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Index

absolute Galois group ............ 261 augmentation representation ...... 520 absolute norm auj3erwesentliche Diskriminanten-- of an ideal .................... 34 teiler ........................ 207

- of an ide1e ................... 361 - of a prime (place) ............. 184 basis

- of a replete ideal .............. 186 - basis of a lattice ............... 24

absolute value - discriminant of a basis .......... 11

- p-adic ....................... 107 - integral basis .................. 12

- product formula ..... 108, 109, 185 battle of Hastings ................. 44

abstract Galois theory ............ 275 Bauer, M., theorem of ............ 548

abstract valuation theory ......... 284 Beilinson conjecture ........ 432,443

admissible mono/epimorphism ... 231 Bernoulli numbers ........... 38,427

adele ........................... 357 - generalized .............. 441, 515

affine scheme .................... 88 Bernoulli polynomial ... 433, 443, 511

algebraic number field ............. 5 big Hilbert class field ............ 399 algebraic numbers ................. 5 Bloch, S ........................ 432 analytic class number formula .... 468 Borel, A. ....................... 432

approximation theorem .......... 117 Brauer, theorem of .............. 522 - strong approximation theorem .. 193 Brumer, A. . .................... 394 Arakelov class group ............ 190 Bruckner,H ................ 338,417 Arakelov divisor ................ 189 Bruckner, theorem of ............ 339 archimedean valuation ........... 118 arithmetic algebraic geometry .... 193 canonical divisor ................ 209 arithmetic progression ... 64,469,545 canonical measure Artin conductor ............. 527, 533 - on Minkowski space ........... 29 -local Artin conductor ......... 532 -on~ ........................ 446 Artin conjecture ................. 525 -onlR~ ....................... 454 Artin, E. . .................. 406,413 canonical metric .................. 29 Artin-Hasse, theorem of .......... 339 canonical module Artin L-series ................... 518 - of a Riemann surface .......... 209 - Artin and Hecke L-series ...... 539 - of a metrized number field ..... 222 - completed Artin L-series ...... 537 - functional equation ....... 540, 541

- of a number field ............. 219 Cebotarev density theorem ....... 545

- zeroes of .................... 541 central function ................. 519 Artin reciprocity law ........ 390,407 centrally symmetric ............... 26 Artin-Schreier theory ............ 281 character Artin symbol ................... 407 - character group .......... 273, 280 associated ........................ 3 - Chern character .......... 244, 246 augmentation ideal - conductor of ........ 434,473,478 - of the Grothendieck ring ...... 243 - Dirichlet character ....... 434,478 - of a group ring ............... 410 - exponent p of ................ 435

560

- GroBencharacter ......... 436, 470 - Hecke character .............. 480 - induced character ............. 521 - irreducible character .......... 520 - primitive Dirichlet character ... 434 - primitive GroBencharacter ..... 472 - principal character ....... 435, 520 - of a representation ............ 519 - trivial character .............. 434 characteristic - Euler-Minkowski .... 212,256,258 - Euler-Poincare ............... 209 characteristic polynomial of a field ele-

ment .......................... 9 Chern character ............. 244, 246 Chern class ..................... 244 Chevalley, C .................... 357 Chinese remainder theorem ........ 21 Chow group ................. 83, 190 Chow theory .................... 193 class field ...................... 304 - big Hilbert class field ......... 399 - global class field .............. 395 - Hilbert class field ........ 399,402 -local class field ............... 322 - problem of class field tower .... 413 - ray class field ................ 396 class field axiom ................ 299 - global class field axiom ....... 383 -local class field axiom ......... 317 class field theory ................ 300 - existence theorem ........ 322, 396 - global ................... 357, 390 - higher-dimensional ........... 310 -local .................... 317,320 - p-class field theory .. 298, 326, 332 - tautological .................. 306 class function ................... 519 class group ...................... 22 - Arakelov class group .......... 190 - connected component of idele class

group ....................... 368 - divisor class group ............. 83 - ideal class group ............... 22 - idele class group .............. 359 - of an order .................... 82

Index

- S-class group ................. 71 - ray class group ........... 363, 365 - replete divisor class group ..... 190 - replete ideal class group ....... 186 class number .............. 34, 36, 81 - finiteness of ................ 36, 81 class number formula ............ 468 closedness relation .......... 108, 185 closure, integral ................... 7 coboundary ..................... 282 cocycle ......................... 282 cohomological dimension ........ 306 cohomology .................... 284 Coleman's norm operator ........ 351 compact - compact group ............... 269 - Pic(B)o is compact ........... 193 compactified Grothendieck group . 233 complementary module (Dedekind) 195 completed L-series . .437,499,503,537 completed zeta function ..... 422,466 - functional equation ....... 425, 466 complete lattice .................. 24 complete valued field ........ 123,131 completely split .................. 49 completion - of a G-modulation ., .......... 308 - profinite completion .......... 274 - of a valued field .............. 123 complex multiplication .......... 402 complex prime .................. 183 conductor ........... 47,79,323,397 - Artin conductor ..... 527, 532, 533 - conductor-discriminant formula 534 - of a Dirichlet character ... 434, 478 - of a GroBencharacter .......... 473 congruence subgroup ............ 363 conjectures - Artin conjecture .............. 525 - Beilinson conjecture ...... 432, 443 -Fermat conjecture .......... 37,38 - Leopoldt conjecture ........... 394 - Lichtenbaum conjecture ....... 516 - Mordell conjecture ........... 207 - Riemann hypothesis .......... 432 - Safarevic conjecture .......... 207

