The BlochKato Conjecture for the Riemann Zeta Function (London Mathematical Society Lecture Note Series, Series Number 418)

"There are still many arithmetic mysteries surrounding the values of the Riemann zeta function at the odd positive integers greater than one. For example, the matter of their irrationality, let alone transcendence, remains largely unknown. However, by extending ideas of Garland, Borel proved that these values are related to the higher K-theory of the ring of integers. Shortly afterwards, Bloch and Kato proposed a Tamagawa number-type conjecture for these values, and showed that it would follow from a result in motivic cohomology which was unknown at the time. This vital result from motivic cohomology was subsequently proven by Huber, Kings, and Wildeshaus. Bringing together key results from K-theory, motivic cohomology, and Iwasawa theory, this book is the first to give a complete proof, accessible to graduate students, of the Bloch-Kato conjecture for odd positive integers. It includes a new account of the results from motivic cohomology by Huber and Kings."--Publisher Cover 1 Series information 3 Title page 5 Copyright information 6 Table of contents 7 List of contributors 9 Preface 10 1 Special Values of the Riemann Zeta Function: Some Results and Conjectures 13 1.1 Values of the Riemann zeta-function and K-groups of mathbb Z 13 1.1.1 Definition and basic analytic properties of zeta(s) 13 1.1.2 Euler’s Theorem 15 1.1.3 Borel’s Theorem 17 1.1.4 K-groups of mathbb Z 19 1.1.5 Lichtenbaum’s conjecture 21 1.2 The Bloch–Kato conjecture for the Riemann zeta-function 23 1.2.1 The Riemann zeta function as a motivic L-function 24 1.2.2 Tamagawa number conjecture of Bloch and Kato 28 1.2.3 Results on the Tamagawa number conjecture for the motive mathbb Q(n) 31 References 32 2 K-theoretic Background 34 2.1 Quillen’s K-theory 34 2.2 K[sub(i)] (R) for small i 36 2.3 K-theory of exact categories 40 2.4 K-theory with nite coefficients 42 2.5 Hurewicz homomorphisms 45 2.6 Étale cohomology and equivariant cohomology 46 2.7 K-groups of rings of integers in number fields 50 2.8 Chern classes á la Grothendieck 51 References 54 3 Values of the Riemann Zeta Function at the Odd Positive Integers and Iwasawa Theory 57 3.1 Introduction 57 3.2 Notation 58 3.3 The Poitou–Tate sequence 58 3.4 The comparison diagram 63 3.5 Theorem of Soulé 67 3.6 The Bloch–Kato conjecture 71 References 75 4 Explicit Reciprocity Law of Bloch–Kato and Exponential Maps 77 4.1 Introduction 77 4.2 The classical explicit reciprocity law 78 4.3 Fontaine’s rings: B[sub(cris)] and B[sub(dR)] 80 4.3.1 Witt vectors 81 4.3.2 Witt vectors over perfection of mathcal O[sub(overline mathbb Qsub[(p)])]/p mathcal O[sub(overline mathbb Qsub[(p)])] 82 4.3.3 The rings B[sub(dR)] [sup(+)] B[sub(dR) ] 84 4.3.4 The rings A[sub(cris)] and B[sub(cris)] [sup(+)] 86 4.3.5 Ideals of A[sub(cris)] 88 4.3.6 The fundamental commutative diagram 89 4.4 Explicit reciprocity law of Bloch and Kato 90 4.4.1 Description of H[sup(1)] (K, mathbb Q[sub(p)] (r)) 91 4.4.2 The Coates–Wiles homomorphism phi [sub(CW)] [sup(r)] for a positive integer r 92 4.4.3 The explicit reciprocity law 92 4.4.4 Proof of the key proposition 4.4.2 94 4.5 The exponential and the dual exponential maps 97 4.5.1 Galois cohomology of