Contemporary Mathematics 300

CONTEMPORARY MATHEMATICS 300

Algebraic Number Theory and Algebraic Geometry Papers Dedicated to A. N. Parshin on the Occasion of his Sixtieth Birthday

Sergei Vostokov Yuri Zorhin Editors http://dx.doi.org/10.1090/conm/300

Algebraic Number Theory and Algebraic Geometry Aleksey Nikolaevich Parshin CoNTEMPORARY MATHEMATICS

300

Algebraic Number Theory and Algebraic Geometry Papers Dedicated to A. N. Parshin on the Occasion of his Sixtieth Birthday

Sergei Vostokov Yuri Zarhin Editors

American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeThrck, managing editor Andreas Blass Andy R. Magid Michael Vogelius

2000 Mathematics Subject Classification. Primary 11815, 11831, 14E22, 14F20, 14H30, 14H40, 14K10, 14K99, 14L05.

Library of Congress Cataloging-in-Publication Data Algebraic number theory and algebraic geometry : papers dedicated to A. N. Parshin on the occasion of his sixtieth birthday / Sergei Vostokov, Yuri Zarhin, editors. p. em. -(Contemporary mathematics; ISSN 0271-4132; v. 300) Includes bibliographical references. ISBN 0-8218-3267-0 (softcover : alk. paper) 1. Algebraic number theory. 2. Geometry, algebraic. I. Parshin, A. N. II. Vostokov, S. V. III. Zarhin, Yuri, 1951- IV. Contemporary mathematics (American Mathematical Society) ; v. 300.

QA247 .A52287 2002 5121.74--dc21 2002074698

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Aleksey Nikolaevich Parshin. Preface by I. R. Shafarevich vii Ramification theory for higher dimensional local fields VICTOR ABRASHKIN 1 U nramified correspondences FEDOR BOGOMOLOV AND YURI TSCHINKEL 17 Local Leopoldt's problem for ideals in totally ramified p-extensions of complete discrete valuation fields M. V. BONDARKO 27 Quotients of algebraic varieties by Zariski dense equivalence relations ALEXANDRU BUIUM 59 A note on arithmetic topology and dynamical systems CHRISTOPHER DENINGER 99 A relation between two moduli spaces studied by V. G. Drinfeld GERD FALTINGS 115 An invariant for varieties in positive characteristic G. VANDER GEER AND T. KATSURA 131 Honda groups and explicit pairings on the modules of Cartier curves FALKO LORENZ AND SERGEI VOSTOKOV 143 Algebraic oriented cohomology theories ALEXANDER MERKURJEV 171 Hyperelliptic jacobians without complex multiplication, doubly transitive permutation groups and projective representations YURI G. ZARHIN 195 Ramification of surfaces: Artin-Schreier extensions IGOR ZHUKOV 211

