Proceedings o f th e St. Petersbur g Mathematical Societ y Volume XII

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American Mathematica l Societ y

TRANSLATIONS Series 2 • Volum e 21 9

Proceedings o f th e St. Petersbur g Mathematical Societ y Volume XII

N. N . Uraltsev a Editor

American Mathematica l Societ y Providence, Rhod e Islan d

http://dx.doi.org/10.1090/trans2/219

EDITORIAL COMMITTE E

AMS Subcommitte e Robert D . MacPherso n

Grigorii A . Margulis James D . Stasheff (Chair )

ASL Subcommitte e Steffe n Lemp p (Chair ) IMS Subcommitte e Mar k I . Freidlin (Chair )

T p y n b i CAHKT-IJETEPEyPrCKOrO

MATEMATMMECKOrO OBIIIECTBA, TO M 1 2

H3^aTejib TaMap a Po^cKOBCKan , H O B O C H 6 H P C K , 200 5

This wor k wa s originally publishe d i n Russi a b y t h e Publ ishe r T a m a r a Rozhkovskaya , Novosibirsk, Russi a unde r t h e t i t l e " T p y ^ t i CaHKT-IIeTepGyprcKor o MaTeMaTH^ecKor o OGniecTBa, T . 12" © CaHKT-IIeTep6yprcKo e MaTeMaTiraecKo e 06m;ecTBO . T h e presen t t rans la t ion wa s created unde r licens e fo r t h e America n Mathemat ica l Societ y an d i s pub -lished b y permission .

2000 Mathematics Subject Classification. P r imar y 11-06 .

ISBN-13: 978-0-8218-4205- 8 ISBN-10: 0-8218-4205- 6

ISSN 0065-929 0

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Contents

Preface

A formulatio n o f conjecture s o n p-adi c zet a function s i n noncommutativ e Iwasawa theor y TAKAKO FUKAY A AN D KAZUY A KAT O

On maxima l 2-extension s o f Q wit h give n ramificatio n B. EIC K AN D H . KOC H

On th e nor m residu e homomorphis m o f degree tw o ALEXANDER MERKURJE V

SKI o f division algebra s an d Galoi s cohomolog y revisite d ANDREI SUSLI N

Measure, integratio n an d element s o f harmoni c analysi s o n generalize d loo p spaces IVAN FESENK O

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Preface

Sergey Vladimirovic h Vostoko v wa s bor n o n Apri l 13 , 194 5 in Leningrad , on e year afte r th e end o f the siege of the cit y during the Secon d World War . Sinc e 196 8 he has bee n workin g a t St . Petersbur g Stat e University .

His scientific interest s were shaped by St. Petersbur g mathematicians Z.I . Bore-vich an d D . K . Faddeev , an d b y th e Mosco w mathematicia n I . R . Shafarevich . I t was Shafarevich' s wor k tha t influence d Vostoko v i n hi s stud y o f arithmetica l prop -erties o f loca l fields an d relate d areas , th e mai n subjec t o f hi s research . Som e o f the best-know n result s o f Vostokov dea l wit h explici t formula s fo r th e wil d Hilber t symbol define d o n th e multiplicativ e grou p an d o n othe r forma l groups , a s wel l a s with explicitl y define d symbol s o n higher-dimensiona l loca l fields.

In 197 8 S . V. Vostoko v establishe d a remarkabl y simpl e an d a t th e sam e tim e powerful explici t formul a fo r th e wil d Hilber t symbol . Thi s formula , togethe r wit h its generalization s t o higher-dimensiona l loca l fields an d t o forma l groups , i s ofte n called the Vostokov symbol Unlik e most of the other explici t formulas , the Vostokov symbol ca n b e defined withou t usin g clas s field theory. Becaus e o f this on e can us e the symbol to study dee p properties o f higher loca l fields, for example the structur e of thei r Milno r K- groups. A simple r versio n o f th e Vostoko v symbol , a linea r algebra version , i s th e tam e symbol , whic h play s a rol e i n part s o f representatio n theory an d mathematica l physics , e.g. , i n verte x operato r algebras .

Vostokov's work on the explici t formul a fo r the Hilber t symbo l is closely relate d to th e solutio n o f th e famou s nint h Hilber t proble m o n explici t reciprocit y laws . Indeed, th e solutio n o f th e proble m cruciall y depend s o n explici t formula s fo r th e local Hilber t symbol . Man y o f the othe r formula s fo r th e Hilber t symbo l obtaine d both befor e an d afte r th e Vostoko v formul a ar e specia l case s o f th e latter , an d i n this sens e th e Vostoko v formul a i s universal . Explici t formula s pla y a ke y rol e i n many fundamenta l paper s on arithmeti c geometry , includin g works of A. Wiles an d K. Kato .