Index 561

- Taniyama-Shimura-Weil ....... 38 Dedekind, R. ..................... 17 conjugate degree - embeddings .................. 161 - of a divisor .................... 96 - prime ideals ................... 53 - of a replete divisor ............ 190 conjugation - of a replete ideal .............. 213 -one ........................ 444 - of a representation ............ 519 -onKe ....................... 226 degree map ..................... 190 connected component of the ide1e class degree valuation .................. 95

group ....................... 368 Deninger, C. .................... 542 convex .......................... 26 density cotangent element ............... 255 - Dirichlet density .............. 542 covering ..................... 92, 93 - natural density ............... 543 - ramified ...................... 92 density theorem - universal ...................... 93 - of Cebotarev ................. 545 covering transformation ........... 93 - of Dirichlet .................. 543 critical strip .................... .432 derivation ...................... 200 crossed homomorphism .......... 282 determinant of a metrized v-module cycle ........................... 193 231,245 cyclotomic different ............... 195,201,254 - Z-extension ............. 385, 386 - of an element ................ 197 - Zp-extension ............ 326, 386 - of a metrized number field ..... 224 cyclotomic field ..... 58, 158,273,398 differential module .......... 200, 254 - generalized cyclotomic fields .. 348 differentials ................ 200, 341 - prime factorization ............. 61 diophantine equation ............ 104 cyclotomic polynomial ....... 59,166 direct (inductive) limit ........... 266 - generalized .................. 348 Dirichlet L-series ........... 435, 496 cyclotomic units .................. 44 - completed ................... 437

- functional equation ........... 440 decomposition field - special values of ..... 442, 443, 515 - of a microprime .............. 290 - zeroes of .................... 442 - of a prime ideal ................ 54 Dirichlet character .......... 434,478 - of a valuation ................ 171 Dirichlet density ................ 542 decomposition group Dirichlet, G.P. Lejeune ............ 42 - of a microprime .............. 290 - density theorem .............. 543 - of a prime ideal ................ 54 - prime number theorem 64,469,543 - of a valuation ................ 167 - unit theorem ........... 42,81,358 decomposition law .............. 409 Dirichlet-character, conductor of .. 434, -of prime numbers ......... 61,409 478 - of prime numbers in the cyclotomic discrete subgroup ................. 24

field .......................... 61 discrete valuation ............ 67,121 - of prime numbers in the ring of - of a function field .............. 95

gaussian integers ............... 4 discriminant ............ 49,201,251 Dedekind domain ................ 18 - auj3erwesentliche Diskriminanten-Dedekind zeta function .......... 457 teiler ........................ 207 - functional equation ........... 467 - of a basis ..................... 11 Dedekind's complementary module 195 - bound ................... 204, 223

562

- conductor-discriminant formula 534 - of an element .................. 11 - of an ideal .................... 14 - of a number field .............. 15 -Minkowski's theorem on ... 38,207 - Stickelberger's relation ......... 15 discriminant and different ........ 201 distribution of prime numbers .... .432 division points of a formal group .. 347 divisor .......................... 82 - Arakelov divisor .............. 189 - Arakelov divisor class group ... 190 - canonical divisor ............. 209 - Chow group ................... 83 - degree of a divisor ............. 96 - divisor class group ............. 83 - divisor group .............. 82, 95 - group of replete divisors ....... 189 - group of replete divisor classes . 190 - principal divisor ............... 83 - replete divisor ................ 189 - replete principal divisor ....... 189 double cosets ................. 55, 58 double functor .................. 307 Dress, A. ....................... 307 duality - dual ideal .................... 194 - Pontryagin dual .............. 273 - Serre duality ................. 209 - Tate duality .............. 326, 404 duplication formula, Legendre's .. 421,

456 Dwork, B. .................. 298, 332 Dwork, theorem of .............. 332

elliptic curve .................... 402 equidistributed prime ideals ...... 545 equivalent representations ........ 519 equivalent valuations ............ 116 etale topology ................ 90, 93 Euler factors at infinity .. 459,527,535,

