič, “The second obstruction for the imbedding problem of algebraic number fields”, Am. Math. Soc. Transl., Ser. 2, 58 (1966), 245–260 | MR | MR

[31] Deuring M., “Imaginäre quadratische Zahlkörper mit der Klassenzahl Eins”, Invent. math., 5:3 (1968), 169–179 | DOI | MR | Zbl

[32] I. V. Dolgačev, “On Severi's conjecture concerning simply connected algebraic surfaces”, Sov. Math., Dokl., 7 (1966), 1169–1172 | MR | Zbl

[33] Dolgachev I., Igor Rostislavovich Shafarevich: In Memoriam, E-print, 2017, arXiv: 1801.00311 [math.HO] | MR

[34] Eyssidieux P., “Sur la convexité holomorphe des revêtements linéaires réductifs d'une variété projective algébrique complexe”, Invent. math., 156:3 (2004), 503–564 | DOI | MR | Zbl

[35] Eyssidieux P., Katzarkov L., Pantev T., Ramachandran M., “Linear Shafarevich conjecture”, Ann. Math. Ser. 2, 176:3 (2012), 1545–1581 | DOI | MR | Zbl

[36] Faltings G., “Endlichkeitssätze für abelsche Varietäten über Zahlkörpern”, Invent. math., 73:3 (1983), 349–366 | DOI | MR | Zbl

[37] Faltings G., Lectures on the arithmetic Riemann–Roch theorem: Notes taken by Shouwu Zhang, Ann. Math. Stud., 127, Princeton Univ. Press, Princeton, NJ, 1992 | MR | Zbl

[38] Fontaine J.-M., “Il n'y pas de variété abélienne sur $\mathbb Z$”, Invent. math., 81:3 (1985), 515–538 | DOI | MR | Zbl

[39] Furter J.-P., Kraft H., On the geometry of the automorphism groups of affine varieties, E-print, 2018, arXiv: 1809.04175 [math.AG] | MR | Zbl

[40] E. Golod, “On the cohomology ring of a finite $p$-group”, Dokl. Akad. Nauk SSSR, 125:4 (1959), 703–706 | MR | Zbl

[41] E. S. Golod, “On nil-algebras and finitely approximable $p$-groups”, Am. Math. Soc. Transl., Ser. 2, 48 (1965), 103–106 | MR | MR | Zbl

[42] E. S. Golod and I. R. Šafarevič, “On class field towers”, Am. Math. Soc. Transl., Ser. 2, 48 (1965), 91–102 | MR | Zbl | Zbl

[43] Grauert H., “Mordells Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenkörper”, Publ. math. Inst. hautes étud. sci., 25 (1965), 131–149 | DOI | MR

[44] Grothendieck A. et al., Séminaire de géometrie algébrique du Bois-Marie 1965–66 (SGA 5): Cohomologie $l$-adique et fonctions $L$, Lect. Notes Math., 589, Springer, Berlin, 1977 | MR

[45] Hasse H., “Existenz und Mannigfaltigkeit abelscher Algebren mit vorgegebener Galoisgruppe über einem Teilkörper des Grundkörpers. I”, Math. Nachr., 1 (1948), 40–61 | DOI | MR | Zbl

[46] Heegner K., “Diophantische Analysis und Modulfunktionen”, Math. Z., 56:3 (1952), 227–253 | DOI | MR | Zbl

[47] V. A. Iskovskih, “Fano 3-folds. I”, Math. USSR, Izv., 11:3 (1977), 485–527 | DOI | MR | Zbl

[48] V. A. Iskovskih, “Fano 3-folds. II”, Math. USSR, Izv., 12:3 (1978), 469–506 | DOI | MR | Zbl

[49] V. A. Iskovskih and Ju. I. Manin, “Three-dimensional quartics and counterexamples to the Lüroth problem”, Math. USSR, Sb., 15:1 (1971), 141–166 | DOI | MR | Zbl

[50] V. A. Iskovskikh and I. R. Shafarevich, “Algebraic surfaces”, Algebraic Geometry–2, Itogi Nauki Tekh., Sovrem. Probl. Mat., Fundam. Napr., 35, VINITI, Moscow, 1989, 131–263 | MR | MR

[51] Algebraic Geometry II, Encycl. Math. Sci., 35, Spinger, Berlin, 1996, 127–254 | MR | MR

