First and second K-groups of an elliptic curve over a global field of positive characteristic
[Sur les premier et second K-groupes d’une courbe elliptique sur un corps global de caractéristique positive]
Annales de l'Institut Fourier, Tome 68 (2018) no. 5, pp. 2005-2067.

On démontre que les plus grands sous-groupes divisibles desgroupes K 1 et K 2 d’une courbe elliptique E sur un corps global de caractéristique positive sont uniquement divisibles et on décrit explicitement les K-groupes modulo leurs plus grands sous-groupes divisibles. On calcule également la cohomologie motivique du modèle minimal de E qui est une surface elliptique sur un corps fini.

In this paper, we show that the maximal divisible subgroup of groups K 1 and K 2 of an elliptic curve E over a function field is uniquely divisible. Further those K-groups modulo this uniquely divisible subgroup are explicitly computed. We also calculate the motivic cohomology groups of the minimal regular model of E, which is an elliptic surface over a finite field.

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DOI : 10.5802/aif.3202
Classification : 11R58, 14F42, 19F27, 11G05
Keywords: K-theory, function field, elliptic curve, motivic cohomology
Mot clés : K-théorie, corps de fonctions, courbe elliptique, cohomologie motivique
Kondo, Satoshi 1, 2 ; Yasuda, Seidai 3

1 National Research University Higher School of Economics Usacheva St., 7, Moscow 119048 (Russia)
2 Kavli Institute for the Physics and Mathematics of the Universe University of Tokyo 5-1-5 Kashiwanoha Kashiwa, 277-8583 (Japan)
3 Department of Mathematics, Osaka University Toyonaka, Osaka 560-0043 (Japan)
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Kondo, Satoshi; Yasuda, Seidai. First and second $K$-groups of an elliptic curve over a global field of positive characteristic. Annales de l'Institut Fourier, Tome 68 (2018) no. 5, pp. 2005-2067. doi : 10.5802/aif.3202. http://archive.numdam.org/articles/10.5802/aif.3202/

[1] Abhyankar, Shreeram S. Resolution of singularities of arithmetical surfaces, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harper & Row, 1965, pp. 111-152 | MR | Zbl

[2] Bass, Hyman; Milnor, John; Serre, Jean-Pierre Solution of the congruence subgroup problem for SL n (n3) and Sp 2n (n2), Publ. Math., Inst. Hautes Étud. Sci. (1967) no. 33, pp. 59-137 | MR | Zbl

[3] Bass, Hyman; Tate, John The Milnor ring of a global field, Algebraic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic (Proc. Conf., Seattle, Wash., Battelle Memorial Inst., 1972) (Lecture Notes in Math.), Volume 342, Springer, 1973, pp. 349-446 | MR | Zbl

[4] Théorie des intersections et théorème de Riemann-Roch. Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6) (Berthelot, Pierre; Grothendieck, Alexander; Illusie, Luc, eds.), Lecture Notes in Math., 225, Springer, 1971, xii+700 pages | MR | Zbl

[5] Bloch, Spencer Algebraic cycles and higher K-theory, Adv. Math., Volume 61 (1986) no. 3, pp. 267-304 | DOI | MR | Zbl

[6] Bloch, Spencer The moving lemma for higher Chow groups, J. Algebr. Geom., Volume 3 (1994) no. 3, pp. 537-568 | MR | Zbl

[7] Chida, Masataka; Kondo, Satoshi; Yamauchi, Takuya On the rational K 2 of a curve of GL 2 type over a global field of positive characteristic, J. K-Theory, Volume 14 (2014) no. 2, pp. 313-342 | DOI | Zbl

[8] Colliot-Thélène, Jean-Louis; Raskind, Wayne On the reciprocity law for surfaces over finite fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 33 (1986) no. 2, pp. 283-294 | MR | Zbl

[9] Colliot-Thélène, Jean-Louis; Sansuc, Jean-Jacques; Soulé, Christophe Torsion dans le groupe de Chow de codimension deux, Duke Math. J., Volume 50 (1983) no. 3, pp. 763-801 | DOI | MR | Zbl

[10] Deligne, Pierre Cohomologie étale. Séminaire de Géométrie Algébrique du Bois-Marie SGA 41/2, Lecture Notes in Math., 569, Springer, 1977, iv+312pp pages | MR | Zbl

[11] Demazure, Michel Lectures on p-divisible groups, Lecture Notes in Math., 302, Springer, 1972, v+98 pages | MR | Zbl

[12] Dolgačev, Igor V. The Euler characteristic of a family of algebraic varieties, Mat. Sb. (N.S.), Volume 89(131) (1972), pp. 297-312 | MR | Zbl

[13] Geisser, Thomas Tate’s conjecture, algebraic cycles and rational K-theory in characteristic p., K-Theory, Volume 13 (1998) no. 2, pp. 109-122 | DOI | Zbl

[14] Geisser, Thomas Motivic cohomology, K-theory and topological cyclic homology, Handbook of K-theory, Volume 1, Springer, 2005, pp. 193-234 | MR | Zbl