541 Euler product ............... 419,435 Euler's identity ............. 419,435 Euler, L. .................... 64, 431

Index

Euler-Minkowski characteristic ... 212, 256,258

Euler-Poincare characteristic ..... 209 existence theorem of class field theory

322,396 expansion - row-column expansion .......... 6 - p-adic expansion ..... 99,101,106 exponent - exponent p of an character ..... 435 - of an operator SJ .............. 278 - of a GroBencharacter ......... .478 exponential function, p-adic ...... 137 exponential valuation 69,107,120,184 extension of a valuation ...... 161, 163 - of a henselian field ....... 144, 147 - of a complete field ............ 131

factorial ring ...................... 7 Faltings, theorem of (Mordell conjec-

ture) ........................ 207 Fermat's last theorem .......... 37,38 Fesenko, I. ...................... 310 Fibonaccinumbers ............... 53 finite prime (place) .............. 183 finiteness of class number ...... 36, 81 Fontaine, I.M., theorem of ........ 207 formal v-module ................ 343 formal group .................... 342 -logarithm of ............. 343, 345 - division points of ............. 347 Fourier transform ............... 446 fractional ideal ................... 21 Frey,G .......................... 38 Frobenius - abstract ................. 285,287 - automorphism ........ 58, 286, 406 - correspondence .............. 226 -reciprocity ................... 521 - on Witt vectors ............... 134 function field .................... 94 functional equation - Artin L-series ........... 540, 541 - Dedekind zeta function ........ 467 - Dirichlet L-series ............. 440 - Hecke L-series .......... 502, 503

Index 563

- Mellin transform ............. 422 -axiom ....................... 383 - Riemann's zeta function .. 425,426 - theory .................. 357, 390 fundamental group ............... 93 global field ..................... 134 - of a G-set .................... 307 global norm residue symbol ...... 391 fundamental identity global reciprocity law ............ 390 - for prime ideals ............... .46 global Tate duality .............. .404 - of valuation theory ... 150, 155, 165 Golod-Safarevic ................ .413 fundamental mesh GroBencharacter ............ 436,470 - of a lattice .................... 24 - exponent of .................. 478 - regulator ............ .43,431,443 - conductor of ................. 473 - volume of the fundamental mesh 26 - primitive .................... 472 - volume of the fundamental mesh of -type of ...................... 478

the unit lattice ................. 43 Grothendieck, A. ........... 225,253 - volume of the fundamental mesh of Grothendieek group, replete ...... 233

anideal ....................... 31 Grothendieek-Riemann-Roch ..... 254 - volume of the fundamental mesh of

a replete ideal ................ 212 group cohomology .............. 284 Grunwald, theorem of ........... .405

fundamental units ................ 42 Furtwangler, P .............. 406,413

Gysin map ...................... 253

Haar measure Gab (maximal abelian quotient) ... 265, - on a p-adic number field ....... 142

274 -onlR. ........................ 446 G-modulation .................. 307 Hasse-Arf, theorem of ....... 355, 530 G-module ...................... 276 Hasse norm theorem ............. 384 -induced ................. 312,374 Hasse-Minkowski, theorem of .... 385 Galois descent .................. 372 Hasse's Zahlbericht ............. 363 Galois group, absolute ........... 261 Heeke character ................. 480 Galois theory Hecke L-series ......... 493, 496, 497 -abstract .................... .'.275 - completed .............. .499,503 - infinite ...................... 261 - functional equation ....... 502, 503 - of valuations ................. 166 - Heeke and Artin L-series ...... 539 gamma function ................ .421 - partial ...................... .496 - higher-dimensional ........... 454 Hecke theta series ............... 489 Gauss sum .......... 51,438, 473, 488 - partial ....................... 489 Gauss's reciprocity law .... 51,64,416 - transformation formula ........ 490 gaussian integers .................. 1 Hensel, K. ....................... 99 gaussian prime numbers ............ 3 Hensel's lemma ............. 129,148 gaussian units ..................... 3 henselian general reciprocity law ........... 300 - field .................... 143,147 generalized cyclotomic theory .... 346 -local ring .................... 152 generic point ..................... 86 - valuation ....... 143,288,309,389 genus - P-valuation .................. 298 - of a number field ......... 214,467 henselization ................... 143 - of a Riemann surface .......... 209 Herbrand ghost components of Witt vectors. 134 -quotient ................. 312, 378 global class field ................ 395 - theorem of ................... 180

564

Hermite, theorem of ............. 206 hermitian scalar product .. 28, 226,444 higher - ramification groups ....... 176, 352 - unit group ................... 122 higher-dimensional - class field theory ............. 310 - gamma function .............. 454 -logarithm .................... 445 Hilbert 90 ......... 278,281,283,284 Hilbert, D ........................ 53 Hilbert class field ....... 399,400,402 Hilbert-Noether, theorem of ...... 283 Hilbert's ramification theory .. 53, 166 Hilbert symbol ......... 305, 333,414 - explicit ...................... 339 - product formula .............. 414 -tame ........................ 335 Hurwitz formula ................ 220

ideal ............................ 16 - absolute norm of ............... 34 - degree of a replete ideal ....... 213 - discriminant of ................ 14 -dual ......................... 194 - fractional ..................... 21 - integral ....................... 21 -invertible ..................... 74 -normof ..................... 186 - principal ideal theorem ........ 410 - replete principal ideal ......... 186 - replete ideal .................. 185 - volume of fundamental mesh .... 31,

212 ideal class group ............. 22, 186 -replete ...................... 186 ideal group .................. 21,408 - defined mod m .............. .408 ideal number ................ 16, 486 idele ........................... 357 - absolute norm of ............. 361 - idele class group .............. 359 - idele group .................. 357 - norm of ..................... 370 - principal ..................... 359 - S-idele ...................... 358

