16
Progress in Mathematics Volume 199
Series Editors
H. Bass 1. Oesterle A. Weinstein
Rational Points on Aigebraic Varieties
Emmanuel Peyre Yuri Tschinkel Editors
Springer Basel AG
Editors:
Emmanuel Peyre Institut Fourier UFR de Mathematiques, UMR 5582 Universit6 de Grenoble 1 et CNRS B.P.74 38402 Saint-Martin d'Heres France
e-mail: [email protected]
2000 Mathematics Subject Classification 14G05, IIG35
Yuri Tschinkel Department of Mathematics Princeton University Washington Road Princeton, NJ 08544-1000 USA
e-mail: [email protected]
A CIP catalogue record for this book is available from the Library of Congress, Washington o.e., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Rational Points on AIgebraic Varieties / Emmanuel Peyre ... ed .. - Basel ; Boston; Berlin: Birkhăuser, 200 l
(Progress in mathematics ; VoI. 199) ISBN 978-3-0348-9536-1 ISBN 978-3-0348-8368-9 (eBook) DOI 10.1007/978-3-0348-8368-9
ISBN 978-3-0348-9536-1
This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concemed, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained.
© 2001 Springer Basel AG Originally published by Birkhăuser Verlag in 2001 Softcover reprint ofthe hardcover 1 st edition 2001 Member of the BertelsmannSpringer Publishing Group Printed on acid-free paper produced of chlorine-free pulp. TCF 00
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INTRODUCTION
This book is devoted to the study of rational and integral points on higherdimensional algebraic varieties. It contains research papers addressing the arithmetic geometry of varieties which are not of general type, with an emphasis on how rational points are distributed with respect to the classical, Zariski and adelic topologies.
The book gives a glimpse of the state of the art of this rapidly expanding domain in arithmetic geometry. The techniques involve explicit geometric constructions, ideas from the minimal model program in algebraic geometry as well as analytic number theory and harmonic analysis on adelic groups.
In recent years there has been substantial progress in our understanding of the arithmetic of algebraic surfaces. Five papers are devoted to cubic surfaces: Basile and Fisher study the existence of rational points on certain diagonal cubics, Swinnerton-Dyer considers weak approximation and Broberg proves upper bounds on the number of rational points on the complement to lines on cubic surfaces. Peyre and Tschinkel compare numerical data with conjectures concerning asymptotics of rational points of bounded height on diagonal cubics of rank ~ 2. Kanevsky and Manin investigate the composition of points on cubic surfaces. Satge constructs rational curves on certain Kummer surfaces. Colliot-Thelene studies the Hasse principle for pencils of curves of genus 1. In an appendix to this paper Skorobogatov produces explicit examples of Enriques surfaces with a Zariski dense set of rational points.
Poonen constructs a (conditional) example of a 3-dimensional complete intersection which violates the Hasse principle but for which all known obstructions to the Hasse principle vanish. The paper of Hassett-Tschinkel is influenced
vi INTRODUCTION
by ideas from the log-minimal model program. It is devoted to the ''potential'' density of integral points on quasi-projective algebraic varieties.
The remaining papers are more analytic. Chambert-Loir and Tschinkel investigate the asymptotics of rational points on compactifications of torsors under linear algebraic groups. Similar fibrations appear in the theory of "partial" Eisenstein series initiated by Strauch. Wooley's analysis of certain exponential sums arising in the circle method allow him to significantly improve asymptotic results concerning the number of integral solutions of sums of binary forms of fixed degree. The paper of Peyre extends the first steps of the classical circle method to hypersurfaces in Fano varieties by lifting the counting of rational points on a variety to the counting of integral points on its universal torsor.
We hope the book conveys some of the excitement shared by participants of the conference at Luminy in September 1999, which was the starting point of this project. Finally, we are very grateful to CIRM and the European network "Arithmetic Geometry" for their support.