[15] Geisser, Thomas; Levine, Marc The K-theory of fields in characteristic p, Invent. Math., Volume 139 (2000) no. 3, pp. 459-493 | DOI | MR | Zbl

[16] Geisser, Thomas; Levine, Marc The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky, J. Reine Angew. Math., Volume 530 (2001), pp. 55-103 | DOI | MR | Zbl

[17] Gillet, Henri Riemann-Roch theorems for higher algebraic K-theory, Adv. Math., Volume 40 (1981) no. 3, pp. 203-289 | DOI | MR | Zbl

[18] Grayson, Daniel R. Finite generation of K-groups of a curve over a finite field (after Daniel Quillen), Algebraic K-theory, Part I (Oberwolfach, 1980) (Lecture Notes in Math.), Volume 966, Springer, 1982, pp. 69-90 | MR | Zbl

[19] Grayson, Daniel R. Weight filtrations via commuting automorphisms, K-Theory, Volume 9 (1995) no. 2, pp. 139-172 | DOI | Zbl

[20] Gros, Michel; Suwa, Noriyuki Application d’Abel-Jacobi p-adique et cycles algébriques, Duke Math. J., Volume 57 (1988) no. 2, pp. 579-613 | DOI | MR | Zbl

[21] Grothendieck, Alexander Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Publ. Math., Inst. Hautes Étud. Sci. (1961) no. 11, 167 pages | MR

[22] Revêtements étales et groupe fondamental (Grothendieck, Alexander; Raynaud, Michèle, eds.), Lecture Notes in Math., 224, Springer, 1971, xxii+447 pages Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1), Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud | MR | Zbl

[23] Hanamura, Masaki Homological and cohomological motives of algebraic varieties, Invent. Math., Volume 142 (2000) no. 2, pp. 319-349 | DOI | Zbl

[24] Harder, Günter Die Kohomologie S-arithmetischer Gruppen über Funktionenkörpern, Invent. Math., Volume 42 (1977), pp. 135-175 | MR | Zbl

[25] Illusie, Luc Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. Éc. Norm. Supér., Volume 12 (1979) no. 4, pp. 501-661 | MR | Zbl

[26] Illusie, Luc Finiteness, duality, and Künneth theorems in the cohomology of the de Rham-Witt complex, Algebraic geometry (Tokyo/Kyoto, 1982) (Lecture Notes in Math.), Volume 1016, Springer, 1983, pp. 20-72 | MR | Zbl

[27] Kahn, Bruno Algebraic K-theory, algebraic cycles and arithmetic geometry, Handbook of K-theory, Volume 1, Springer, 2005, pp. 351-428 | MR | Zbl

[28] Kato, Kazuya A generalization of local class field theory by using K-groups. II, Proc. Japan Acad. Ser. A, Volume 54 (1978) no. 8, pp. 250-255 http://projecteuclid.org/getrecord?id=euclid.pja/1195517586 | MR | Zbl

[29] Kato, Kazuya; Saito, Shuji Unramified class field theory of arithmetical surfaces, Ann. Math., Volume 118 (1983) no. 2, pp. 241-275 | DOI | MR | Zbl

[30] Kondo, Satoshi; Yasuda, Seidai On the second rational K-group of an elliptic curve over global fields of positive characteristic., Proc. Lond. Math. Soc., Volume 102 (2011) no. 6, pp. 1053-1098 | DOI | Zbl

[31] Kondo, Satoshi; Yasuda, Seidai The Riemann-Roch theorem without denominators in motivic homotopy theory, J. Pure Appl. Algebra, Volume 218 (2014) no. 8, pp. 1478-1495 | DOI | Zbl

[32] Kondo, Satoshi; Yasuda, Seidai On two higher Chow groups of schemes over a finite field., Doc. Math., Volume 20 (2015), pp. 737-752 | Zbl

[33] Levine, Marc Mixed motives, Mathematical Surveys and Monographs, 57, American Mathematical Society, 1998, x+515 pages | MR | Zbl

[34] Levine, Marc Techniques of localization in the theory of algebraic cycles, J. Algebr. Geom., Volume 10 (2001) no. 2, pp. 299-363 | MR | Zbl

[35] Levine, Marc Mixed motives, Handbook of K-theory. Vol. 1 and 2, Springer, 2005, pp. 429-521 | Zbl

[36] Levine, Marc The homotopy coniveau tower, J. Topol., Volume 1 (2008) no. 1, pp. 217-267 | MR | Zbl

[37] Lipman, Joseph Desingularization of two-dimensional schemes, Ann. Math., Volume 107 (1978) no. 1, pp. 151-207 | MR | Zbl

[38] Liu, Qing Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, 6, Oxford University Press, 2002, xvi+576 pages (Translated from the French by Reinie Erné, Oxford Science Publications) | MR | Zbl