Index

imaginary quadratic field ......... 402 index of specialty ............... 218 induced - character .................... 521 - G-module .......... 312,374,521 - representation ................ 521 inductive limit .................. 266 inductive system ................ 265 inertia degree ........ 46,49,184,285 - abstract ................. 285,309 - of a metrized number field ..... 224 - of a prime ideal ............ 46, 49 - of a primes (place) ............ 184 - of a valuation ............ 150, 165 inertia field - of a prime ideal ................ 57 - of a valuation ................ 173 inertia group - abstract ...................... 285 - of a prime ideal ................ 57 - of a valuation ................ 168 infinite Galois theory ............ 261 infinite prime ................... 183 infinite prime number ............ 184 integer - algebraic ....................... 5 - gaussian ....................... 1 - p-adic .............. 100, 104, 111 integral -basis ......................... 12 -closure ........................ 7 -ideal ......................... 21 - integrally closed ................ 7 - ring extension .................. 6 inverse different ................. 195 invertible ideal ................... 74 invertible o-module ......... 229,230 irreducible character ............. 520 irreducible representation ........ 519 irregular prime number ............ 38 isometric ....................... 229 Iwasawa, K. ..................... 37 Iwasawa theory .................. 63

Jacobisymbol .................. 417 Jacobi's theta series ......... 422,424

Index

J annsen, U. ..................... 221 j-invariant ..................... .402

Kahler differentials .............. 200 Kahler, E. . ..................... 200 Kato, K. ................... 310, 432 K I K, maximal unramified extension

154,285 Krasner's lemma ................ 152 Kronecker's lugendtraum ........ 401 Kronecker's programme ......... 548 Kronecker-Weber theorem ... 324, 398 Krull dimension .................. 73 Krull topology .............. 167,262 Krull valuation .................. 123 Krull-Akizuki, theorem of ......... 77 K-theory .............. 193,310,431 Kummer extension .......... 278, 380 Kummer theory ........ 277,279,340 Kummer, E ................... 16,38 Ktirschak, J. .................... 107

L-function of a G(ClIR)-set ...... 455 L-series - Artin L-series ................ 518 - Artin and Hecke L-series ...... 539 - completed Artin L-series ...... 537 - completed Dirichlet L-series ... 437 - completed Hecke L-series .499,503 - Dirichlet L-series ........ 435,496 - functional equation .. 440, 502, 540 - Hecke L-series .......... 493, 496 - p-adic L-series ............... 516 - partial L-series .............. .496 Langlands philosophy ........... 549 lattice ......................... 2, 23 - basis of ....................... 24 - complete lattice ............... 24 - fundamental mesh ............. 24 - Minkowski's lattice point theorem

27 - unit lattice .................... 40 - volume of fundamental mesh .... 26 - Z-structure .................... 24 Legendre's duplication formula ... 421,

456

565

Legendresymbol ............ 50,336 lemma (renowned lemmas) - Hensel's lemma .............. 129 - Krasner's lemma ............. 152 - Nakayama's lemma ............ 72 - snake lemma .................. 79 length of a module ................ 82 Leopoldt conjecture ............. 394 Lichtenbaum conjecture .......... 516 limit - inductive (direct) ............. 266 - projective ............... 103, 266 line bundle ................. 208,255 local class field .................. 322 -axiom ....................... 317 - theory ....................... 317 local field ...................... 134 - 2-1ocal field .................. 310 localization ................... 65,71 - of a valued field .............. 160 local norm residue symbol ....... 321 local reciprocity law ............. 320 local ring ........................ 66 local-to-global principle ..... 161, 357,

384,385,391 logarithm - of a formal group ......... 343, 345 - higher-dimensional ........... 445 - p-adic .................. 136,142 Lubin-Tate - extension .................... 348 - module ...................... 343 - series ....................... 328

Mackey functor ................. 307 maximal -order ......................... 72 - tamely ramified extension ..... 157 - unramified extension ..... 154, 285 - unramified extension of Qp .... 176 -unramified extension oflFp((t)) 176 measure - Haar measure on a p-adic number

field ......................... 142 - Haar measure on R ........... 446 - Minkowski measure .......... 221

