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On the rational \(K_2\) of a curve of \(\mathrm{GL}_{2}\) type over a global field of positive characteristic. (English) Zbl 1322.19002

Let \(Z\) be a proper smooth scheme over a global field \(k\) of characteristic \(p.\) For a \(k\)-algebra \(M\) let \(Z_{M}\) denote the base change \(Z_{M}=Z{\times}_{{\text{Spec}} k}M.\) Let \(l\) be a prime different from \(p.\) Denote by \(G_{k}={\text{Gal}} (k^{\text{sep}})\) the absolute Galois group of \(k.\) The authors introduce the following \((\mathrm{GL}_{2})\)-condition on \(Z:\)
The \(G_{k}\)-representation \(H^!_{\text{ét}}(Z_{k^{\text{sep}}}, {\mathbb Q}_{l})\) is a direct sum of \(2\)-dimensional irreducible representations.
Assume that characteristic of \(k\) is greater than \(2.\) Let \(X\) be a proper smooth curve over \(k\) and \(\mathcal X\) an integral regular model of \(X.\) Suppose that \(X\) satisfies the condition (\(\mathrm{GL}_{2}\)). The main result of the paper is that under these assumptions the boundary map of the localization sequence in the \(G\)-theory \[ K_{2}(X)_{\mathbb Q}@>{{\oplus}{\partial}_{\mathcal P}}>> {\bigoplus}_{{\mathcal P}\in C} G_{1}({\mathcal X}_{{\kappa}(\mathcal P)})_{\mathbb Q} \] is surjective. is surjective. As a corollary the authors obtain examples where surjectivity holds.
(1)
an elliptic curve, which is not isotrivial
(2)
a Drinfeld modular curve
(3)
the moduli of \(\mathcal D\)-elliptic sheaves of rank \(2\)
(4)
a genus \(2\) curve constructed in the paper.

MSC:

19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
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References:

[1] DOI: 10.5802/afst.1065 · Zbl 1074.11030 · doi:10.5802/afst.1065
[2] Subgroups of semi-abelian varieties
[3] Algebraic geometry and arithmetic curves 6 (2002)
[4] DOI: 10.1007/BF01244308 · Zbl 0809.11032 · doi:10.1007/BF01244308
[5] Complex multiplication 255 (1983) · Zbl 0536.14029
[6] DOI: 10.1007/s002220100174 · Zbl 1038.11075 · doi:10.1007/s002220100174
[7] Proc. Symp. Pure Math. 55 pp 365– (1994)
[8] DOI: 10.1112/plms/pdq045 · Zbl 1222.19001 · doi:10.1112/plms/pdq045
[9] Mixed motives and algebraic K-theory pp 246–
[10] DOI: 10.2307/2374050 · Zbl 0491.10020 · doi:10.2307/2374050
[11] Automorphic forms on GL(2) 114 (1970) · Zbl 0236.12010
[12] Automorphic forms, representations, and L-functios pp 63– (1979)
[13] Number theoretic background pp 3– (1977)
[14] DOI: 10.1007/BF01404549 · Zbl 0147.20303 · doi:10.1007/BF01404549
[15] Langlands’ conjecture for GL(2) over functional fields pp 565– (1980)
[16] DOI: 10.1070/SM1974v023n04ABEH001731 · Zbl 0321.14014 · doi:10.1070/SM1974v023n04ABEH001731
[17] DOI: 10.2977/prims/1195164439 · Zbl 1045.11510 · doi:10.2977/prims/1195164439
[18] J. Reine Angew. Math. 422 pp 201– (1991)
[19] DOI: 10.1007/BF01457062 · Zbl 0573.14001 · doi:10.1007/BF01457062
[20] Arcata/Calif. 1985, Contemp. Math. 67 pp 25– (1987)
[21] Classical motives 55 pp 163– (1994)
[22] Representation theory of finite groups and associative algebras pp 685– (1962)
[23] Galois cohomology pp 210– (2002)
[24] DOI: 10.1007/BF01428197 · Zbl 0239.10015 · doi:10.1007/BF01428197
[25] The local Langlands conjecture for GL(2) 335 (2006) · Zbl 1100.11041
[26] Introduction to the arithmetic theory of automorphic functions 11 pp 267–
[27] DOI: 10.1007/BFb0087916 · doi:10.1007/BFb0087916
[28] DOI: 10.2748/tmj/1178207751 · Zbl 0993.11031 · doi:10.2748/tmj/1178207751
[29] Algebraic geometry 52 (1977)
[30] Modèles de Néron et monodromie Groupes de monodromie en geometrie algebrique. I 288 pp 313– (1972)
[31] Zetafunctios of simple algebras 260 (1972)
[32] DOI: 10.1016/S0001-8708(81)80006-0 · Zbl 0478.14010 · doi:10.1016/S0001-8708(81)80006-0
[33] Izv. Akad. Nauk SSSR Ser. Mat. 39 pp 272– (1975)
[34] J. Reine Angew. Math. 476 pp 27– (1996)
[35] Vector bundles on curves – new directions, Lectures given at the 3rd session of the Centro Internazionale Matematico Estivo (CIME) held in Cetraro (Cosenza), Italy, June 19-27, 1995 pp 110– (1997)
[36] Ann. of Math. Studies 147 (2000)
[37] Ann. Sci. Éc. Norm. Supér. 7 pp 181– (1974)
[38] Modular curves and abelian varieties 224 pp 241– (2004)
[39] DOI: 10.1007/BF01394268 · Zbl 0498.14010 · doi:10.1007/BF01394268
[40] Higher regulators on quaternionic Shimura curves and values of L-functions pp 377– (1986)
[41] Higher algebraic K-theory. I (2007)
[42] Astérisque 129 (1985)
[43] Parshin’s conjecture revisited pp 413– (2008)
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