CONTENTS
Introduction .......................................................... v
Abstracts .............................................................. Xl
CARMEN LAURA BASILE & THOMAS ANTHONY FISHER - Diagonal cubic equations in four variables with prime coefficients ...................... 1
References ............................................................ 11
NIKLAS BROBERG - Rational points on cubic surfaces ................ 13 Introduction .......................................................... 13 1. Notations and preliminaries ........................................ 15 2. Ternary quadratic forms ............................................ 20 3. Proof of the main theorem ........................................ 28 References ............................................................ 34
ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL - Torseurs arithmetiques et espaces fibres ........................................................ 37
Introduction .......................................................... 37 Notations et conventions .............................................. 40 1. Torseurs arithmetiques ............................................ 40 2. Espaces fibres ...................................................... 51 References ............................................................ 69
ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL - Fonctions zeta des hauteurs des espaces fibres .............................................. 71
Introduction .......................................................... 71 Notations et conventions .............................................. 74 3. Fonctions holomorphes dans un tube .............................. 75 4. Varietes toriques .................................................. 87
viii CONTENTS
5. Application aux fibrations en varietes toriques .................... 101 Appendice A. Un theoreme tauberien ................................ 107 Appendice B. Demonstration de quelques inegalites .................. 109 References ............................................................ 114
JEAN-LoUIS COLLIOT-THELENE - Hasse principle for pencils of curves of genus one whose Jacobians have a rational 2-division point, close variation on a paper of Bender and Swinnerton-Dyer ............................ 117
Statement of the Theorems .......................................... 119 1. Selmer groups associated to a degree 2 isogeny .................... 125 2. Proof of Theorem A ................................................ 142 3. Proof of Theorem B ................................................ 156 References ............................................................ 160
ALEXEI SKOROBOGATOV - Enriques surfaces with a dense set of rational points, Appendix to the paper by J.-L. Colliot- Thelene .................. 163
References ............................................................ 168
BRENDAN HASSETT & YURI TSCHINKEL - Density of integral points on algebraic varieties ...................................................... 169
Introduction .......................................................... 169 1. Generalities ........................................................ 171 2. Geometry .......................................................... 172 3. The fibration method and nondegenerate multisections ............ 177 4. Approximation techniques .......................................... 181 5. Conic bundles and integral points .................................. 183 6. Potential density for log K3 surfaces .............................. 193 References ............................................................ 195
DIMITRI KANEVSKY & YURI MANIN - Composition of points and the Mordell- Weil problem for cubic surfaces ................................ 199
1. Introduction ........................................................ 199 2. Cardinality of generators of subgroups in a reflection group ........ 202 3. Structure of universal equivalence .................................. 206 4. A group-theoretic description of universal equivalence ............ 208 5. Birationally trivial cubic surfaces: a finiteness theorem ............ 213 References ............................................................ 218
EMMANUEL PEYRE - Torseurs universels et methode du cercle ........ 221 Introduction .......................................................... 221
CONTENTS ix
1. Vne version raffinee d'une conjecture de Manin .................... 223 2. Passage au torseur universe I ........................................ 233 3. Intersections completes ............................................ 254 4. Conclusion ........................................................ 271 References ............................................................ 272
EMMANUEL PEYRE & YURI TSCHINKEL - Tamagawa numbers of diagonal cubic surfaces of higher rank ............................................ 275
Introduction .......................................................... 275 1. Description of the conjectural constant ............................ 277 2. The Galois module Pic(V) ........................................ 280 3. Euler product for the good places .................................. 286 4. Density at the bad places .......................................... 288 5. The constant a(V) ................................................ 291 6. Some statistical formulae .......................................... 297 7. Presentation of the results .......................................... 298 References ............................................................ 304
BJORN POONEN - The Hasse principle for complete intersections in projective space .................................................................... 307
References ............................................................ 310
PHILIPPE SATGE - Une construction de courbes k-rationnelles sur les surfaces de Kummer d 'un produit de courbes de genre 1. . ............... 313