566 Index

measure, canonical norm ............................. 8 - on Minkowski space ........... 29 - absolute ......... 34,184,186,361 -onlR ........................ 446 -onC ........................ 444 -onlR~ ....................... 454 - Coleman's norm operator ...... 351 Mellin principle ................ .423 - of a gaussian integer ............ 2 Mellin transform ................ 422 - Hasse norm theorem .......... 384 metric - of an ideal ................... 186 - canonical on Minkowski space .. 29 - of an idele ................... 370 - hermitian .................... 226 - universal norms .......... 304, 308 - Minkowski .................... 31 normalization ............... 7, 76, 91 - standard .................. 28, 228 normalized valuation ........ 121, 184 - trivial ................... 227, 229 norm-one surface ............ 39,460 metrized norm residue group .............. 277 - number field ................. 222 norm residue symbol ............ 302 - v-module ................... 227 -global ....................... 391 - projective resolution .......... 234 -local ........................ 321 microprime ................ 290, 299 -overQp ..................... 331 Minkowski, H. . .................. 24 - product formula .............. 393 -bound ........................ 34 norm theorem, of Hasse .......... 384 - lattice point theorem ........... 27 norm topology .................. 303 - measure ..................... 221 n-th ramification group .......... 176 -metric ........................ 31 number field - space .................... 29,444 - algebraic ....................... 5 - theorem on discriminant ... 38, 207 - discriminant of ................ 15 - theorem on linear forms ........ 28 - genus of ................. 214,467 -theory ........................ 28 - imaginary quadratic .......... .402 - theory, multiplicative version ... 32 - metrized ..................... 222 Minkowski-Hasse, theorem of .... 385 - p-adic ....................... 136 Mobius function ................ 474 - quadratic ..................... 50 Mobius inversion formula ........ 484 numbers modular form ................... 434 - algebraic ....................... 5 modular function ............... .402 - Bernoulli ........ 38,427,441,515 modulation ..................... 307 - Fibonacci ..................... 53 module of definition ............. 407 - ideal ..................... 16, 486 module m ...................... 363 - p-adic .............. 100, 101, 111 - of a Hecke character .......... 480 - p-adic .................. 128, 136 monogenous ring extension ....... 178 Mordell conjecture .............. 207 Odlyzko, A.M ................... 223 mUltiplicity of a representation .... 519 order of a number field ............ 72

- maximal ...................... 72 Nakayama's lemma ............... 72 Ostrowski, theorem of ........... 124 Nart, E ......................... 149 natural density .................. 543 p-adic Newton polygon ................ 144 - absolute value ................ 107 Noether, E. ..................... 282 - expansion ............ 99,101,114 nonarchimedean valuation ........ 118 - exponential valuation ......... 107

Index 567

- L-series ..................... 516 primitive -numbers ........ 100,101,111,271 - character ................ 434, 472 - periodic p-adic expansion ..... 106 - polynomial .................. 129 - unit rank ..................... 394 - root of unity ................... 59 -units ........................ 112 principal character .......... 435, 520 - valuation ..................... 69 principal divisor .................. 83 - Weierstrass preparation theorem 116 - replete ...................... 189 - zeta function ................. 516 principal ideal, replete ........... 186 p-adic principal ideal theorem .......... .410 - exponential function .......... 137 principal idele .................. 359 -logarithm ............... 136, 142 principal units .................. 122 - number field ................. 136 procyclic groups ................ 273 -numbers ............ 128, 136,271 product formula partial - for absolute values ... 108, 109, 185 - Heeke theta series ............ 489 - for the Hilbert symbol ......... 414 - L-series ..................... 496 - for the norm residue symbol ... 393 - zeta function ................. 458 product, restricted ............... 357 p-c1ass field theory ..... 298, 326, 332 profinite completion ............. 274 Pell's equation ................ 43,84 profinite group .................. 264 periodic p-adic expansion ........ 106 projection formula ............... 248 s;:>-function,ofWeierstrass ....... .401 projective Picard group ................. 75,185 -limit .................... 103, 266 - replete .................. 186, 239 -line .......................... 97 place ........................... 183 - v-module ................... 228 Poincare homomorphism .... 234,237 - resolution .................... 234 Poisson summation formula ...... 447 - system ...................... 266 polyhedric cone ................. 504 Priifer ring Z .................... 272 Pontryagin dual ................. 273 p-Sylow subgroup ............... 274 power residue symbol ....... 336,415 purely ramified .......... 49,158,286 power series field ........... 127,136 pythagorean triples ................ 5 power sum ................. 433, 443 preparation theorem (Weierstrass) .. 116 quadratic number field ............ 50 presheaf ......................... 87 quadratic residue ................. 50 prime (place) quadratic residue symbol ..... 50, 417 - absolute norm of ............. 184 - complex ..................... 183 ramification field ................ 175 - finite ........................ 183 ramification group ............... 168 - infinite ...................... 183 - Herbrand's theorem ........... 180 - microprime .............. 290, 299 - higher .................. 176, 352 -real ......................... 183 - upper numbering of ........... 180 prime decomposition ......... 18, 409 ramification index

- in the cyclotomic field .......... 61 - abstract ................. 285, 309

- of gaussian integers ............. 1 - of metrized number fields ..... 224 prime element .............. 121,289 - of prime ideals ................ 45 prime number theorem, of Dirichlet 64, - of primes (places) ............ 184

469,543 - of valuations ............. 150, 165

568

ramification points ................ 92 ramification theory - Hilbert's ...................... 53 - higher .................. 176, 354 - of valued fields ............... 166 ramified ......................... 49 - covering ...................... 92 -tamely ...................... 154 -totally ............... 49, 158,286 - wildly ....................... 158 rank of coherent a-module ... 229,243 rationally equivalent .............. 83 ray class field .......... 366, 396,403 ray class group ............. 363, 365 real prime ...................... 183 reciprocity, Frobenius ............ 521 reciprocity homomorphism ....... 294 reciprocity law - Artin reciprocity law ..... 390, 407 - for power residues ............ 415 -Gauss'sreciprocitylaw .... 51,416 - general ...................... 300 - global reciprocity law ......... 390 -local reciprocity law .......... 320 reciprocity map ................. 291 regular prime ideal ............... 79 regular prime number ............. 38 regular representation ............ 520 regulator ............... .43,431,443 replete - divisor ...................... 189 - divisor class group ............ 190 - Grothendieck group ........... 233 - Picard group ................. 186 - principal divisor .............. 189 - principal ideal ................ 186 replete ideal .................... 185 - absolute norm of ............. 186 - degree of .................... 213 - replete principal ideal ......... 186 - volume of fundamental mesh .. 212 replete ideal class group .......... 186 representation of a group ......... 518 - augmentation representation ... 520 - character of ......... , ........ 519 -degree ........ , .......... , ..... 519

