July 17 — August 11, 2006
the Mathematisches Institut Georg-August-Universität
The program will run Monday through Friday, 10:00 to 17:00, with lectures at 10am, 11:30 and 14:30. There will also be seminars/workshops on computational aspects of arithmetic algebraic geometry at 16:00. Lunch will be organized for all participants Monday through Friday at the University Cafeteria (Mensa) at 12:45. There will be an opening reception on Tuesday, July 18, and a closing banquet on Thursday, August 10, 2006. Evening meals will not be organized by the school, but all participants will receive a modest per diem to supplement the cost of weekend and evening meals.
The key finiteness theorems
This course will focus on the study of rational points on curves of
higher genus. The schedule will run approximately as follows:
2. The Mordell-Weil theorem
3. Faltings' Theorem
4. Modular curves and Mazur's Theorem
5. Fermat curves and Wiles' Theorem
This course is aimed at graduate students entering the field and I will try to assume a minimal background. Comfort with basic notions of algebraic geometry, particularly algebraic curves, and algebraic number theory, such as can be found for example in
- Arbarello, Cornalba, Griffiths, and Harris, J., Geometry of algebraic curves. Vol. I. Grundlehren der Mathematischen Wissenschaften, 267. Springer-Verlag, New York, 1985,
- Fröhlich and Taylor, Algebraic number theory Cambridge Studies in Advanced Mathematics, 27. Cambridge University Press, Cambridge, 1993
on Elliptic Curves
This course will be somewhat more specialised, focussing
on elliptic curves and their rational points with special emphasis on the
Heegner point construction arising from modularity and the theory of
complex multiplication. A selection of topics will include:
1. Heegner points
2. The theorems of Gross-Zagier and Kolyvagin
3. Shimura curve parametrisations
4. p-adic uniformisation
5. Stark-Heegner points.
- H. Darmon, Rational points on modular elliptic curves, CBMS Regional Conference Series in Mathematics, 101. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004.
- M. Bertolini and H. Darmon, The rationality of Stark-Heegner points over genus fields of real quadratic fields
- M. Bertolini and H. Darmon, Hida families and rational points on elliptic curves
This course will introduce the geometry of rational surfaces,
with a view toward arithmetic applications.
- First we will review the classification of smooth projective rational surfaces over algebraically closed fields, offering numerous concrete examples: cubic surfaces, complete intersections of two quadrics, and toric surfaces. The surfaces are realized as blow-ups of the projective plane or rational ruled surfaces.
- Next we sketch the classification of rational surfaces over nonclosed fields, following Iskovskih, Manin, and others. Minimal Del Pezzo surfaces and conic bundles are the main building blocks. One key invariant is the Galois action on the Néron-Severi lattice, which encodes some of the arithmetic information about the surface. (Specific instrances will be explored in more detail in Professor Kresch's lectures.) We will foreshadow the birational techniques developed later in Professor Abramovich's lectures.
- We will also discuss singular Del Pezzo surfaces. These arise naturally in classification - given a surface with nef and big anticanonical class, there is a natural `anticanonical' model with rational double points. The presence of singularities appears to facilitate the counting of rational points, e.g., all the cubic surfaces which have been analyzed successfully are singular. Furthermore, for Del Pezzo surfaces over Dedekind domains, the reduction modulo primes are generally singular (cf. Corti's work on Del Pezzo surfaces over Dedekind domains.) A detailed understanding of the fibers can yield arithmetic results, like weak approximation over function fields.
- We will conclude with a discussion of Cox rings and universal torsors. We review the formalism of universal torsors, following Colliot-Thélène, Sansuc, Skorobogatov and others. Cox rings (or total coordinate rings) are useful for producing concrete equations of universal torsors for special classes of varieties. We discuss recent work of Derenthal, Popov, and Batyrev on effective computation of generators and relations for the Cox ring.
the Hasse Principle
These lectures will cover the theory of descent
and the Brauer-Manin obstruction to the
Hasse principle and weak approximation,
particularly as applied to rational surfaces.
One of the main goals will be to outline,
thoroughly and explicity, the passage from
defining equation to effective computation of
the Brauer-Manin obstruction, for certain
del Pezzo surfaces.
