HIGHER-DIMENSIONAL GEOMETRY OVER FINITE FIELDS NATO Science for Peace and Security Series This Series presents the results of scientific meetings supported under the NATO Programme: Science for Peace and Security (SPS). The NATO SPS Programme supports meetings in the following Key Priority areas: (1) Defence Against Terrorism; (2) Countering other Threats to Security and (3) NATO, Partner and Mediterranean Dialogue Country Priorities. The types of meeting supported are generally “Advanced Study Institutes” and “Advanced Research Workshops”. The NATO SPS Series collects together the results of these meetings. The meetings are co-organized by scientists from NATO countries and scientists from NATO’s “Partner” or “Mediterranean Dialogue” countries. The observations and recommendations made at the meetings, as well as the contents of the volumes in the Series, reflect those of participants and contributors only; they should not necessarily be regarded as reflecting NATO views or policy. Advanced Study Institutes (ASI) are high-level tutorial courses to convey the latest developments in a subject to an advanced-level audience. Advanced Research Workshops (ARW) are expert meetings where an intense but informal exchange of views at the frontiers of a subject aims at identifying directions for future action. Following a transformation of the programme in 2006 the Series has been re-named and re- organised. Recent volumes on topics not related to security, which result from meetings supported under the programme earlier, may be found in the NATO Science Series. The Series is published by IOS Press, Amsterdam, and Springer Science and Business Media, Dordrecht, in conjunction with the NATO Public Diplomacy Division. Sub-Series A. Chemistry and Biology Springer Science and Business Media B. Physics and Biophysics Springer Science and Business Media C. Environmental Security Springer Science and Business Media D. Information and Communication Security IOS Press E. Human and Societal Dynamics IOS Press http://www.nato.int/science http://www.springer.com http://www.iospress.nl Sub-Series D: Information and Communication Security – Vol. 16 ISSN 1874-6268 Higher-Dimensional Geometry over Finite Fields Edited by Dmitry Kaledin Steklov Institute, Moscow, Russia and Yuri Tschinkel Courant Institute, NYU and Mathematisches Institut, Göttingen, Germany Amsterdam • Berlin • Oxford • Tokyo • Washington, DC Published in cooperation with NATO Public Diplomacy Division Proceedings of the NATO Advanced Study Institute on Higher-Dimensionals Geometry over Finite Fields Göttingen, Germany 25 June – 6 July 2007 © 2008 IOS Press. All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without prior written permission from the publisher. ISBN 978-1-58603-855-7 Library of Congress Control Number: 2008924572 Publisher IOS Press Nieuwe Hemweg 6B 1013 BG Amsterdam Netherlands fax: +31 20 687 0019 e-mail: [email protected] Distributor in the UK and Ireland Distributor in the USA and Canada Gazelle Books Services Ltd. IOS Press, Inc. White Cross Mills 4502 Rachael Manor Drive Hightown Fairfax, VA 22032 Lancaster LA1 4XS USA United Kingdom fax: +1 703 323 3668 fax: +44 1524 63232 e-mail: [email protected] e-mail: [email protected] LEGAL NOTICE The publisher is not responsible for the use which might be made of the following information. PRINTED IN THE NETHERLANDS Higher-Dimensional Geometry over Finite Fields v D. Kaledin and Y. Tschinkel (Eds.) IOS Press, 2008 © 2008 IOS Press. All rights reserved. Preface Number systems based on a finite collection of symbols, such as the 0s and 1s of com- puter circuitry, are ubiquitous in the modern age. Finite fields are the most important such number systems, playing a vital role in military and civilian communications through coding theory and cryptography. These disciplines have evolved over recent decades, and where once the focus was on algebraic curves over finite fields, recent developments have revealed the increasing importance of higher-dimensional algebraic varieties over finite fields. These are the proceedings of the NATO Advanced Study Institute “Higher- dimensional geometry over finite fields” held at the University of Göttingen in June– July 2007. They introduce the reader to recent developments in algebraic geometry over finite fields with particular attention to applications of geometric techniques to the study of rational points on varieties over finite fields of dimension at least 2. Dmitry Kaledin Yuri Tschinkel This page intentionally left blank vii Contents Preface v Dmitry Kaledin and Yuri Tschinkel Finite Field Experiments 1 Hans-Christian Graf v. Bothmer K3 Surfaces of Picard Rank One Which Are Double Covers of the Projective Plane 63 Andreas-Stephan Elsenhans and Jörg Jahnel Beilinson Conjectures in the Non-Commutative Setting 78 Dmitry