Arithmetic Geometry Clay Mathematics Institute Summer School Arithmetic Geometry July 17 ...

570 Pages·2009·4.67 MB·English

Arithmetic Geometry

Clay Mathematics Proceedings
Volume 8
Arithmetic Geometry
Clay Mathematics Institute Summer School
Arithmetic Geometry
July 17–August 11, 2006
Mathematisches Institut,
Georg-August-Universität,
Göttingen, Germany
Henri Darmon
David Alexandre Ellwood
Brendan Hassett
Yuri Tschinkel
Editors
American Mathematical Society
Clay Mathematics Institute
2000 Mathematics Subject Classification. Primary 14E30,14G05, 14D10.
The figure on the front cover, “Rational Curves on a K3 Surface,”
is courtesy of Noam D. Elkies.
Library of Congress Cataloging-in-Publication Data
Arithmeticgeometry/HenriDarmon...[etal.],editors.
p.cm. —(Claymathematicsproceedings;v.8)
Includesbibliographicalreferences.
ISBN978-0-8218-4476-2(alk.paper)
1.Arithmeticalalgebraicgeometry. I.Darmon,Henri,1965–
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Contents
About the cover: Rational curves on a K3 surface
Noam D. Elkies 1
Curves
Rational points on curves
Henri Darmon 7
Non-abelian descent and the generalized Fermat equation
Hugo Chapdelaine 55
Merel’s theorem on the boundedness of the torsion of elliptic curves
Marusia Rebolledo 71
Generalized Fermat equations (d’apr`es Halberstadt-Kraus)
Pierre Charollois 83
Heegner points and Sylvester’s conjecture
Samit Dasgupta and John Voight 91
Shimura curve computations
John Voight 103
Computing Heegner points arising from Shimura curve parametrizations
Matthew Greenberg 115
The arithmetic of elliptic curves over imaginary quadratic fields and
Stark-Heegner points
Matthew Greenberg 125
Lectures on modular symbols
Yuri I. Manin 137
Surfaces
Rational surfaces over nonclosed fields
Brendan Hassett 155
Non-abelian descent
David Harari 211
Mordell-Weil Problem for Cubic Surfaces, Numerical Evidence
Bogdan Vioreanu 223
v
vi CONTENTS
Higher-dimensional varieties
Algebraic varieties with many rational points
Yuri Tschinkel 243
Birational geometry for number theorists
Dan Abramovich 335
Arithmetic over function fields
Jason Starr 375
Galois + E´quidistribution=Manin-Mumford
Nicolas Ratazzi and Emmanuel Ullmo 419
The Andr´e-Oort conjecture for products of modular curves
Emmanuel Ullmo and Andrei Yafaev 431
Moduli of abelian varieties and p-divisible groups
Ching-Li Chai and Frans Oort 441
Cartier isomorphism and Hodge Theory in the non-commutative case
Dmitry Kaledin 537
Introduction
Classically, arithmetic is the study of rational or integral solutions of Diophan-
tine equations. From a modern standpoint, this is a particular case of the study
of schemes over algebraically nonclosed fields and more general commutative rings.
The geometric viewpoint, dating back to ancient Greece, has been a source of in-
spiration to generations of mathematicians. The guiding principle is that
geometry determines arithmetic.
The tremendous power of this principle has been amply demonstrated in the
works of Faltings on the Mordell conjecture and Wiles on Fermat’s last theorem.
Thisvolumegrewoutofthe2006ClaySummerSchoolheldattheMathemati-
schesInstitutoftheUniversityofG¨ottingen. Thegoaloftheschoolwastointroduce
participants to the wealth of new techniques and results in arithmetic geometry.
The first three weeks of the school were devoted to three main courses, covering
curves, surfaces and higher-dimensional varieties, respectively; the last week was
dedicated to more advanced topics. An important component of the school was a
seminar focused on computational and algorithmic aspects of arithmetic geometry.