Vostokov i s th e autho r o f mor e tha n 7 5 paper s an d a boo k (coauthore d wit h I. Fesenko ) o n loca l fields. Thi s boo k wa s publishe d i n Englis h b y th e America n Mathematical Societ y i n 199 3 (secon d revise d editio n i n 2002) . Containin g clea r and explici t description s o f fundamenta l arithmetica l structure s o f loca l numbe r theory, th e boo k ha s prove d t o b e ver y usefu l t o student s an d specialists , no t only i n numbe r theory , bu t i n severa l othe r areas , suc h a s algebrai c topolog y an d mathematical physics .

The scientific schoo l created b y Vostokov in St . Petersbur g i s one of the leadin g centers o f researc h i n numbe r theor y i n Russia . S . V . Vostoko v i s a n organize r o f numerous internationa l conference s i n numbe r theor y a t th e Eule r Institut e i n St . Petersburg.

vii

viii P R E F A C E

In the 1960s , S. V. Vostokov was one of the organizers of the summer mathemat -ical schoo l affiliate d wit h th e St . Petersbur g Stat e University . Fo r severa l decade s he activel y collaborate d wit h th e specialize d hig h schoo l No . 4 5 affiliate d wit h the St . Petersbur g Stat e University , man y o f whose graduate s becom e professiona l mathematicians.

Vostokov's scientific an d teaching qualitites go together with his personal char m and friendly attitud e to his colleagues, students, and young mathematicians startin g their scientifi c careers .

For hi s 60t h birthda y anniversary , mathematician s fro m differen t part s o f th e world contribute d thei r paper s writte n o n topic s clos e t o Vostokov' s scientifi c in -terests. Thes e paper s for m tw o volumes (X I and XII ) o f the Proceeding s o f the St . Petersburg Mathematica l Society .

The author s an d editor s o f thes e volume s congratulat e Serge y Vladimirovic h Vostokov o n hi s birthda y an d wis h hi m continue d success .

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Titles i n Thi s Serie s

219 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society , Volume XI I

218 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume X I

217 V . Kaimanovic h an d A . Lodkin , Editors , Representatio n theory , dynamica l systems ,

and asymptoti c combinatoric s

216 M . V . Karasev , Editor , Quantu m Algebra s an d Poisso n Geometr y i n Mathematica l Physics

214 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume X

213 Ernes t Vinberg , Editor , Li e Group s an d Invarian t Theor y

212 V . M . Buchstabe r an d I . M . Krichever , Editors , Geometry , Topology , an d

Mathematical Physics . S . P . Novikov' s Seminar : 2002-200 3

211 K . Nomizu , Editor , Selecte d Paper s o n Analysi s an d Differentia l Equation s

210 S . G . Gindikin , Editor , Li e Group s an d Symmetri c Spaces . I n Memor y o f F . I .

Karpelevich

209 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume I X

208 M . V . Karasev , Editor , Asymptoti c Method s fo r Wav e an d Quantu m Problem s

207 Yu . M . Suhov , Editor , Analyti c Method s i n Applie d Probabilit y

206 M . A . Shubi n an d M . S . Agranovich , Editors , Partia l Differentia l Equation s (Mar k

Viskik's Seminar )

205 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society , Volume VII I

204 K . N o m i z u , Editor , Selecte d Paper s o n Classica l Analysi s 203 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume VI I 202 V . Turae v an d A . Vershik , Editors , Topology , Ergodi c Theory , Rea l Algebrai c

Geometry. Rokhlin' s Memoria l

201 Michae l Semenov-Tian-Shansky , Editor , L . D . Faddeev' s Semina r o n Mathematica l

Physics

200 L . Lerman , G . Polotovskn , an d L . Shilnikov , Editors , Method s o f Qualitativ e Theory o f Differentia l Equation s an d Relate d Topic s