[52] Kaplansky I., “Topological rings”, Bull. Amer. Math. Soc., 54:9 (1948), 809–826 | DOI | MR | Zbl

[53] Katzarkov L., “On the Shafarevich maps”, Algebraic geometry, Proc. Summer Res. Inst. (Santa Cruz, 1995), Proc. Symp. Pure Math., 62, Pt. 2, Amer. Math. Soc., Providence, RI, 1997, 173–216 | DOI | MR | Zbl

[54] Koch H., Galoissche Theorie der $p$-Erweiterungen, Mit einem Geleitwort von I.R. Šafarevič, Springer, Berlin, 1970 | MR | Zbl

[55] Kollár J., “Shafarevich maps and plurigenera of algebraic varieties”, Invent. math., 113:1 (1993), 177–215 | DOI | MR | Zbl

[56] Kollár J., Shafarevich maps and automorphic forms, Princeton Univ. Press, Princeton, NJ, 1995 | MR | Zbl

[57] Kolyvagin V.A., “Euler systems”, The Grothendieck Festschrift: A collection of articles written in honor of the 60th birthday of A. Grothendieck, v. 2, Prog. Math., 87, Birkhäuser, Boston, MA, 1990, 435–483 | MR | Zbl

[58] A. I. Kostrikin, “The Burnside problem”, Am. Math. Soc. Transl., Ser. 2, 36 (1964), 63–99 | MR | Zbl | Zbl

[59] Kostrikin A.I., “Variations modulaires sur un thème de Cartan”, Actes Congr. Int. Math. (Nice, 1970), v. 1, Gauthier-Villars, Paris, 1971, 285–292 | MR

[60] A. I. Kostrikin, Around Burnside, Nauka, Moscow, 1986 | MR | MR | Zbl | Zbl

[61] Springer, Berlin, 1990 | MR | MR | Zbl | Zbl

[62] A. I. Kostrikin, V. A. Iskovskikh, and A. N. Parshin, “Introduction”, Proc. Steklov Inst. Math., 208 (1995), 1–4 | MR

[63] A. I. Kostrikin and I. R. Shafarevich, “The homology groups of nilpotent algebras”, Dokl. Akad. Nauk SSSR, 115:6 (1957), 1066–1069 | MR | Zbl

[64] A. I. Kostrikin and I. R. Shafarevich, “Cartan pseudogroups and Lie $p$-algebras”, Sov. Math., Dokl., 7 (1966), 715–718 | MR | Zbl

[65] A. I. Kostrikin and I. R. Šafarevič, “Graded Lie algebras of finite characteristic”, Math. USSR, Izv., 3:2 (1969), 237–304 | DOI | MR

[66] Vik. S. Kulikov, “Degenerations of K3 surfaces and Enriques surfaces”, Math. USSR, Izv., 11:5 (1977), 957–989 | DOI | MR | Zbl | Zbl

[67] Vik. S. Kulikov, “Surjectivity of the period mapping for K3 surfaces”, Usp. Mat. Nauk, 32:4 (1977), 257–258 | MR | Zbl

[68] Vik. S. Kulikov and G. B. Shabat, “Igor' Rostislavovich Shafarevich: Great Mathematician and Teacher”, Mat. Prosveshchenie, Ser. 3,, 22 (2018), 37–63

[69] L. V. Kuz'min, “The Tate module for algebraic number fields”, Math. USSR, Izv., 6:2 (1972), 263–321 | DOI | Zbl

[70] Lackenby M., “Covering spaces of 3-orbifolds”, Duke Math. J., 136:1 (2007), 181–203 | DOI | MR | Zbl

[71] Lackenby M., “New lower bounds on subgroup growth and homology growth”, Proc. London Math. Soc. Ser. 3, 98:2 (2009), 271–297 | DOI | MR | Zbl

[72] Lang S., Tate J., “Principal homogeneous spaces over abelian varieties”, Amer. J. Math., 80:3 (1958), 659–684 | DOI | MR | Zbl

[73] A. I. Lapin, “On the theory of Shafarevich's symbol”, Izv. Akad. Nauk SSSR, Ser. Mat., 18:2 (1954), 145–158 | MR | MR | Zbl