The present proceedings volume reflects this structure.
Curves:
The main geometric invariant of a curve is its genus; the arithmetic is very
different for curves of genus 0,1 and ≥ 2 respectively. In genus 0, we can answer,
completelyandeffectively,whetherornotacurvecontainsrationalpointsandhow
thesepoints are distributed. The theoryof genus 1curvesis oneof the richest sub-
jects in mathematics, with spectacular recent theorems, e.g., modularity of elliptic
curves over the rationals, and with many outstanding open questions, such as the
Birch/Swinnerton-Dyer conjecture. In higher genus, the most fundamental result
is the proof of the Mordell conjecture by Faltings, and the most challenging open
question is to give an effective version of this result.
The lecture notes by Darmon cover the following topics:
• Faltings’ proof of the Mordell Conjecture;
• Rational points on modular curves and Mazur’s approach to bounding
them;
• Rational points on Fermat curves and Wiles’ proof of Fermat’s Last The-
orem;
vii
viii INTRODUCTION
• Elliptic curves and the Birch and Swinnerton-Dyer conjecture, following
Gross-Zagier and Kolyvagin.
Contributions by Chapdelaine, Charollois, Dasgupta, Greenberg, Rebolledo, and
Voight discuss more specialised topics that grew out of these lectures, such as
• Generalised Fermat equations (Chapdelaine);
• Merel’s extension of Mazur’s techniques to study rational points on mod-
ular curves over number fields, and the uniform boundedness conjecture
for torsion of elliptic curves (Rebolledo);
• Natural generalisations of Fermat’s Last Theorem due to Kraus and Hal-
berstadt, building on Frey’s approach (Charollois);
• CM points on modular curves and their applications to elliptic curves
(Dasgupta, Voight);
• Shimura curves with a focus on computational aspects (Voight, Green-
berg);
• Stark-Heegner points (Greenberg).
Inaddition,apaperbyManintreatsmodularsymbols(whichplayanimportant
roleinMerel’sproofoftheuniformboundednessconjectureexplainedinRebolledo’s
article) and discusses higher dimensional generalizations.
Surfaces:
Thegeometryofsurfacesoverthecomplexnumbersismuchmoreinvolved,and
theirbirationalclassificationwasamilestoneinalgebraicgeometry. Hassett’spaper
gives a thorough introduction to this classification over nonclosed fields, and its
implicationsforDiophantinequestionsliketheexistenceofrationalpointsandweak
approximation. It also touches on geometric descent constructions generalizing
Fermat’sdescent(universaltorsors)andalgebraicapproachestotheseobjects(Cox
rings).
Harari’s paper discusses non-abelian versions of descent, which have yielded
new counterexamples to local-global principles for rational points on surfaces over
number fields. Once rational points exist, one can ask whether they are Zariski
dense and analyze their distribution with respect to heights; these questions are
addressed,forbothsurfacesandhigher-dimensionalvarieties,inTschinkel’ssurvey.
Vioreanu offers tantalizing computational evidence for conjectures about the
algebraic structure of rational points on cubic surfaces. He explores whether all
points can be generated from a small number using elementary geometric opera-
tions.
Higher-dimensional varieties:
Some of the most interesting higher-dimensional varieties from the arithmetic
pointofviewarelow-degree hypersurfacesandvarietiescloselyrelatedtoalgebraic
groups: toric varieties, homogeneous spaces, and equivariant compactifications of
groups. Hereoneisinterestedinexistencequestions,densityofrationalpoints,and
counting points of bounded height. For the last problem, height zeta functions are
animportanttoolandtechniquesofharmonic analysiscanbeprofitablyemployed.
A selection of recent results in this direction appears in the survey of Tschinkel.