Introduction .......................................................... 313 1. Relevement des courbes de P1,k x P1,k sur la surface de Kummer .. 316 2. Exemples .......................................................... 320 References ............................................................ 333
MATTHIAS STRAUCH - Arithmetic Stratifications and Partial Eisenstein Series .................................................................. 335
Introduction .......................................................... 335 1. The fibre bundles: geometric-arithmetic preliminaries .............. 338 2. Height zeta functions .............................................. 342 3. Arithmetic stratification ............................................ 351 References ............................................................ 355
SIR PETER SWINNERTON-DYER - Weak Approximation and R-equivalence on Cubic Surfaces ...................................................... 357
1. Introduction ........................................................ 358
x CONTENTS
2. Geometric background ............................................ 361 3. Approximation at an infinite prime ................................ 370 4. Approximation at a finite prime .................................... 371 5. The lifting process .................................................. 380 6. The dense lifting process .......................................... 386 7. Adelic results ...................................................... 394 8. Surfaces Xf + xg + xi - dx3 = 0 ................................ 395 References ............................................................ 403
TREVOR D. WOOLEY - Hua's lemma and exponential sums over binary forms .................................................................. 405
1. Introduction ........................................................ 405 2. Preliminary reductions ............................................ 411 3. Integral points on affine plane curves .............................. 421 4. The inductive step ................................................ 430 5. The completion of the proof of Theorem 1.1 ........................ 441 References ............................................................ 445
ABSTRACTS
Diagonal cubic equations in four variables with prime coefficients CARMEN LAURA BASILE & THOMAS ANTHONY FISHER ................ 1
The aim of this paper is to give an alternative proof of a theorem of R. Heath-Brown regarding the existence of non-zero integral solutions of the equation
PIX: + P2X~ + P3X : + P4xl = 0,
where the Pj are prime integers congruent to 2 modulo 3.
Rational points on cubic surfaces NIKLAS BROBERG ...................................................... 13
Let k be an algebraic number field and F (xo, Xl, X2, X3) a nonsingular cubic form with coefficients in k. Suppose that the projective cubic k-surface X C JPt given by F = 0 contains three coplanar lines defined over k, and let U(k) be the set of those k-rational points on X which do not lie on any line on X. We show that the number of points in U(k), with height at most B, is OF,e(B4/He) for any c > o.
Torseurs arithmetiques et espaces fibres ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL ........................ 37
We study the compatibility of Manin's conjecture with natural geometric constructions, like fibrations induced from torsors under linear algebraic groups. The main problem it to understand the variation of
xii ABSTRACTS
metrics from fiber to fiber. For this we introduce the notions of "arithmetic torsors", "adelic torsion" and "Arakelov L-functions". We discuss concrete examples, like horospherical varieties and equivariant compactifications of semi-abelian varieties. These techniques are applied to prove "going up" and "descent" theorems for height zeta functions on such fibrations.
Fonctions zeta des hauteurs des espaces fibres ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL ........................ 71
In this paper we study the compatibility of Manin's conjectures concerning asymptotics of rational points on algebraic varieties with certain natural geometric constructions. More precisely, we consider locally trivial fibrations constructed from torsors under linear algebraic groups. The main problem is to understand the behaviour of the height function as one passes from fiber to fiber - a difficult problem, even though all fibers are isomorphic. We will be mostly interested in fibrations induced from torsors under split tori. Asymptotic properties follow from analytic properties of height zeta functions. Under reasonable assumptions on the analytic behaviour of the height zeta function for the base we establish analytic properties of the height zeta function of the total space.
Hasse principle for pencils of curves of genus one whose Jacobians have a rational 2-division point, close variation on a paper of Bender and SwinnertonDyer JEAN-LoUIS COLLIOT-THELENE ........................................ 117
Une serie d'articles exploite une nouvelle technique qui mime 11 des conditions suffisantes d'existence et de densite des points rationnels sur certaines surfaces fibrees en courbes de genre un au-dessus de la droite projective. Dans les premiers articles de cette serie (plusieurs articles de Swinnerton-Dyer, un article en collaboration de Skorobogatov, Swinnerton-Dyer et l'auteur), la jacobienne de la fibre generique des surfaces considerees a tous ses points d'ordre 2 rationnels. Un article recent de Bender et Swinnerton-Dyer traite de cas OU cette jacobienne possede seulement un point d'ordre 2 non trivial (pour que la methode fonctionne, il semble necessaire que la jacobienne possede un point de torsion rationnel non trivial). Le present article est une reecriture de celui de Bender et Swinnerton-Dyer. La principale contribution est une reformulation plus abstraite des hypotheses principales des theoremes.