[74] A. I. Lapin, “The general duality law and a new foundation of the theory of class fields”, Izv. Akad. Nauk SSSR, Ser. Mat., 18:4 (1954), 335–378 | MR | Zbl

[75] Looijenga E., Peters C., “Torelli theorems for Kähler K3 surfaces”, Compos. math., 42:2 (1980), 145–186 | MR | Zbl

[76] Lubotzky A., “Group presentation, $p$-adic analytic groups and lattices in $\mathrm {SL}_2(\mathbf C)$”, Ann. Math. Ser. 2, 118:1 (1983), 115–130 | DOI | MR | Zbl

[77] Ju. I. Manin, “Rational points of algebraic curves over function fields”, Am. Math. Soc. Transl., Ser. 2, 50 (1966), 189–234 | Zbl

[78] Yu. I. Manin, Cubic Forms: Algebra, Geometry, Arithmetic, Nauka, Moscow, 1972 | MR | MR | Zbl | Zbl

[79] Cubic Forms: Algebra, Geometry, Arithmetic, North-Holland Math. Libr., 4, North-Holland, Amsterdam, 1986 | MR | MR | Zbl | Zbl

[80] Ju. I. Manin, “Periods of parabolic forms and $p$-adic Hecke series”, Math. USSR, Sb., 21:3 (1973), 371–393 | DOI | MR | Zbl

[81] Yu. I. Manin, “Non-Archimedean integration and Jacquet–Langlands $p$-adic $L$-functions”, Russ. Math. Surv., 31:1 (1976), 5–57 | DOI | MR | Zbl | Zbl

[82] Miyaoka Y., “On the Chern numbers of surfaces of general type”, Invent. math., 42 (1977), 225–237 | DOI | MR | Zbl

[83] B. G. Moĭšezon, “Remarks on projective imbeddings of algebraic varieties”, Sov. Math., Dokl., 3 (1962), 1146–1148 | MR | Zbl

[84] B. G. Moĭšezon, “A criterion for projectivity of complete algebraic abstract varieties”, Am. Math. Soc. Transl., Ser. 2, 63 (1967), 1–50 | Zbl

[85] V. V. Nikulin, “Finite automorphism groups of Kähler $K3$ surfaces”, Trans. Moscow Math. Soc., 2 (1980), 71–135 | MR | Zbl | Zbl

[86] V. V. Nikulin, “Factor groups of groups of automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections. Algebrogeometric applications”, J. Sov. Math., 22:4 (1983), 1401–1475 | DOI | MR | Zbl

[87] Nikulin V.V., “Discrete reflection groups in Lobachevsky spaces and algebraic surfaces”, Proc. Int. Congr. Math. (Berkeley, CA, 1986), v. 1, Amer. Math. Soc., Providence, RI, 1987, 654–671 | MR

[88] V. V. Nikulin and I. R. Shafarevich, Geometries and Groups, Nauka, Moscow, 1983 | MR | MR | Zbl | Zbl

[89] Geometries and Groups, Universitext, Springer Ser. Sov. Math., Springer, Berlin, 1987 | MR | MR | Zbl | Zbl

[90] P. S. Novikov and S. I. Adjan, “Infinite periodic groups. I”, Math. USSR, Izv., 2:1 (1968), 209–236 | DOI | MR | MR

[91] P. S. Novikov and S. I. Adjan, “Infinite periodic groups. II”, Math. USSR, Izv., 2:2 (1968), 241–479 | DOI | MR

[92] P. S. Novikov and S. I. Adjan, “Infinite periodic groups. III”, Math. USSR, Izv., 2:3 (1968), 665–685 | DOI | MR

[93] O'Connor J.J., Robertson E.F., Igor Rostislavovich Shafarevich http://mathshistory.st-andrews.ac.uk/Biographies/Shafarevich.html

[94] Ogg A.P., “Cohomology of abelian varieties over function fields”, Ann. Math. Ser. 2, 76 (1962), 185–212 | DOI | MR | Zbl

[95] Ogus A., “Supersingular K3 crystals”, Journées de géométrie algébrique de Rennes (Juillet 1978): Groupes formels, représentations galoisiennes et cohomologie des variétés de caractéristique positive. II, Astérisque, 64, Soc. math. France, Paris, 1979, 3–86 | MR

[96] Orlov D.O., “Equivalences of derived categories and K3 surfaces”, J. Math. Sci., 84:5 (1997), 1361–1381 | DOI | MR | Zbl