Index

- equivalent representations ..... 519 -induced ..................... 521 - irreducible ................... 519 - multiplicity of ................ 519 - regular ...................... 520 residue - of L-series .................. .423 - of a p-adic differential ........ 341 - of zeta functions ............. .425 residue class field ............... 121 restricted product ................ 357 Ribet, K. ........................ 38 Riemann,B. - hypothesis ................... 432 - surface ...................... 208 Riemann's zeta function ......... .419 - completed ............... 422, 466 - Euler's identity .......... 419, 435 - functional equation ....... 425, 426 - special values of ..... 427,431,432 - trivial zeroes ................. 432 Riemann-Hurwitz formula ... 220, 221,

224 Riemann-Roch - for number fields .... 213,214,218 - Grothendieck-Riemann-Roch for

number fields ................ 254 - theorem of ................... 209 ring - of adeles ..................... 357 - Dedekind domain .............. 18 - factorial ....................... 7 -henselian .................... 152 - henselian valuation ring ....... 143 -local ring ..................... 66 - of p-adic integers .... 104, 111,271

- Priifer ring Z ................. 272 - valuation ring ................ 121 - of Witt vectors ........... 134, 283 row-column expansion ............. 6

Safarevic conjecture ............. 207 Safarevic, I.R. .................. .413 scheme ...................... 88, 96 Schmidt, F.K. ................ 58, 152 Schwartz function ............... 446

Index 569

S -class group .................... 71 - maximal extension of Qp ...... 176 section of a sheaf ................. 87 Tamme, G .................. 225,240 Serre duality ............... 209,214 Taniyama-Shimura-Weil conjecture 38 sesquilinear ..................... 226 Tate duality ................ 326,404 sheaf ............................ 88 Tate's thesis .................... 503 - presheaf ...................... 87 tautological class field theory ..... 306 - section of ..................... 87 Taylor, R. ........................ 38 - stalk of ....................... 88 theorem (renowned theorems) - structure sheaf ................. 88 - Artin-Hasse .................. 339 Shintani's unit theorem .......... 507 - Artin reciprocity law .......... 390 S-idele ......................... 358 - Brauer ...................... 522 Siegel-Klingen, theorem of ....... 515 - Bruckner .................... 339 simplicial cone .................. 504 - Cebotarev ................... 545 singularity .................... 73,91 - Dirichlet density theorem ...... 543 - resolution of singularities ....... 91 - Dirichlet unit theorem .. 42, 81, 358 snake lemma ..................... 79 - Dirichlet prime number theorem solenoid ........................ 368 469 spectrum of a ring ................ 85 -Dwork ...................... 332 stalk of a sheaf ................... 88 - extension theorem (for valuations) standard metric .............. 28, 228 161,163 Stickelberger's discriminant relation 15 - Faltings (Mordell conjecture) .. 207 Stirling's formula ............... 206 - F.K. Schmidt (henselian valuation) strict cohomological dimension ... 306 152

strong approximation theorem .... 193 structure sheaf ................... 88

- F.K. Schmidt (prime decomposition) 58

S-units ...................... 71,358 - Fontaine ..................... 207

supplementary theorems ..... 340, 416 - Gauss reciprocity law .. 51, 64, 416

Sylow subgroup ................. 274 - Grothendieck-Riemann-Roch .. 254

symbols - Grunwald ................... .405

- Artin symbol ................ .407 - Hasse-Arf ................... 355

- Hilbert symbol ........... 305,333 - Hasse-Minkowski ............ 385

- Jacobi symbol ............... .417 - Hasse norm theorem .......... 384

- Legendre symbol .......... 50, 336 - Herbrand .................... 180

-norm residue symbol. 302, 321, 331, - Hermite ..................... 206 391,393 - Hilbert-Noether .............. 283

- power residue symbol .... 336,415 - Hilbert theorem 90 ............ 281 - quadratic residue symbol ...... 417 - Kronecker-Weber ........ 324, 398 - tame Hilbert symbol .......... 335 - Krull-Akizuki ................. 77 -[x,a) ........................ 341 - Minkowski-Hasse ............ 385

- Minkowski lattice point theorem 27 Takagi, T. . ..................... 406 - Minkowski theorem on the discrimi-Tamagawa measure .............. 432 nant ......................... 207 tame Hilbert symbol ............. 335 - Ostrowski ................... 124 tamely ramified ................. 154 - principal ideal theorem ........ 410 - maximal extension ........... : 157 -Riemann-Roch ........... 209,218 - maximal extension of lFp«t)) .. 176 - Rierilann-Roch-Grothendieck .. 254

