Le K1 des courbes sur les corps globaux : conjecture de Bloch et noyaux sauvages

Michael Laske

Description

Sous la direction de Karim Belabas
Thèse soutenue le 19 novembre 2009: Bordeaux 1
Pour X une courbe sur un corps global k, lisse, projective et géométriquement connexe, nous déterminons la Q-structure du groupe de Quillen K1(X) : nous démontrons que dimQ K1(X) ? Q =2r, où r désigne le nombre de places archimédiennes de k (y compris le cas r = 0 pour un corps de fonctions). Cela con?rme une conjecture de Bloch annoncée dans les années 1980. Dans le langage de la K-théorie de Milnor, que nous dé?nissons pour les variétés algébriques via les groupes de Somekawa, le premier K-groupe spécial de Milnor SKM1 (X) est de torsion. Pour la preuve, nous développons une théorie des hauteurs applicable aux K-groupes de Milnor, et nous généralisons l’approche de base de facteurs de Bass-Tate. Une structure plus ?ne de SKM 1 (X) émerge en localisant le corps de base k, et une description explicite de la décomposition correspondante est donnée. En particulier, nous identi?ons un sous-groupe WKl(X):= ker (SKM 1 (X) ? Zl ? Lv SKM 1 (Xv) ? Zl) pour chaque entier rationnel l, nommé noyau sauvage, dont nous croyons qu’il est ?ni.
-K-théorie
-K-groupes de Milnor
-Noyaux Sauvages
-Géométrie Algébrique
-Conjecture de Bloch
-Groupes de Somekawa
For a smooth projective geometrically connected curve X over a global ?eld k, we determine the Q-structure of its ?rst Quillen K-group K1(X) by showing that dimQ K1(X) ? Q =2r, where r denotes the number of archimedean places of k (including the case r = 0 for k a function ?eld). This con?rms a conjecture of Bloch. In the language of Milnor K-theory, which we de?ne for varieties via Somekawa groups, the ?rst special Milnor K-group SKM 1 (X) is torsion. For the proof, we develop a theory of heights applicable to Milnor K-groups, and generalize the factor basis approach of Bass-Tate. A ?ner structure of SKM 1 (X) emerges when localizing the ground ?eld k, and we give an explicit description of the resulting decomposition. In particular, we identify a potentially ?nite subgroup WKl(X):= ker (SKM 1 (X) ? Zl ? Lv SKM 1 (Xv) ? Zl) for each rational prime l, named wild kernel.
-K-theory
-Wild Kernel
-Bloch’s Conjecture
-Somekawa groups
-Milnor K-groups
-Algebraic Geometry
Source: http://www.theses.fr/2009BOR13861/document

Sujets

• K-théorie
• Géométrie algébrique

Informations

• Publié par: Thesee
• Nombre de lectures: 34
• Langue: English

Extrait

N ◦ d’ordre: 3861

` THESE presentee`a ´ ´ ´ L’ UNIVERSITE BORDEAUX I ´ ´ ECOLE DOCTORALE DE MATH EMATIQUES ET INFORMATIQUE

par Michael LASKE POUR OBTENIR LE GRADE DE DOCTEUR ´ ´ ´ SPECIALITE: MATHEMATIQUES PURES

LeK 1 des Courbes sur les Corps Globaux. Conjecture de Bloch et Noyaux Sauvages

The`sedirig´eeparKarimBELABAS

Devantlacommissiond’examenform´eede: M.BELABASKarim,Professeur,IMB-Universite´Bordeaux M. DE JEU Rob, Professeur, Afdeling Wiskunde - Vrije Universiteit Amsterdam M.ELBAZ-VINCENTPhilippe,Professeur,InstitutFourier-Universite´Grenoble M.JAULENTJean-Francois,Professeur,IMB-Universit´eBordeaux

Soutenue le 19 novembre 2009

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Contents

Introduction 1 MilnorK-theory of varieties 1.1 Somekawa Groups. . . . . . . . . . . . . . . . . . . .. . . . . . 1.2 Comparison Theorems. . . . . . . . . . . . . . . . . .. . . . . . 1.3 Further Properties. . . . . . . . . . . . . . . . . . . .. . . . . . 1.4 SpecialK M -theory. . . . . . . . . . . . . . . . . . . . . . . . . . 2 Heights 2.1 Proofs of Mordell-Weil type. . . . . . . . . . . . . . .. . . . . . 2.2 Heights on Abelian Groups. . . . . . . . . . . . . . .. . . . . . 2.3 Linear Heights. . . . . . . . . . . . . . . . . . . . .. . . . . . . 2.4 Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Heights in Algebraic Geometry. . . . . . . . . . . . . .. . . . . 3 Decomposable Elements 3.1 Multiplication. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Point Degree. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Factor Basis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Bounds for Multiplication. . . . . . . . . . . . . . . .. . . . . . 4 Bloch’s Conjecture 4.1 Heights onK-groups. . . . . . . . . . . . . . . . . . . . . . . . . 4.2 An Approach via Heights. . . . . . . . . . . . . . . . . . . . . . References

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v 1 1 3 5 6 9 9 10 12 16 27 31 31 34 35 41 43 43 47 51

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Acknowledgments The author wishes to express his sincere gratitude to Wayne Raskind and Michael Spiess for their interest in this work and thoughtful advise how to im- prove it. Further it is a pleasure to thank Bruno Kahn and Jean-Louis Colliot- Th´ele`neforhelpfuldiscussionsandcomments. The author is greatly indebted to the referees Rob de Jeu and Philippe Elbaz-Vincent for their support and patience during numerous corrections of the manuscript. Special thanks are due to Karim Belabas for his encouragement and guidance throughout the preparation of this thesis.