[97] Orr M., Skorobogatov A.N., “Finiteness theorems for K3 surfaces and abelian varieties of CM type”, Compos. Math., 154:8 (2018), 1571–1592 | DOI | MR | Zbl

[98] Orr M., Skorobogatov A.N., Zarhin Yu.G., On uniformity conjectures for abelian varieties and K3 surfaces, E-print, 2018, arXiv: 1809.01440 [math.NT]

[99] A. N. Paršin, “Algebraic curves over function fields. I”, Math. USSR, Izv., 2:5 (1968), 1145–1170 | DOI | MR | Zbl

[100] A. N. Paršin, “Modular correspondences, heights and isogenies of Abelian varieties”, Proc. Steklov Inst. Math., 132 (1975), 243–270 | MR | Zbl | Zbl

[101] A. N. Parshin, “On the application of ramified coverings in the theory of Diophantine equations”, Math. USSR, Sb., 66:1 (1990), 249–264 | DOI | MR | Zbl

[102] Premet A., Strade H., “Simple Lie algebras of small characteristic. VI: Completion of the classification”, J. Algebra, 320:9 (2008), 3559–3604 | DOI | MR | Zbl

[103] I. I. Pjatecki\u i-Šapiro and I. R. Šafarevič, “Galois theory of transcendental extensions and uniformization”, Am. Math. Soc. Transl., Ser. 2, 69 (1968), 111–145 | MR

[104] I. I. Pjatecki\u i-Šapiro and I. R. Šafarevič, “A Torelli theorem for algebraic surfaces of type $K3$”, Math. USSR, Izv., 5:3 (1971), 547–588 | DOI | MR

[105] I. I. Pjatecki\u i-Šapiro and I. R. Šafarevič, “The arithmetic of K3 surfaces”, Proc. Steklov Inst. Math., 132 (1975), 45–57 | MR | Zbl

[106] Reichardt H., “Konstruktion von Zahlkörpern mit gegebener Galoisgruppe von Primzahlpotenzordnung”, J. reine angew. Math., 177 (1937), 1–5 | MR

[107] Roquette P., “On class field towers”, Algebraic number theory, Proc. Instructional Conf. (Brighton, 1965), Acad. Press, London, 1967, 231–249 | MR

[108] Rubin K., “Tate–Shafarevich groups and $L$-functions of elliptic curves with complex multiplication”, Invent. math., 89:3 (1987), 527–559 | DOI | MR

[109] A. N. Rudakov and I. R. Shafarevich, “Irreducible representations of a simple three-dimensional Lie algebra over a field of finite characteristic”, Math. Notes, 2:5 (1967), 760–767 | DOI | MR | Zbl

[110] A. N. Rudakov and I. R. Šafarevič, “Inseparable morphisms of algebraic surfaces”, Math. USSR, Izv., 10:6 (1976), 1205–1237 | DOI | MR | Zbl

[111] A. N. Rudakov and I. R. Shafarevich, “Quasi-elliptic surfaces of type K3”, Russ. Math. Surv., 33:1 (1978), 215–216 | DOI | MR | MR | Zbl | Zbl

[112] A. N. Rudakov and I. R. Shafarevich, “Vector fields on elliptic surfaces”, Russ. Math. Surv., 33:6 (1978), 255–256 | DOI | MR | Zbl | Zbl

[113] A. N. Rudakov and I. R. Šafarevič, “Supersingular $K3$ surfaces over fields of characteristic 2”, Math. USSR, Izv., 13:1 (1979), 147–165 | DOI | MR | Zbl | Zbl

[114] A. N. Rudakov and I. R. Šafarevič, “On the degeneration of surfaces of type $K3$”, Sov. Math., Dokl., 24 (1981), 163–165 | MR | Zbl

[115] A. N. Rudakov and I. R. Šafarevič, “On the degeneration of $K3$ surfaces over fields of finite characteristic”, Math. USSR, Izv., 18:3 (1982), 561–574 | DOI | MR | Zbl | Zbl

[116] A. N. Rudakov and I. R. Shafarevich, “Surfaces of type K3 over fields of finite characteristic”, J. Sov. Math., 22:4 (1983), 1476–1533 | DOI | MR