v ALEKSEY NIKOLAEVICH PARSHIN

Aleksey Nikolaevich Parshin was born on November 7, 1942. In 1959 he en- tered the Mechanics and Mathematics faculty (Mekh-Mat) of Moscow State Uni- versity. His outstanding mathematical capabilities became apparent already during his student years. Then he wrote a paper where he suggested a construction of a "nonabelian jacobian" of Riemann surface as the quotient of a certain (infinite- dimensional) manifold modulo the image of the fundamental group of the surface. This construction was a complex analogue of the theory of "iterated integrals" that was being developed independently by Chen. Unfortunately, A. N. Parshin never returned to this intriguing subject. Later it turned out that this theory provides impressive results for covers with nilpotent Galois groups but even in this case a lot still has to be done. This paper was inspired by attempts to find a "nonabelian" generalization of class field theory. And all posterior papers of A. N. Parshin went back to various number-theoretical questions, though sometimes they went away to completely dif- ferent fields. Mostly his research is related to the following two directions. First, it is a study of two-dimensional schemes that are "fibered" over a one-dimensional base. Parshin's attention was attracted by so called "finiteness conjecture" accord- ing to which there are only finitely many of those schemes if the base, degenerations, and the genus of the fiber are fixed (modulo obvious exceptions). His first results dealt with the case of schemes over a field, i.e., with algebraic surfaces. Parshin proved that if the conditions of the "finiteness conjecture" are fulfilled then the corresponding surfaces consist of a a finite number of algebraic families (the fact that these families are zero-dimensional, i.e. boil down to finitely many surfaces, was be- fore long proved by S. Yu. Arakelov with help of a special deformation technique). But even Parshin's result allowed already to prove the "finiteness conjecture" for surfaces over finite fields. In his ICM talk in Nice, A. N. Parshin reported on a quite unexpected result: the validity of "finiteness conjecture" implies the cele- brated "Mordell conjecture". In particular, this provides a proof of an analogue of the Mordell conjecture for schemes over finite fields. Thirteen years later Faltings proved the "finiteness conjecture" for arithmetical schemes. Thanks to the result of Parshin, this gave a proof of the Mordell conjecture. A conjectural "Parshin's inequality" (an analogue of Bogomolov-Miyaoka-Yao inequality for algebraic surfaces) lies in the same circle of ideas related to two-dimensional schemes. The validity of Parshin's inequality would imply a plentitude of arithmetical corollaries. Another direction that A. N. Parshin continues to develop to this day is related to a generalization of class field theory and classical arithmetic of one-dimensional schemes on the whole to the higher-dimensional case. An idea of this generalization is that the basic notion is neither a point (an irreducible zero-dimensional scheme) vii viii ALEKSEY NIKOLAEVICH PARSHIN nor an irreducible divisor (a subscheme of codimension 1) but a flag, i.e, a point, an irreducible curve passing through this point, an irreducible surface passing through this curve etc. This conception turned out to be extremely fruitful, circulating in a plenty of subjects of mathematics. Several years were devoted by Parshin to the development of an analogue of class field theory for higher-dimensional fields. Here the main novelty was that the role of the multiplicative group of a field is played by the group Kn (more precisely, a certain topological variant of K n defined in terms of "symbols" ) . This leads to a natural generalization of the classical reciprocity law. In the "geometric" case one gets new relations between residues of differential forms. During the following years A. N. Parshin discovered and studied other interrelations between the problems described above. In particular, the building of class field theory had required the construction of an analogue of the group of adeles (or fdeles) for a n-dimensional field. It turned out that this construction can be gen- eralized to the case of adele groups of algebraic groups. In particular, the "adelic" description of vector bundles over an algebraic curve (Weil - Serre) has a precise n-dimensional analogue (for instance, for algebraic surfaces). Thus he discovered an analogue of Serre's theorem about interrelations between vector bundles and Bruhat-Tits buildings. In the same circle of ideas lies a "fixed point formula" for actions of tori on manifolds with arbitrary singularities. In number-theoretical sit- uation these considerations allow one to study local constants in the functional equations for L-functions. Recently many interrelations with other ideas were discovered. For example, even in the case of algebraic varieties over the field of complex numbers, the generalizations of reciprocity laws turn out to be related with so called "complex linking numbers", local symbols are related with topological conformal field theory, etc. It seems that there is a very broad area for study, whose boundary is still not visible. Some of Parshin's papers do not fit into the two directions described above. Thus he discovered the possibility of an application of Sh. Kobayashi's hyperbolic geometry to number theoretical problems (for the first time, to the best of my knowledge). This was a beginning of now widely ramified direction: proofs of arithmetical properties of algebraic varieties (or statements of conjectures). Parshin's interests are much broader that one may judge by his research in number theory and algebraic geometry. This is demonstrated by his participation in editing (in Russian) the collected papers of H. Weyl and D. Hilbert, his paper on complementarity principle in "Voprosy philosophii", or by his "Reflections on Godel's Theorem" published in "Istoriko-Matematicheskie Issledovaniya". Aleksey Nikolaevich Parshin is a corresponding member of the Russian Acad- emy of Sciences and doctor honoris causa of the Universite Paris-Nord. I. R. Shafarevich Titles in This Series