[39] Merkurʼev, Aleksandr S.; Suslin, Andreĭ A. K-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Dokl. Akad. Nauk SSSR, Volume 264 (1982) no. 3, pp. 555-559 | MR | Zbl

[40] Merkurʼev, Aleksandr S.; Suslin, Andreĭ A. The group K 3 for a field, Izv. Akad. Nauk SSSR Ser. Mat., Volume 54 (1990) no. 3, pp. 522-545 | MR | Zbl

[41] Milne, James Stuart Duality in the flat cohomology of a surface, Ann. Sci. Éc. Norm. Supér., Volume 9 (1976) no. 2, pp. 171-201 | MR | Zbl

[42] Milne, James Stuart Values of zeta functions of varieties over finite fields, Am. J. Math., Volume 108 (1986) no. 2, pp. 297-360 | DOI | MR | Zbl

[43] Milnor, John Algebraic K-theory and quadratic forms, Invent. Math., Volume 9 (1969/1970), pp. 318-344 | MR | Zbl

[44] Morel, Fabien; Voevodsky, Vladimir 𝔸 1 -homotopy theory of schemes, Publ. Math., Inst. Hautes Étud. Sci., Volume 90 (1999), pp. 45-143 | DOI | Zbl

[45] Müller-Stach, Stefan J. Algebraic cycle complexes: Basic properties, The arithmetic and geometry of algebraic cycles. Proceedings of the NATO Advanced Study Institute, Banff, Canada, June 7–19, 1998, Kluwer Academic Publishers, 2000, pp. 285-305 | Zbl

[46] Nagata, Masayoshi Imbedding of an abstract variety in a complete variety, J. Math. Kyoto Univ., Volume 2 (1962), pp. 1-10 | MR | Zbl

[47] Nesterenko; Suslin, Andreĭ A. Homology of the general linear group over a local ring, and Milnor’s K-theory, Izv. Akad. Nauk SSSR Ser. Mat., Volume 53 (1989) no. 1, pp. 121-146 | MR | Zbl

[48] Nygaard, Niels O. Slopes of powers of Frobenius on crystalline cohomology, Ann. Sci. Éc. Norm. Supér., Volume 14 (1981) no. 4, pp. 369-401 | MR | Zbl

[49] Ogg, Andrew P. Elliptic curves and wild ramification, Am. J. Math., Volume 89 (1967), pp. 1-21 | MR | Zbl

[50] Oguiso, Keiji An elementary proof of the topological Euler characteristic formula for an elliptic surface, Comment. Math. Univ. St. Pauli, Volume 39 (1990) no. 1, pp. 81-86 | MR | Zbl

[51] Pushin, Oleg Higher Chern classes and Steenrod operations in motivic cohomology, K-Theory, Volume 31 (2004) no. 4, pp. 307-321 | DOI | MR | Zbl

[52] Quillen, Daniel On the cohomology and K-theory of the general linear groups over a finite field, Ann. Math., Volume 96 (1972), pp. 552-586 | MR | Zbl

[53] Quillen, Daniel Finite generation of the groups K i of rings of algebraic integers, Algebr. K-Theory I, Proc. Conf. Battelle Inst. 1972 (Lecture Notes in Math.), Volume 341, 1973, pp. 179-198 | Zbl

[54] Riou, Joël Algebraic K-theory, A 1 -homotopy and Riemann-Roch theorems, J. Topol., Volume 3 (2010) no. 2, pp. 229-264 | DOI | Zbl

[55] Shioda, Tetsuji On the Mordell-Weil lattices, Comment. Math. Univ. St. Pauli, Volume 39 (1990) no. 2, pp. 211-240 | MR | Zbl

[56] Soulé, Christophe Groupes de Chow et K-théorie de variétés sur un corps fini, Math. Ann., Volume 268 (1984) no. 3, pp. 317-345 | DOI | MR | Zbl

[57] Suslin, Andreĭ A. On the Grayson spectral sequence, Number theory, algebra, and algebraic geometry. Collected papers dedicated to the 80th birthday of Academician Igor’ Rostislavovich Shafarevich., Maik Nauka/Interperiodika, 2003, pp. 202-237 | Zbl

[58] Tate, John Symbols in arithmetic, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, Gauthier-Villars, 1971, pp. 201-211 | MR | Zbl

[59] Totaro, Burt Milnor K-theory is the simplest part of algebraic K-theory, K-Theory, Volume 6 (1992) no. 2, pp. 177-189 | DOI | MR | Zbl

[60] Voevodsky, Vladimir Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic, Int. Math. Res. Not. (2002) no. 7, pp. 351-355 | DOI | MR | Zbl

[61] Voevodsky, Vladimir; Suslin, Andreĭ A.; Friedlander, Eric M. Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, 143, Princeton University Press, 2000, vi+254 pages | MR | Zbl

[62] Weibel, Charles Algebraic K-theory of rings of integers in local and global fields., Handbook of K-theory. Vol. 1 and 2, Springer, 2005, pp. 139-190 | Zbl

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