INTRODUCTION Introduction This thesis studies the first algebraicK-group of curves over global fields.

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K-theory of Curves LetXprojective geometrically connected curve over a perfectbe a smooth fieldk.

K 1 in a nutshell.The firstK-group of a curve splits along the Adams eigenspaces, K 1 (X) = ∼ K 1 (k)⊕K 1M (X),(1) whereK 1 (k) =k × , andK 1M (Xbe referred to as the first Milnor) will K-group ofX(since this is not standard notation, see1.1.2for the precise definition). This group is described by a short exact sequence, providedX(k)￿=∅, 0−→SK 1M (X)−→K 1M (X)− N →K 1M (k)−→0 whereK 1M (k) =k × , andNdenotes the transfer in MilnorK-theory for smooth projective varieties (generalized via Somekawa groups [28], [1]). IfXhas ak- rational point, this sequence splits. Thus the nontrivial part ofK 1 (X) is the special MilnorK 1 -groupSK 1M (X). Ad Hoc definition ofSK 1M (X).Denote byk(X) the function field ofX; its second MilnorK-group is the quotient ofk(X) × ⊗k(X) × by the Steinberg relations, K 2M (k(X)) :=k(X) × ⊗k(X) × ￿ a⊗(1−a)|a∈k(X) × −{1}. Forf, g∈k(X) × , let{f, g}denote the equivalence class off⊗ginK 2M (k(X)). Identifying Weil divisorsX (1) of the curveXwith its Zariski-closed pointsX (0) , the residue fieldk(x), forx∈X (1) , is a finite extension ofk. The boundary map ∂:K 2M (k(X))→  x∈X(1) k(x) × , {f, g}→  x∈X(1)  (−1 g ord x (f)  (x), ) ord x (f)ord x (g) f ord x (g) given by the tame symbols is well-defined, i. e. factors through the Steinberg relations. The norm  x∈X(1) (x) × →k × ,⊕ x∈X(1) a x →  x∈X(1) N k(x)/k (a x ) N:k satisfies the Weil reciprocity formulaN◦∂= 1, hence yields a mapN: K 1M (X) := coker∂→k ×  define. We SK 1M (X) := ker  N: coker∂→k ×  .

viINTRODUCTION Ground Field.This explicit description ofSK 1M (X) involves the underlying ground field which we are now going to specify. (i) Letkbe a finite field. ThenSK 1M (X) = 0. (ii) Letkbe a local field. In this case, the groupSK 1M (X), under the name V(Xto intense study concerning generalizations of class field), has been subject theory, initiated by Kato and Saito [15], [26], Bloch [5], Katz and Lang [16] in the beginning of the 1980ies. Briefly, there is a reciprocity map τ:SK 1M (X)→π a 1 b (X) geo which is surjective on the torsion subgroup. Hereπ 1ab (Xd´zealetelabniiai)ehtse fundamental groupπ 1 (Xassifyin)clfsongrievocelate´etinfigX, with geometric partπ 1ab (X) geo := ker  π ab (X)→G a k b  . 1 (iii) Letkbe a global field. Unlike the situation of local ground field as in (ii), the global case is not well-understood. •The local case suggests thatSK 1M (X) carries geometrical information about the curveXinformation precisely and how is it encoded?. Which  •What is the structure ofSK 1M (X has been subject This) as abelian group? to a conjecture of Bloch [5 far it has been So] dating back to the 1980ies. known, by a result of Raskind [24, Cor. 0.3 & Lemmas 2.1, 2.2] from 1990, that M SK 1 (X)⊗Q/Z= 0.(2) Bloch’s Conjecture An Analogy.A heuristic guideline for our investigation ofK 1 of curves, is thatSK 1M  resemble” “shouldof a curveK 2M of the underlying ground field. As first confirmation of this principle we observe that Matsumoto’s Theorem, stating thatK 2 (F) = ∼ K 2M (F) for a fieldF, finds in (1 Milnor) its counterpart: K-theory captures all ofK 1 (X). The MilnorK-theory of fields is well-understood, and one of its fundamental theorems states thatK 2M  One can ask theof a global field is a torsion group. analog question in a geometrical context. Conjecture(Bloch, [5, 1.24]).For a smooth projective geometrically connected curveXover a global fieldk, the groupSK 1M (X)is torsion. This thesis provides a strategy for the demonstration of Bloch’s Conjecture, though we are not able to complete the proof. As suggested by analogy, it is instructive to review the classical approaches to determine the structure ofK 2M of a global fieldF. (i) First,K 2M (Fis decomposed into smaller pieces via the tame symbols,) ∂:K 2M (F)→  vfinite K 1M (F(v)) =  F(v) × v withF(v) denoting the residue field ofFat the finite placev homomor-. The phism∂is surjective, and thetame kernelker∂equalsK 2 (O F ) ifFis a number field andK 2 (Y) ifF=F(Y) is a function field. ThusK 2M (F) is an extension of the tame kernel by a torsion group.