[117] A. N. Rudakov and I. R. Shafarevich, “On degeneration of K3 surfaces”, Proc. Steklov Inst. Math., 166 (1986), 247–259 | MR | Zbl | Zbl

[118] A. N. Rudakov, T. Tsink (Zink), and I. R. Shafarevich, “The influence of height on degenerations of algebraic surfaces of type $K3$”, Math. USSR, Izv., 20:1 (1983), 119–135 | DOI | MR | Zbl | Zbl

[119] G. Shabat, “Calculating and drawing Belyi pairs”, J. Math. Sci., 226:5 (2017), 667–693 ; Zap. Nauchn. Semin. POMI, 446 (2016), 182–220 | DOI | MR | Zbl

[120] Scholz A., “Über die Bildung algebraischer Zahlkörper mit auflösbarer Galoisscher Gruppe”, Math. Z., 30 (1929), 332–356 | DOI | MR | Zbl

[121] Scholz A., “Konstruktion algebraischer Zahlkörper mit beliebiger Gruppe von Primzahlpotenzordnung. I”, Math. Z., 42 (1937), 161–188 | DOI | MR

[122] Schur J., “Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen”, J. reine angew. Math., 132 (1907), 85–137 | MR | Zbl

[123] I. Shafarevich, “On the normalizability of topological fields”, C. R. (Dokl.) Acad. Sci. URSS, 40 (1943), 133–135 | MR | Zbl

[124] I. R. Shafarevich, “On Galois groups of $y$-adic fields”, C. R. (Dokl.) Acad. Sci. URSS, 53 (1946), 15–16 | MR | Zbl

[125] I. R. Šafarevič, “On $p$-extensions”, Am. Math. Soc. Transl., Ser. 2,, 4 (1956), 59–72 | Zbl

[126] I. R. Šafarevič, “A general reciprocity law”, Am. Math. Soc. Transl., Ser. 2, 4 (1956), 73–106 | Zbl

[127] I. R. Šafarevič, “On the construction of fields with a given Galois group of order $l^\alpha $”, Am. Math. Soc. Transl., Ser. 2, 4 (1956), 107–142 | MR | Zbl

[128] I. R. Šafarevič, “On the problem of imbedding fields”, Am. Math. Soc. Transl., Ser. 2, 4 (1956), 151–183 | MR | Zbl

[129] I. R. Šafarevič, “Construction of fields of algebraic numbers with given solvable Galois group”, Am. Math. Soc. Transl., Ser. 2, 4 (1956), 185–237 | MR | Zbl

[130] I. R. Shafarevich, “Galois theory and arithmetic of number fields”, Proc. Third All-Union Math. Congr., v. 2, Akad. Nauk SSSR, Moscow, 1956, 8 (in Russian)

[131] I. R. Shafarevich, “Birational equivalence of elliptic curves”, Dokl. Akad. Nauk SSSR, 114:2 (1957), 267–270 | MR | Zbl

[132] I. R. Shafarevich, “Exponents of elliptic curves”, Dokl. Akad. Nauk SSSR, 114:4 (1957), 714–716 | MR | Zbl

[133] I. R. Shafarevich, “The imbedding problem for splitting extensions”, Dokl. Akad. Nauk SSSR, 120:6 (1958), 1217–1219 | MR | Zbl

[134] I. R. Shafarevich, “Group of principal homogeneous algebraic manifolds”, Dokl. Akad. Nauk SSSR, 124:1 (1959), 42–43 | MR | Zbl

[135] I. R. Shafarevich, “Impression of International Congress of Mathematicians in Edinburgh”, Usp. Mat. Nauk, 14:2 (1959), 243–246

[136] I. R. Šafarevič, “Principal homogeneous spaces defined over a function field”, Am. Math. Soc. Transl., Ser. 2, 37 (1964), 85–114 | Zbl

[137] I. R. Šafarevič, “Algebraic number fields”, Proc. Int. Congr. Math. (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, 163–176 (in Russian) | MR

[138] Am. Math. Soc. Transl., Ser. 2, 31 (1963), 25–39 | MR

[139] I. R. Shafarevich, “Extensions with given points of ramification”, Publ. Math., Inst. Hautes Étud. Sci., 18:1 (1963), 71–92 | DOI