ABSTRACTS xiii
La premiere hypothese est formulee de fa<;;on entierement algebrique (certains groupes de Selmer algebrico-geometriques sont supposes petits) et la seconde hypothese est simplement : I'll n'y a pas d'obstruction de Brauer-Manin verticale". Comme dans la plupart des articles de cette serie, les resultats dependent de deux conjectures difficiles : l'hypothese de Schinzel et la finitude des groupes de Tate-Shafarevich.
Enriques surfaces with a dense set of rational points, Appendix to the paper by J.-L. Colliot- Thelene ALEXEI SKOROBOGATOV ................................................ 163
Density of integral points on algebraic varieties BRENDAN HASSETT & YURI TSCHINKEL .............................. 169
We study the Zariski density of integral points on quasi-projective algebraic varieties.
Composition of points and the Mordell- Weil problem for cubic surfaces DIMITRI KANEVSKY & YURI MANIN .................................... 199
Let V be a plane smooth cubic curve over a finitely generated field k. The Mordell-Weil theorem for V states that there is a finite subset Pc V(k) such that the whole V(k) can be obtained from P by drawing secants and tangents through pairs of previously constructed points and consecutively adding their new intersection points with V. Equivalently, the group of birational transformations of V generated by reflections with respect to k-points is finitely generated. In this paper, we establish a Mordell-Weil type finite generation result for some birationally trivial cubic surfaces W. To the contrary, we prove that the birational automorphism group generated by reflections cannot be finitely generated if W(k) is infinite.
Torseurs universels et methode du cercle EMMANUEL PEYRE ...................................................... 221
Ce texte decrit les premieres etapes d'une generalisation de la methode du cercle au cas d'une hypersurface lisse dans une variete presque de Fano.
En effet, sous certaines conditions, il est possible d'exprimer dans ce cas les deux membres d'une version raffinee de la conjecture de Manin sur Ie comportement asymptotique du nombre de points de hauteur bornee
xiv ABSTRACTS
de l'hypersurface en termes des torseurs universels de la variete ambiante qui jouent, dans ce cadre, Ie role de l'espace affine.
Tamagawa numbers of diagonal cubic surfaces of higher rank EMMANUEL PEYRE & YURI TSCHINKEL ................................ 275
We consider diagonal cubic surfaces defined by an equation of the form
ax3 + by3 + cz3 + dt3 = O.
Numerically, one can find all rational points of height :::; B for B in the range of up to 105 , thanks to a program due to D. J. Bernstein. On the other hand, there are precise conjectures concerning the constants in the asymptotics of rational points of bounded height due to Manin, Batyrev and the authors. Changing the coefficients one can obtain cubic surfaces with rank of the Picard group varying between 1 and 4. We check that numerical data are compatible with the above conjectures. In a previous paper we considered cubic surfaces with Picard groups of rank one with or without Brauer-Manin obstruction to weak approximation. In this paper, we test the conjectures for diagonal cubic surfaces with Picard groups of higher rank.
The Hasse principle for complete intersections in projective space BJORN POONEN ........................................................ 307
Assuming the existence of a smooth geometrically integral complete intersection X of dimension :::: 3 in pn over a number field K, such that the Zariski closure of X (K) is nonempty but of codimension :::: 2 in X, we construct a 3-dimensional smooth geometrically integral complete intersection X' in PK that violates the Hasse principle. Such a violation could not be explained by the Brauer-Manin obstruction or Skorobogatov's generalization thereof.