570 Index

- Shintani unit theorem ......... 507 universal covering ................ 93 - Siegel-Klingen ............... 515 universal norms ............. 304, 308 - Weierstrass preparation theorem 116 unramified ..... 49,184,202,286,309 -Wilson ........................ 2 - q IR is unramified ... 184, 220, 224 theta series ..................... 443 - extension of algebraic number fields - of an algebraic number field ... 484 202 -Hecke ....................... 489 - extension of hens eli an fields ... 153 - Jacobi .................. 422,424 - maximal extension ....... 154,285 - of a lattice ................... 450 upper half-plane ................ .425 - transformation formula ... 425,437, upper half-space ................ 445

438,439,452,490 upper numbering of ramification groups Todd class ...................... 254 180 topology -etale ......................... 90 valuation .................... 67, 116 - Krull ................... 167,262 - abstract henselian .... 288,309,389 -norm ........................ 303 - archimedean ................. 118 - Zariski ....................... 85 - degree valuation ............... 95 totally disconnected ............. 264 - discrete ............... 67,95,121 totally ramified .......... 49, 158,286 - discrete valuation ring .......... 67 totally split ...................... 49 - equivalent valuations .......... 116 trace ......................... 8, 444 - exponential ...... 69, 107, 120, 184 trace form ...................... 194 - extension of ............. 131, 144 trace-zero hyperplane ............. 39 - henselian ....... 143,288,309, 389 trace-zero space ................ .460 - henselian P -valuation ......... 298 transfer .................... 296, 410 - Krull valuation ............... 123 - on Witt vectors ............... 134 - nonarchimedean .............. 118 transformation formula for theta series -normalized .............. 121, 184

425,437,438,439,452,490 - p-adic ........................ 69 trivial character ................. 434 valuation ring ................... 121 trivial metric ............... 227, 229 -henselian .................... 143 trivial zeroes of the Riemann's zeta valuation theory .................. 99

function ..................... 432 - abstract ................. 284, 309 type of a GroBencharacter ........ 478 values (special)

- of Dirichlet L-series .. 442,443, 515 unique prime decomposition ....... 18 -of Riemann's zeta function ... 427, units ............................ 39 431,432 - cyclotomic ................... .44 Vandermonde matrix .............. 11 -Dirichlet's unit theorem .42,81,358 vector bundle ............... 193,255 - fundamental .................. 42 Ver ........................ 296,410 - gaussian ....................... 3 vp , for a prime p ................ 184 - p-adic ....................... 112 - principal ..................... 122 Weber function ................. 403 - Shintani' s unit theorem ........ 507 Weber, H. .................. 366,405 - S-units ................... 71,358 Weierstrass ,p-function .......... .401 - unit lattice .................... 40 Weierstrass preparation theorem .. 116 - unit rank, p-adic .............. 394 wildly ramified .................. 158

Index 571

Wiles, A ......................... 38 - of Dirichlet L-series .......... 442 Wilson's theorem .................. 2 - of Riemann's zeta function .... 432 Witt, E ......................... 410 zeta function ................... .419 Witt vectors ............ 134,283,340 - completed ............... 422, 466

- Dedekind .................... 457 Z .............................. 272 - partial ....................... 458 Zagier, D ....................... 433 - p-adic ....................... 516 Zahlbericht, Hasse's ............. 363 - Riemann .................... 419 Zariski topology .................. 85 Zp-extension, cyclotomic ........ 326 zeroes Z-structure ...................... 24 - of Artin L-series ............. 541

Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics

A Selection

217. Stenstrom: Rings of Quotients 218. GihmaniSkorohod: The Theory of Stochastic Processes II 219. DuvautlLions: Inequalities in Mechanics and Physics 220. Kirillov: Elements of the Theory of Representations 221. Mumford: Algebraic Geometry I: Complex Projective Varieties 222. Lang: Introduction to Modular Forms 223. BerghlUifstrom: Interpolation Spaces. An Introduction 224. Gilbargffrudinger: Elliptic Partial Differential Equations of Second Order 225. Schiitte: Proof Theory 226. Karoubi: K-Theory. An Introduction 227. GrauertlRemmert: Theorie der Steinschen Riiume 228. Segal/Kunze: Integrals and Operators 229. Hasse: Number Theory 230. Klingenberg: Lectures on Closed Geodesics 231. Lang: Elliptic Curves. Diophantine Analysis 232. GihmaniSkorohod: The Theory of Stochastic Processes III 233. StroocklVaradhan: Multidimensional Diffusion Processes 234. Aigner: Combinatorial Theory 235. DyukinlYushkevich: Controlled Markov Processes 236. GrauertlRemmert: Theory of Stein Spaces 237. Kothe: Topological Vector Spaces II 238. GrahamlMcGehee: Essays in Commutative Harmonic Analysis 239. Elliott: Probabilistic Number Theory I 240. Elliott: Probabilistic Number Theory II 241. Rudin: Function Theory in the Unit Ball of en 242. HuppertlBlackbum: Finite Groups II 243. HuppertlBlackbum: Finite Groups III 244. KubertJLang: Modular Units 245. ComfeldIFominiSinai: Ergodic Theory 246. NaimarkiStem: Theory of Group Representations 247. Suzuki: Group Theory I 248. Suzuki: Group Theory II 249. Chung: Lectures from Markov Processes to Brownian Motion 250. Arnold: Geometrical Methods in the Theory of Ordinary Differential Equations 251. ChowlHale: Methods of Bifurcation Theory 252. Aubin: Nonlinear Analysis on Manifolds. Monge-Ampere Equations 253. Dwork: Lectures on p-adic Differential Equations 254. Freitag: Siegelsche Modulfunktionen 255. Lang: Complex Multiplication 256. Hormander: The Analysis of Linear Partial Differential Operators I 257. Hormander: The Analysis of Linear Partial Differential Operators II 258. Smoller: Shock Waves and Reaction-Diffusion Equations 259. Duren: Univalent Functions 260. FreidlinlWentzell: Random Perturbations of Dynamical Systems 261. BoschiGiintzerlRemmert: Non Archimedian Analysis - A System Approach