300 Sergei Vostokov and Yuri Zarhin, Editors, Algebraic number theory and algebraic geometry: Papers dedicated to A. N. Parshin on the occasion of his sixtieth birthday, 2002 299 George Kamberov, Peter Norman, Franz Pedit, and Ulrich Pinkall, Quaternions, spinors, and surfaces, 2002 298 Robert Gilman, Alexei G. Myasnikov, and Vladimir Shpilrain, Editors, Computational and statistical group theory, 2002 297 Stephen Berman, Paul Fendley, Yi-Zhi Huang, Kailash Misra, and Brian Parshall, Editors, Recent developments in infinite-dimensional Lie algebras and conformal field theory, 2002 296 Sean Cleary, Robert Gilman, Alexei G. Myasnikov, and Vladimir Shpilrain, Editors, Combinatorial and geometric group theory, 2002 295 Zhangxin Chen and Richard E. Ewing, Editors, Fluid flow and transport in porous media: Mathematical and numerical treatment, 2002 294 Robert Coquereaux, Ariel Garda, and Roberto Trinchera, Editors, Quantum symmetries in theoretical physics and mathematics, 2002 293 Donald M. Davis, Jack Morava, Goro Nishida, W. Stephen Wilson, and Nobuaki Yagita, Editors, Recent progress in homotopy theory, 2002 292 A. Chenciner, R. Cushman, C. Robinson, and Z. Xia, Editors, Celestial Mechanics, 2002 291 Bruce C. Berndt and Ken Ono, Editors, q-series with applications to combinatorics, number theory, and physics, 2001 290 Michel L. Lapidus and Machiel van Frankenhuysen, Editors, Dynamical, spectral, and arithmetic zeta functions, 2001 289 Salvador Perez-Esteva and Carlos Villegas-Blas, Editors, Second summer school in analysis and mathematical physics: Topics in analysis: Harmonic, complex, nonlinear and quantization, 2001 288 Marisa Fernandez and Joseph A. Wolf, Editors, Global differential geometry: The mathematical legacy of Alfred Gray, 2001 287 Marlos A. G. Viana and Donald St. P. Richards, Editors, Algebraic methods in statistics and probability, 2001 286 Edward L. Green, Serkan Ho§ten, Reinhard C. Laubenbacher, and Victoria Ann Powers, Editors, Symbolic computation: Solving equations in algebra, geometry, and engineering, 2001 285 Joshua A. Leslie and Thierry P. Robart, Editors, The geometrical study of differential equations, 2001 284 Gaston M. N'Guerekata and Asamoah Nkwanta, Editors, Council for African American researchers in the mathematical sciences: Volume IV, 2001 283 Paul A. Milewski, Leslie M. Smith, Fabian Waleffe, and Esteban G. Tabak, Editors, Advances in wave interaction and turbulence, 2001 282 Arlan Ramsay and Jean Renault, Editors, Groupoids in analysis, geometry, and physics, 2001 281 Vadim Olshevsky, Editor, Structured matrices in mathematics, computer science, and engineering II, 2001 280 Vadim Olshevsky, Editor, Structured matrices in mathematics, computer science, and engineering I, 2001 279 Alejandro Adem, Gunnar Carlsson, and Ralph Cohen, Editors, Topology, geometry, and algebra: Interactions and new directions, 2001 278 Eric Todd Quinto, Leon Ehrenpreis, Adel Faridani, Fulton Gonzalez, and Eric Grinberg, Editors, Radon transforms and tomography, 2001 277 Luca Capogna and Loredana Lanzani, Editors, Harmonic analysis and boundary value problems, 2001 TITLES IN THIS SERIES