Kaledin Looking for Rational Curves on Cubic Hypersurfaces 92 János Kollár (Notes by Ulrich Derenthal) Abelian Varieties over Finite Fields 123 Frans Oort How to Obtain Global Information from Computations over Finite Fields 189 Michael Stoll Geometry of Shimura Varieties of Hodge Type over Finite Fields 197 Adrian Vasiu Lectures on Zeta Functions over Finite Fields 244 Daqing Wan De Rham Cohomology of Varieties over Fields of Positive Characteristic 269 Torsten Wedhorn Homomorphisms of Abelian Varieties over Finite Fields 315 Yuri G. Zarhin Author Index 345 This page intentionally left blank Higher-Dimensional Geometry over Finite Fields 1 D. Kaledin and Y. Tschinkel (Eds.) IOS Press, 2008 © 2008 IOS Press. All rights reserved. Finite Field Experiments e-mail: [email protected] Hans-Christian GRAF V. BOTHMER Abstract. We show how to use experiments over finite fields to gain infor- mation about the solution set of polynomial equations in characteristic zero. Introduction Let X be a variety defined over Z. According to Grothendieck we can picture X as a family of varieties Xp over Spec Z with fibers over closed points of Spec Z corresponding to reductions modulo p and the generic fiber over (0) corresponding to the variety XQ defined by the equations of X over Q. The generic fiber is related to the special fibers by semicontinuity theorems. For example, the dimension of Xp is upper semicontinuous with dim XQ =mindim Xp. p>0 This allows us to gain information about XQ by investigating Xp which is often computationally much simpler. Even more surprising is the relation between the geometry of Xp and the number of Fp rational points of Xp discovered by Weil: ⊂ Pn Theorem 0.1. Xp Fp gN Fp Xp √ |1 − N + p|≤2g p. He conjectured even more precise relations for varieties of arbitrary dimension which were proved by Deligne using l-adic cohomology. In this tutorial we will use methods which are inspired by Weil’s ideas, but are not nearly as deep. Rather we will rely on some basic probabilistic estimates which 2 H.-C. Graf v. Bothmer / Finite Field Experiments Figure 1. A variety over Spec Z are nevertheless quite useful. I have learned these ideas from my advisor Frank Schreyer, but similar methods have been used independently by other people, for example Joachim von zur Gathen and Igor Shparlinski [1], Oliver Labs [2]and Noam Elkies [3]. The structure of these notes is as follows: We start in Section 1 by evaluat- ing the polynomials defining a variety X at random points. This can give some heuristic information about the codimension c of X and about the number d of codimension-c components of X. In Section 2 we refine this method by looking at the tangent spaces of X in random points. This gives a way to also estimate the number of components in every codimension. As an application we show how this can be applied to gain new information about the Poincar´e center problem. In Section 3 we explain how it is often possible to prove that a solution found over Fp actually lifts to Q. This is applied to the construction of new surfaces in P4. Often one would like not only to prove the existence of a lift, but explicitly find one. It is explained in Section 4 how this can be done if the solution set is zero dimensional. We close in Section 5 with a beautiful application of these lifting techniques found by Oliver Labs, showing how he constructed a new septic with 99 real nodes 3 in PR. For all experiments in this tutorial we have used the computer algebra system Macaulay 2 [4]. The most important Macaulay 2 commands used are explained in Appendix A, for more detailed information we refer to the online help of Macaulay H.-C. Graf v. Bothmer / Finite Field Experiments 3 2[4]. In Appendix B Stefan Wiedmann provides a MAGMA translation of the Macaulay 2 scripts in this tutorial. All scripts are available online at [5]. We would like to include translations to other computer algebra packages, so if you are for example a Singular-expert, please contact us. Finally I would like to thank the referee for many valuable suggestions. 1. Guessing We start by considering the most simple case, namely that of a hypersurface n n X ⊂ A defined by a single polynomial f ∈ Fp[x1,...,xn]. If a ∈ A is a point we have 0 one possibility f(a)= =0( p − 1) possibilities 1 Naively we would therefore expect that we obtain zero for about p of the points. Experiment 1.1. We evaluate a given polynomial in 700 random points, using Macaulay 2: R = ZZ[x,y,z,w] -- work in AA^4 F = x^23+1248*y*z+w+129269698 -- a Polynomial K = ZZ/7 -- work over F_7 L = apply(700, -- substitute 700 i->sub(F,random(K^1,K^4))) -- random points tally L -- count the results obtaining: o5 = Tally{-1 => 100} -2 => 108 -3 => 91 0=>98 1=>102 2=>101 3=>100 Indeed, all elements of F7 occur about 700/7 = 100 times as one would expect naively.