Une construction de courbes k-rationnelles sur les surfaces de Kummer d 'un produit de courbes de genre 1. PHILIPPE SATGE ........................................................ 313
k etant un corps de caracteristique differente de 2, nous decrivons une methode permettant de construire des courbes k-rationnelles (i.e. k-birationnellement equivalentes a la droite projective) sur les surfaces de Kummer associees a un produit de courbes de genre 1 munies d'involutions k-hyperelliptiques. Nous ramenons ce probleme a un probleme de
ABSTRACTS xv
geometrie enumerative sur Ie produit P1,k x P1,k de la droite projective par elle meme. Bien que la resolution generale du probleme de geometrie enumerative auquel nous arrivons soit hors de portee des methodes que nous connaissons, la recherche de solutions particulieres dans des systemes lineaires convenablement choisis permet d'obtenir des exemples interessants. On constate par exemple que l'on retrouve ainsi, de maniere assez systematique, plusieurs resultats qui apparaissent de maniere isolee dans la litterature.
Arithmetic Stratifications and Partial Eisenstein Series MATTHIAS STRAUCH .................................................... 335
Let P\ G and Q\H be generalized flag varieties over a number field F. In this paper we study certain locally trivial fibre bundles Y'7 over P\ G having Q\H as general fibre, and determine the arithmetic stratification of Y'7 with respect to a line bundle. The arithmetic stratification is defined in terms of height zeta functions and the height zeta function of a stratum is of the form
L e(SA,Hp(-y)) E2w - 1QO (S/-L, 'T](P-y)) , -YEP{F)\G{F)
where E2w - 1Qo is a ''partial Eisenstein series" associated to the Schubert
cell Q\Qw-1Qo. The computation of the constant term of these gives estimates that allow one to determine the abcissa of convergence of the height zeta function of the stratum.
Weak Approximation and R-equivalence on Cubic Surfaces SIR PETER SWINNERTON-DYER ........................................ 357
Let V be a nonsingular cubic surface defined over an algebraic number field K, and assume that V has points in every completion Kv' There is a long-standing problem of finding the obstructions to the Hasse principle and to weak approximation on V, the conjecture in each case (due to Colliot-Thelene and Sansuc) being that the obstruction is just the Brauer-Manin obstruction. The latter is known to be computable, though the algorithm is somewhat ugly and a heuristic process is usually preferable. Another way of phrasing the same problem is to ask what is the adelic closure of the set V (K).
A partial answer to this question is given by the following theorem: to each place v of bad reduction one can associate a finite disjoint union
V(Kv) = UWj{v)
xvi ABSTRACTS
which is easily computable in any particular case. The v-adic closure of any R-equivalence class in V(K) is a set Uiv which is the union of
some of the W j( v); and the adelic closure of any R-equivalence class is of
the form TI'Uiv x TI"V(Kv), where i depends on v, the first product is over all places of bad reduction and the second product is over all places of good reduction for V. Thus the adelic closure of V(K) is a union of
sets TI'W}v) x TI"V(Kv). For specific V a search program will give those products which contain a point of V(K) and in which points of V(K) are therefore everywhere dense. For a product which appears not to contain a point of V(K), it is reasonable to hope that there is a Brauer-Manin obstruction. For all V for which this process has been used, it turns out that one can indeed find the exact adelic closure of V(K) in this way. This is illustrated in the final section.
Hua's lemma and exponential sums over binary forms TREVOR D. WOOLEY .................................................. 405
We establish mean value estimates for exponential sums over binary forms of strength comparable with the bounds attainable via classical, single variable estimates for diagonal forms. These new mean value estimates strengthen earlier bounds of the author when the degree d of the form satisfies 5 ::; d ::; 10, the improvements stemming from a basic lemma which provides uniform estimates for the number of integral points on affine plane curves in mean square. Exploited by means of the Hardy-Littlewood method, these estimates permit one to establish asymptotic formulae for the number of integral zeros of equations defined as sums of binary forms of the same degree d, provided that the number of variables exceeds ~~ 2d , improving significantly on what is attainable either by classical additive methods, or indeed the general methods of Birch and Schmidt.