276 Emma Previato, Editor, Advances in algebraic geometry motivated by physics, 2001 275 Alfred G. Noel, Earl Barnes, and Sonya A. F. Stephens, Editors, Council for African American researchers in the mathematical sciences: Volume III, 2001 274 Ken-ichi Maruyama and John W. Rutter, Editors, Groups of homotopy self-equivalences and related topics, 2001 273 A. V. Kelarev, R. Gobel, K. M. Rangaswamy, P. Schultz, and C. Vinsonhaler, Editors, Abelian groups, rings and modules, 2001 272 Eva Bayer-Fluckiger, David Lewis, and Andrew Ranicki, Editors, Quadratic forms and their applications, 2000 271 J.P. C. Greenlees, Robert R. Bruner, and Nicholas Kuhn, Editors, Homotopy methods in algebraic topology, 2001 270 Jan Denef, Leonard Lipschitz, Thanases Pheidas, and Jan Van Geel, Editors, Hilbert's tenth problem: Relations with arithmetic and algebraic geometry, 2000 269 Mikhail Lyubich, John W. Milnor, and Yair N. Minsky, Editors, Laminations and foliations in dynamics, geometry and topology, 2001 268 Robert Gulliver, Walter Littman, and Roberto Triggiani, Editors, Differential geometric methods in the control of partial differential equations, 2000 267 Nicolas Andruskiewitsch, Walter Ricardo Ferrer Santos, and Hans-Jiirgen Schneider, Editors, New trends in Hopf algebra theory, 2000 266 Caroline Grant Melles and Ruth I. Michler, Editors, Singularities in algebraic and analytic geometry, 2000 265 Dominique Arlettaz and Kathryn Hess, Editors, Une degustation topologique: Homotopy theory in the Swiss Alps, 2000 264 Kai Yuen Chan, Alexander A. Mikhalev, Man-Keung Siu, Jie-Tai Yu, and Efim I. Zelmanov, Editors, Combinatorial and computational algebra, 2000 263 Yan Guo, Editor, Nonlinear wave equations, 2000 262 Paul Igodt, Herbert Abels, Yves Felix, and Fritz Grunewald, Editors, Crystallographic groups and their generalizations, 2000 261 Gregory Budzban, Philip Feinsilver, and Arun Mukherjea, Editors, Probability on algebraic structures, 2000 260 Salvador Perez-Esteva and Carlos Villegas-Blas, Editors, First summer school in analysis and mathematical physics: Quantization, the Segal-Bargmann transform and semiclassical analysis, 2000 259 D. V. Huynh, S. K. Jain, and S. R. L6pez-Permouth, Editors, Algebra and its applications, 2000 258 Karsten Grove, Ib Henning Madsen, and Erik Kjaer Pedersen, Editors, Geometry and topology: Aarhus, 2000 257 Peter A. Cholak, Steffen Lempp, Manuel Lerman, and Richard A. Shore, Editors, Computability theory and its applications: Current trends and open problems, 2000 256 Irwin Kra and Bernard Maskit, Editors, In the tradition of Ahlfors and Bers: Proceedings of the first Ahlfors-Bers colloquium, 2000 255 Jerry Bona, Katarzyna Saxton, and Ralph Saxton, Editors, Nonlinear PDE's, dynamics and continuum physics, 2000 254 Mourad E. H. Ismail and Dennis W. Stanton, Editors, q-series from a contemporary perspective, 2000

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.orgfbookstoref. A. N. Parshin is a world-renowned mathematician who has made significant contributions to number theory through the use of algebraic geometry. Articles in this volume present new research and the latest developments in algebraic number theory and algebraic geometry and are dedicated to Parshin's sixtieth birthday. Well-known mathematicians contributed to this volume, including, among others, F. Bogomolov, C. Deninger, and G. Faltings. The book is intended for graduate students and research mathematicians interested in number theory, algebra, and algebraic geometry.

ISBN 0-8218-3267-0