to Rigid Analytic Geometry 262. Doob: Classical Potential Theory and Its Probabilistic Counterpart 263. Krasnosel'skiilZabreiko: Geometrical Methods of Nonlinear Analysis 264. AubiniCellina: Differential Inclusions 265. GrauertlRemmert: Coherent Analytic Sheaves 266. de Rham: Differentiable Manifolds

267. Arbarello/CornalbalGriffiths/Harris: Geometry of Algebraic Curves, Vol. I 268. Arbarello/CornalbalGriffithslHarris: Geometry of Algebraic Curves, Vol. II 269. Schapira: Microdifferential Systems in the Complex Domain 270. Scharlau: Quadratic and Hermitian Forms 271. Ellis: Entropy, Large Deviations, and Statistical Mechanics 272. Elliott: Arithmetic Functions and Integer Products 273. Nikol'skiI: Treatise on the Shift Operator 274. Hormander: The Analysis of Linear Partial Differential Operators ill 275. Hormander: The Analysis of Linear Partial Differential Operators IV 276. Liggett: Interacting Particle Systems 277. FultonlLang: Riemann-Roch Algebra 278. BarrlWells: Toposes, Triples and Theories 279. BishoplBridges: Constructive Analysis 280. Neukirch: Class Field Theory 281. Chandrasekharan: Elliptic Functions 282. Lelong/Gruman: Entire Functions of Several Complex Variables 283. Kodaira: Complex Manifolds and Deformation of Complex Structures 284. Finn: Equilibrium Capillary Surfaces 285. BuragolZalgaller: Geometric Inequalities 286. Andrianaov: Quadratic Forms and Hecke Operators 287. Maskit: Kleinian Groups 288. Jacod/Shiryaev: Limit Theorems for Stochastic Processes 289. Manin: Gauge Field Theory and Complex Geometry 290. Conway/Sloane: Sphere Packings, Lattices and Groups 291. Hahn/O'Meara: The Classical Groups and K-Theory 292. KashiwaralSchapira: Sheaves on Manifolds 293. RevuzIYor: Continuous Martingales and Brownian Motion 294. Knus: Quadratic and Hermitian Forms over Rings 295. DierkeslHildebrandtIKiisterlWohlrab: Minimal Surfaces I 296. DierkeslHildebrandtIKiisterlWohlrab: Minimal Surfaces II 297. PasturlFigotin: Spectra of Random and Almost-Periodic Operators 298. Berline/GetzlerNergne: Heat Kernels and Dirac Operators 299. Pommerenke: Boundary Behaviour of Conformal Maps 300. OrliklTerao: Arrangements of Hyperplanes 301. Loday: Cyclic Homology 302. LangelBirkenhake: Complex Abelian Varieties 303. DeVorelLorentz: Constructive Approximation 304. Lorentz/v. GolitschekIMakovoz: Construcitve Approximation. Advanced Problems 305. Hiriart-UrrutyILemarechal: Convex Analysis and Minimization Algorithms I.

Fundamentals 306. Hiriart-UrrutylLemarechal: Convex Analysis and Minimization Algorithms II.

Advanced Theory and Bundle Methods 307. Schwarz: Quantum Field Theory and Topology 308. Schwarz: Topology for Physicists 309. AdemlMilgram: Cohomology of Finite Groups 310. GiaquintaIHildebrandt: Calculus of Variations I: The Lagrangian Formalism 311. GiaquintalHildebrandt: Calculus of Variations II: The Hamiltonian Formalism 312. Chung/Zhao: From Brownian Motion to SchrOdinger's Equation 313. Malliavin: Stochastic Analysis 314. AdamslHedberg: Function Spaces and Potential Theory 315. Biirgisser/ClausenlShokrollahi: Algebraic Complexity Theory 316. Saff/Totik: Logarithmic Potentials with External Fields 317. RockafellarlWets: Variational Analysis 318. Kobayashi: Hyperbolic Complex Spaces 319. BridsonlHaefiiger: Metric Spaces of Non-Positive Curvature 320. KipnisILandim: Scaling Limits of Interacting Particle Systems 321. Grimmett: Percolation 322. Neukirch: Algebraic Number Theory