[140] Shafarevich I.R., Lectures on minimal models and birational transformations of two dimensional schemes, Tata Inst. Fundam. Res. Lect. Math. Phys. Math., 37, Tata Inst. Fundam. Res., Bombay, 1966 | MR | Zbl

[141] Shafarevich I.R., “On some infinite-dimensional groups”, Rend. Mat. Appl. Ser. 5, 25:1–2 (1966), 208–212 | MR | Zbl

[142] I. R. Shafarevich, Zeta Function, Mosk. Gos. Univ., Moscow, 1969 (in Russian)

[143] Shafarevich I.R., “Le théorème de Torelli pour les surfaces algébriques de type K3”, Actes Congr. Int. Math. (Nice, 1970), v. 1, Gauthier-Villars, Paris, 1971, 413–417 | MR

[144] I. R. Shafarevich, Foundations of Algebraic Geometry, Nauka, Moscow, 1972 | MR | MR | MR | Zbl

[145] Basic Algebraic Geometry, Grundl. Math. Wiss., 213, Springer, Berlin, 1974 | MR | MR | MR | Zbl

[146] I. R. Šafarevič, “On some infinite-dimensional groups. II”, Math. USSR, Izv., 18:1 (1982), 185–194 | DOI | MR | Zbl

[147] I. R. Shafarevich, Basic Notions of Algebra, Itogi Nauki Tekh. Sovr. Probl. Mat. Fundam. Napr., 11, Algebra–1, VINITI, Moscow, 1986

[148] I. R. Shafarevich, “Deformations of commutative algebras of class 2”, Leningr. Math. J., 2:6 (1991), 1335–1351 | MR | MR

[149] Shafarevich I.R., “Abelian and nonabelian mathematics”, Math. Intell., 13:1 (1991), 67–75 | DOI | MR | Zbl

[150] Shafarevich I.R., “On the arithmetic of singular $K3$-surfaces”, Algebra and analysis: Proc. Int. Centen. Chebotarev Conf. (Kazan, 1994), W. de Gruyter, Berlin, 1996, 103–108 | MR | Zbl

[151] I. R. Shafarevich, “Some families of Abelian surfaces”, Izv. Math., 60:5 (1996), 1083–1093 | DOI | DOI | MR | Zbl

[152] I. R. Shafarevich, “Families of reducible algebraic varieties”, Math. Notes, 60:6 (1996), 717–720 | DOI | DOI | MR | MR | Zbl

[153] Shafarevich I.R., “Degeneration of semisimple algebras”, Commun. Algebra, 29:9 (2001), 3943–3960 | DOI | MR | Zbl

[154] I. R. Shafarevich, “On the group $GL(2,K[t])$”, Proc. Steklov Inst. Math., 246 (2004), 308–314 | MR | Zbl

[155] I. R. Shafarevich, “On the problem of the 10th discriminant”, St. Petersburg Math. J., 25:4 (2014), 699–711 | DOI | MR | Zbl

[156] Shafarevich I.R., Collected mathematical papers, Springer Coll. Works Math., Springer, Berlin, 2015 | MR | Zbl

[157] I. R. Šafarevič, B. G. Averbuh, Ju. R. Vainberg, A. B. Žižčenko, Ju. I. Manin, B. G. Moišezon, G. N. Tjurina, and A. N. Tjurin, Algebraic Surfaces, Tr. Mat. Inst. Steklova, 75, Nauka, Moscow, 1965 | MR | MR

[158] Algebraic Surfaces, Proc. Steklov Inst. Math., 75, Am. Math. Soc., Providence, RI, 1967 | MR | MR

[159] Shestakov I.P., Umirbaev U.U., “The tame and the wild automorphisms of polynomial rings in three variables”, J. Amer. Math. Soc., 17:1 (2004), 197–227 | DOI | MR | Zbl

[160] Siu Y.-T., “Every K3 surface is Kähler”, Invent. math., 73:1 (1983), 139–150 | DOI | MR | Zbl

[161] Stark H.M., “A complete determination of the complex quadratic fields of class-number one”, Mich. Math. J., 14:1 (1967), 1–27 | DOI | MR | Zbl

[162] Strade H., “The classification of the simple modular Lie algebras. VI: Solving the final case”, Trans. Amer. Math. Soc., 350:7 (1998), 2553–2628 | DOI | MR | Zbl