B[sub(dR)] 98 4.5.2 The dual exponential 99 4.5.3 The exponential map 100 4.5.4 Tate duality 100 4.5.5 The relation between exp and exp* 101 4.5.6 Relation between exp and Coates–Wiles homomorphism 103 4.5.7 The image of Soulé–Deligne elements under the dual exponential 104 References 107 5 The Norm Residue Theorem and the Quillen-Lichtenbaum Conjecture 109 5.1 Introduction 109 5.2 Background 110 5.3 Merkurjev–Suslin 113 5.4 Motivic cohomology 118 5.5 Voevodsky 121 5.6 Quillen–Lichtenbaum 124 References 129 6 Regulators and Zeta-functions 133 6.1 Introduction 133 6.2 Regulators 134 References 140 7 Soulé’s Theorem 142 7.1 Introduction 142 7.2 Proof of Theorem A 143 7.3 Proof of a lemma of Soulé 146 7.4 The theorem of Dwyer and Friedlander 148 References 150 8 Soulé’s Regulator Map 152 8.1 Introduction 152 8.2 The Z[sub(p)]-ranks of H[sup(1)] (Q[sub(S)]/Q, Z[sub(p)] (i)) and H[sub(2)] (Q[sub(S)]/Q, Z[sub(p)] (i)) 153 8.3 Some results from Iwasawa theory 156 8.4 The proof of Soulé’s theorem 159 References 164 9 On the Determinantal Approach to the Tamagawa Number Conjecture 166 9.1 Motives and avatars 167 9.1.1 167 9.2 mathfrak l-adic representations and the fundamental line 169 9.2.1 A hierarchy of mathfrak l-adic representations 169 9.2.2 The f-cohomology 170 9.2.3 Pseudo-geometric representations and the fundamental line 172 9.2.4 Local Tamagawa numbers 175 9.2.5 The fundamental line and Galois cohomology 176 9.3 Motivic cohomology and conjectures on special values 180 9.3.1 Mixed Hodge structures 180 9.3.2 Motivic structures and L-functions 181 9.3.3 Formulation of the conjecture 183 9.4 Bloch and Kato versus Fontaine and Perrin–Riou 186 9.4.1 Regulators of Tate motives and archimedean Tamagawa numbers 186 9.4.2 Global Tamagawa numbers 189 9.4.3 Shafarevich–Tate groups. 192 9.5 The TNC for the Tate motives over abelian number elds 196 9.5.1 Reformulation of the conjecture 197 9.5.2 First step 197 9.5.3 Second step: The Lichtenbaum conjecture 199 9.5.3.1 Analytic part 199 9.5.3.2 Special motivic elements 200 9.5.3.3 Going to the archimedean world via the Beilinson regulator 201 9.5.3.4 Going to the p-adic world via the étale regulator 201 References 203 10 Motivic Polylogarithm and Related Classes 205 10.1 Introduction 205 10.2 Motivic polylogarithm classes 207 10.2.1 Motivic cohomology 207 10.2.2 Basic schemes 207 10.2.3 Sign character 208 10.2.4 The operator tr[sub([a])] 208 10.2.5 Residue sequence 209 10.2.6 Residue isomorphism 210 10.2.7 Base changed version 211 10.2.8 Definition of motivic polylogarithm 211 10.3 Motivic Eisenstein classes 211 10.3.1 Definition of x* mathcal Pol[sub(mathcal M)] 212 10.3.2 A key eigenspace 212 10.3.3 Splitting lemma and consequence 212 10.3.4 Contraction 213 10.3.5 Motivic Eisenstein class 213 10.4 Comparison of motivic, l-adic and archimedean classes 214 10.4.1 l-adic polylog 214 10.4.2 Alternative construction of mathcal Pol[sup((k))] [sub(l)] as parallel to the construction of mathcal Pol[sup((k))] [sub(mathcal M)] 215 10.4.3 l-adic regulator r[sub(l)] 217 10.4.4 l-adic Eisenstein class 217 10.4.5 Another construction of Eis[sub(l)] [sup(k)] and proof of Theorem 1 218 10.4.6 Eisenstein symbol 220 References 220 11 The Comparison Theorem for the Soulé–Deligne Classes 222 11.1 Introduction 222 11.1.1 Statement of