ABOUT THE COVER: RATIONAL CURVES ON A K3 SURFACE 3
we have a basis (x ,x ,x ) for Γ(D) for which S can be given by
0 1 2
S(x ,x ,x ) = 3686400x6
0 1 2 0
−256(33975x2−8569x x −45x2)x4
1 1 2 2 0
+(5130225x4−1860100x3x +138414x2x2−4180x x3+9x4)x2
1 1 2 1 2 1 2 2 0
−8(5643x5x −2495x4x2+209x3x3−5x2x4).
1 2 1 2 1 2 1 2
The symmetry takes (x : x : x ) to (−x : x : x ), and the node is at
0 1 2 0 1 2
(x : x : x ) = (0 : 0 : 1). Our model of X/Q is obtained from the double
0 1 2
cover y2 = S(x ,x ,x ) by blowing up the preimage of this node. In the picture,
0 1 2
the real locus of the sextic curve C :S =0 is plotted in black in the (x /x ,x /x )
0 2 1 2
plane; it consists of nine components, together with an isolated point at the node.
Aline‘⊂P2 istritangenttoC ifandonlyifitliftstoapairofsmoothrational
curves ‘ on X. Then ‘ +‘ = D in NS(X). Orthogonal projection to L⊗Q
± + −
then maps NS(X) to a lattice L0 containing L with index 2, taking the curves ‘
±
to a pair of vectors ±v = ‘ − 1D ∈ L0 of norm 5/2 that are orthogonal to R .
± 2 L
Conversely, every such pair comes from a tritangent line. There are 43 such lines;
oneoftheseisx =0,whichisthelineatinfinityinourpicture,andtheremaining
2
42areplottedingreen. (Someofthetangencypointsarenotinthepicturebecause
they are either complex conjugate or real but outside the picture frame.)
Each of these lines has the property that the restriction S| is the square of a
‘
cubicpolynomial. Thesameistrueif‘isalinepassingthroughthenodeofC and
tangent to C at two other points. There are nine such lines, plotted in gray. They
correspond to norm-(5/2) vectors in L0 not orthogonal to R , up to multiplication
L
by −1 and translation by R .
L
A generic line λ⊂P2 meets these 43+9 lines in 52 distinct points that lift to
52 pairs of rational points on the genus-2 curve y2 = S| . This already improves
λ
on the previous record for an infinite family of genus-2 curves over Q (which was
24 pairs, due to Mestre). We do better yet by exploiting rational curves of higher
degree in P2 on which S restricts to a perfect square.
There are 1240 conics c ⊂ P2 for which S| is a square; geometrically these
c
are the conics such that each point in the intersection c∩C has even multiplicity
(eitherthenodeofC orapointoftangency). Suchaconicliftstoapairofrational
curves c on X with c +c = 2D. These c come from vectors c −D ∈ L
± + − ± ±
of norm 4 up to translation by R , except for norm-4 vectors of the form v−v0
L
with v,v0 ∈L0 of norm 5/2. The conics c are all rational over Q, because for each
c we can find c0 such that c ·c0 is odd. In general the intersections of c with a
+ +
generic line λ ⊂ P2 need not be rational, but we can choose λ so as to gain a few
rational points. Most notably, 18 of the conics happen to pass through the point
P : (x : x : x ) = (0 : 1 : 3) on the axis of symmetry x = 0 of the sextic C.
0 0 1 2 0
These conics are plotted in purple on our picture. If λ is a generic line through P
0
then the genus-2 curve y2 = S| gains 18 more pairs of points above the second
λ
intersections of λ with the purple conics. We also lose one pair because two of
our 52 tritangent lines pass through P , but we gain two more pairs by finding two
0
rationalcubiccurvesκ⊂P2 forwhichS| isasquareandP isthenodeofκ. This
κ 0
bringsthetotalto52+18−1+2=71. Ifc ,c aretwooftheremaining1222conics
1 2
such that (c ) ·(c ) is odd then we have infinitely many choices (parametrized
1 + 2 +
by an elliptic curve of positive rank) of lines λ 3 P for which each of λ∩c and
0 1
λ∩c consists of two further rational points, bringing our total to 75. This is the
2