[163] Soulé C., Abramovich D., Burnol J.-F., Kramer J., Lectures on Arakelov geometry, Cambridge Stud. Adv. Math., 33, Cambridge Univ. Press, Cambridge, 1992 | MR | Zbl

[164] Sury B., “Igor Rostislavovich Shafarevich: III (may we say “Shah”?) of Number Theory”, Resonance, 22:5 (2017), 441–454 | DOI

[165] J. T. Tate and I. R. Shafarevich, “The rank of elliptic curves”, Sov. Math., Dokl., 8 (1967), 917–920 | MR | Zbl

[166] The Grothendieck theory of dessins d'enfants, LMS Lect. Note Ser., 200, ed. by L. Schneps, Cambridge Univ. Press, Cambridge, 1994 | MR | Zbl

[167] Todorov A.N., “Applications of the Kähler–Einstein–Calabi–Yau metric to moduli of K3 surfaces”, Invent. math., 61:3 (1980), 251–265 | DOI | MR | Zbl

[168] Treger R., Metrics on universal covering of projective variety, E-print, 2014, arXiv: 1209.3128v5 [math.AG]

[169] Treger R., On uniformization of compact Kahler spaces, E-print, 2016, arXiv: 1601.02413 [math.AG]

[170] A. N. Tjurin, “Classification of vector bundles over an algebraic curve of arbitrary genus”, Am. Math. Soc. Transl., Ser. 2, 63 (1967), 245–279 | MR | Zbl

[171] A. N. Tjurin, “On the classification of $n$-dimensional vector-bundles over an algebraic curve of arbitrary genus”, Am. Math. Soc. Transl., Ser. 2, 73 (1968), 196–211 | MR

[172] A. N. Tjurin, “Analogs of Torelli's theorem for multidimensional vector bundles over an arbitrary algebraic curve”, Math. USSR, Izv., 4:2 (1970), 343–370 | DOI | MR | Zbl

[173] G. N. Tjurina, “Deformation of complex structures on algebraic manifolds”, Sov. Math., Dokl., 4 (1964), 1567–1570 | Zbl

[174] Ulmer D., “Elliptic curves with large rank over function fields”, Ann. Math. Ser. 2, 155:2 (2002), 295–315 | DOI | MR | Zbl

[175] Verbitsky M., “Mapping class group and a global Torelli theorem for hyperkähler manifolds”, Duke Math. J., 162:15 (2013), 2929–2986 | DOI | MR | Zbl

[176] È. B. Vinberg, “On a theorem on infinite dimensionality of an associative algebra”, Am. Math. Soc. Transl., Ser. 2, 82 (1969), 237–242 | Zbl | Zbl

[177] Vojta P., “Siegel's theorem in the compact case”, Ann. Math. Ser. 2, 133:3 (1991), 509–548 | DOI | MR | Zbl

[178] S. V. Vostokov, “Explicit form of the law of reciprocity”, Math. USSR, Izv., 13:3 (1979), 557–588 | DOI | MR | Zbl | Zbl

[179] O. N. Vvedenskiĭ, “On quasi-local ‘class fields’ of elliptic curves. I”, Math. USSR, Izv., 10:5 (1976), 913–936 | DOI | MR

[180] Weil A., “On algebraic groups and homogeneous spaces”, Amer. J. Math., 77 (1955), 493–512 | DOI | MR | Zbl

[181] Yau S.-T., “Calabi's conjecture and some new results in algebraic geometry”, Proc. Natl. Acad. Sci. USA, 74:5 (1977), 1798–1799 | DOI | MR | Zbl

[182] Yau S.-T., “On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I”, Commun. Pure Appl. Math., 31:3 (1978), 339–411 | DOI | MR | Zbl

[183] Yu. G. Zarkhin, “Finiteness theorem for isogenies of Abelian varieties over function fields of finite characteristic”, Funkt. Anal. Appl., 8:4 (1974), 301–303 | DOI | MR | Zbl

[184] E. I. Zel'manov, “Solution of the restricted Burnside problem for groups of odd exponent”, Math. USSR, Izv., 36:1 (1991), 41–60 | DOI | MR | Zbl | Zbl

[185] E. I. Zel'manov, “A solution of the restricted Burnside problem for 2-groups”, Math. USSR, Sb., 72:2 (1992), 543–565 | DOI | MR | Zbl | Zbl