the Main Theorem 222 11.1.2 Proofs of the Main Theorem 224 11.1.3 Structure of the paper 226 11.2 Notation 226 11.3 The elliptic polylogarithm 228 11.3.1 Logarithm 228 11.3.2 Polylogarithm 231 11.4 Eisenstein classes 233 11.4.1 Geometric setup 233 11.4.2 The l-adic Eisenstein class 234 11.4.3 The residue of the Eisenstein class 235 11.4.4 Rigidity 236 11.5 The cup-product construction 237 11.5.1 The motivic cup-product construction 238 11.5.2 Sheaf theoretic interpretation 239 11.5.3 The Main Theorem, l-adic part 241 11.6 The Hodge theoretic story 244 11.6.1 The construction 244 11.6.2 The Main Theorem, Hodge theoretic part 246 11.7 The case of the Riemann zeta-function 247 References 248 12 Eisenstein Classes, Elliptic Soulé Elements and the ell–Adic Elliptic Polylogarithm 251 Introduction 251 Notations 253 12.1 Statement of the main results 255 12.1.1 The residue at infty of the Eisenstein class 255 12.1.2 Evaluation of the cup-product construction 255 12.1.3 Eisenstein classes and elliptic units 257 12.2 Sheaves of Iwasawa modules and the moment map 259 12.2.1 Iwasawa algebras 259 12.2.2 The moment map 260 12.2.3 Étale sheaves of Iwasawa modules 265 12.2.4 The case of torsors 266 12.2.5 The shea ed moment map 267 12.2.6 Soulé’s twisting construction and the moment map 269 12.3 Three examples 272 12.3.1 The Bernoulli measure and its moments 272 12.3.2 Modified cyclotomic Soulé–Deligne elements 273 12.3.3 Elliptic Soulé elements 276 12.4 Eisenstein classes, elliptic Soulé elements and the integral ell-adic elliptic polylogarithm 279 12.4.1 A brief review of the elliptic logarithm sheaf 280 12.4.2 The elliptic polylogarithm and Eisenstein classes 281 12.4.3 A variant of the elliptic polylogarithm 282 12.4.4 The variant of the elliptic polylogarithm and Eisenstein classes 284 12.4.5 Sheaves of Iwasawa modules and the elliptic logarithm sheaf 285 12.4.6 The elliptic polylogarithm and elliptic units 289 12.4.7 Eisenstein classes and elliptic Soulé elements 291 12.5 The residue at infty of the elliptic Soulé elements 291 12.5.1 Definition of the residue at infty 291 12.5.2 Computation of the residue at infty of the elliptic Soulé element 295 12.6 The evaluation of the cup-product construction for elliptic Soulé elements 298 12.6.1 A different description of the cup-product construction 298 12.6.2 The auxiliary class mathcal B mathcal S[sup(langle t rangle)][sub(c)] 299 12.6.3 Evaluation at infty of the modi ed elliptic Soulé element 301 12.6.4 Proof of Theorem 12.6.9 304 References 307 13 Postscript 309 13.1 Leitfaden 309 13.2 Tate–Shafarevich groups 311 13.3 The case of a totally real base field 314 References 316 Edited By John Coates, A. Raghuram, Anupan Saikia, And R. Sujatha. These Are The Proceedings Of A Week-long Workshop Entitled 'the Bloch-kato Conjectures For The Riemann Zeta Function At The Odd Positive Integers', Which Was Held At The Indian Institute Of Science Education And Research (iiser), Pune, India, In July 2012--preface. Includes Bibliographical References. An account of a significant body of recent work that resolves some long-standing mysteries concerning special values of the Riemann zeta function. It brings together many important results from K-theory, motivic cohomology, and Iwasawa theory, accessible at graduate level and above.