199 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume V I

198 R . A . Minlos , Seny a Shlosman , an d Yu . M . Suhov , Editors , O n Dobrushin' s Way .

Prom Probabilit y Theor y t o Statistica l Physic s

197 Vladimi r Arnold , Maxi m Kontsevich , an d A n t o n Zorich , Editors , Pseudoperiodi c

Topology

196 Ya . Eliashberg , D . Fuchs , T . Rat iu , an d A . Weinste in , Editors , Norther n

California Symplecti c Geometr y Semina r

195 O . V . Belegrade k e t al. , Mode l Theor y an d Application s

194 A . Astashkevic h an d S . Tabachnikov , Editors , Differentia l Topology , Infinite-Dimensional Li e Algebras , an d Application s (D . B . Fuchs ' 60t h Anniversar y Collection)

193 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society , Volume V

192 Le v Beklemishev , Mat i Pentus , an d Nikola i Vereshchagin , Provability , Complexity, Grammar s

TITLES I N THI S SERIE S

191 A . Yu . Morozo v an d M . A . Olshanetsky , Editors , Mosco w Semina r i n Mathematical Physic s

190 S . Tabachnikov , Editor , Differentia l an d Symplecti c Topolog y o f Knot s an d Curve s

189 V . Buslaev , M . Solomyak , an d D . Yafaev , Editors , Differentia l Operator s an d

Spectral Theor y (M . Sh . Birman' s 70t h Anniversar y Collection )

188 O . A . Ladyzhenskaya , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume I V

187 M . V . Karasev , Editor , Coheren t Transform , Quantization , an d Poisso n Geometr y

186 A . Khovanskii , A . Varchenko , an d V . Vassiliev , Editors , Geometr y o f Differentia l

Equations

185 B . Feigi n an d V . Vassiliev , Editors , Topic s i n Quantu m Group s an d Finite-Typ e

Invariants (Mathematic s a t th e Independen t Universit y o f Moscow )

184 Pete r Kuchmen t an d Vladimi r Lin , Editors , Voronez h Winte r Mathematica l School s

(Dedicated t o Seli m Krein )

183 K . Nomizu , Editor , Selecte d Paper s o n Harmoni c Analysis , Groups , an d Invariant s

182 V . E . Zakharov , Editor , Nonlinea r Wave s an d Wea k Turbulenc e

181 G . I . Olshanski , Editor , Kirillov' s Semina r o n Representatio n Theor y

180 A . Khovanskii , A . Varchenko , an d V . Vassiliev , Editors , Topic s i n Singularit y

Theory

179 V . M . Buchstabe r an d S . P . Novikov , Editors , Solitons , Geometry , an d Topology :

On th e Crossroa d

178 V . Kreinovic h an d G . Mints , Editors , Problem s o f Reducin g th e Exhaustiv e Searc h

177 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors ,

Topics i n Statistica l an d Theoretica l Physic s (F . A . Berezi n Memoria l Volume )

176 E . V . Shikin , Editor , Som e Question s o f Differentia l Geometr y i n th e Larg e

175 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors ,

Contemporary Mathematica l Physic s (F . A . Berezi n Memoria l Volume )

174 A . A . Bolibruch , A . S . Merkur'ev , an d N . Yu . Netsvetaev , Editors , Mathematic s

in St . Petersbur g

173 V . Kharlamov , A . Korchagin , G . Polotovskii , an d O . Viro , Editors , Topolog y o f

Real Algebrai c Varietie s an d Relate d Topic s

172 K . Nomizu , Editor , Selecte d Paper s o n Numbe r Theor y an d Algebrai c Geometr y

171 L . A . Bunimovich , B . M . Gurevich , an d Ya . B . Pesin , Editors , Sinai' s Mosco w

Seminar o n Dynamica l System s

170 S . P . Novikov , Editor , Topic s i n Topolog y an d Mathematica l Physic s

169 S . G . Gindiki n an d E . B . Vinberg , Editors , Li e Group s an d Li e Algebras : E . B .

Dynkin's Semina r

168 V . V . Kozlov , Editor , Dynamica l System s i n Classica l Mechanic s

167 V . V . Lychagin , Editor , Th e Interpla y betwee n Differentia l Geometr y an d Differentia l

Equations

166 O . A . Ladyzhenskaya , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume II I

165 Yu . Ilyashenk o an d S . Yakovenko , Editors , Concernin g th e Hilber t 16t h Proble m

164 N . N . Uraltseva , Editor , Nonlinea r Evolutio n Equation s

163 L . A . Bokut' , M . Hazewinkel , an d Yu . G . Reshetnyak , Editors , Thir d Siberia n School "Algebr a an d Analysis "

For a complet e lis t o f t i t le s i n thi s series , visi t t h e AMS Bookstor e a t w w w . a m s . o r g / b o o k s t o r e / .