Roughly, the topics of the lectures will be:
1. Brauer groups, Galois cohomology
2. Brauer-Manin obstruction with quaternion algebras: examples
3. Descent, torsors
4. Hasse principle and Brauer-Manin obstruction
5. Further examples
- J.-L. Colliot-Thélène, D. Kanevsky, and J.-J. Sansuc, Arithmétique des surfaces cubiques diagonales, Diophantine approximation and transcendence theory (Bonn, 1985), Lecture Notes in Math. 1290, Springer, Berlin, 1987, 1-108
- J.-L. Colliot-Thélène and J.-J. Sansuc, La descente sur les variétés rationnelles, Journées de Géometrie Algébrique d'Angers (July 1979), Sijthoff & Noordhoff, Alphen aan den Rijn, 1980, 223-237
- A. Kresch and Yu. Tschinkel, On the arithmetic of del Pezzo surfaces of degree 2, Proc. London Math Soc. (3) 89 (2004), 545-569
- A. Skorobogatov, Torsors and rational points, Cambridge Univ. Press, Cambridge, 2001
Descent and Weak Approximation
The goal of the lectures is to present concrete applications of the theory of
descent. We will focus on qualitative questions: existence of a rational point
for a (smooth and projective) surface X defined over a number field k,
density (for Zariski or adelic topology) of the set X(k) of rational points.
In particular we will deal with non-abelian cohomology.
The schedule will run approximately as follows:
1. The general framework of non-abelian descent
2. Application to bielliptic surfaces
3. Application to Enriques surfaces
- D. Harari, Weak approximation and non-abelian fundamental groups, Ann. Sci. Ecole Norm. Sup. (4), 33, (2000), no. 4, 467-484
- D. Harari, A.N. Skorobogatov, Non-abelian descent and the arithmetic of Enriques surfaces, Int. Math. Res. Not. 52, (2005), 3203--3228
- J-P. Serre, Cohomologie Galoisienne/Galois Cohomology, Springer-Verlag, 1994/2002
- A.N. Skorobogatov, Torsors and rational points, Cambridge University Press 2001
Varieties with many rational points
In this course, we study the distribution of rational points
with respect to heights. We focus on varieties closely related to
linear algebraic groups, e.g., equivariant compactifications of groups
and homogeneous spaces. In these cases, questions concerning
the asymptotic distribution of points of bounded height are
closely related to adelic harmonic analysis on the groups.
On the other hand, analytic techniques lead naturally to
investigations of global geometric invariants of the underlying
varieties, studied in the context of the minimal model program.
The schedule will run approximately as follows:
1. General introduction
2. Circle method and hypersurfaces
3. Toric varieties
4. Height zeta functions of toric varieties
5. Flag varieties
6. Compactifications of additive groups
7. Spherical varieties
8. Conjectures on rational and integral points
- E. Bombieri, W. Gubler, Heights in diophantine geometry, Cambridge University Press 2006
- A. Chambert-Loir, Y. Tschinkel, On the distribution of points of bounded height on equivariant compactifications of vector groups, Inventiones Math. 148, no. 2, 421-452, (2002)
- J. Shalika, R. Takloo-Bighash, Y. Tschinkel, Rational points on compactifications of semi-simple groups, (2006)
I shall describe the classical application of the circle
method to the Waring problem, following , and then
try to explain how Deligne's estimates of exponential
lead to Heath-Brown's theorem on cubic forms in ten
variables . In the course of the lectures further
references will be given.
- R.C. Vaughan, The Hardy-Littlewood method, Cambridge, 1997
- D.R. Heath-Brown, Proc. London Math. Soc., 47 (1983), 225-257
- D.R. Heath-Brown, J. Reine Angew. Math., 481 (1996), 149-206
Lecture 1: The Tsen-Lang Theorem
Tsen and Lang studied conditions on a field generalizing algebraic closure. They proved function fields over algebraically closed fields satisfy the conditions. I will discuss the proof, a corollary for the Brauer group and some related issues.
References: Sections II.3 and II.4 of Serre's "Galois cohomology". Section IV.6 of Kollár's "Rational curves on algebraic varieties".
Lecture 2: Arithmetic over function fields of curves
By Tsen's theorem, the function field of a curve over an algebraically closed field is C1. In the early 1990's, it was conjectured the field satisfies a stronger condition: every rationally connected variety over the field has a rational point. I will discuss the proof, some corollaries, and some related issues like weak approximation.
References: "Lectures on rationally connected varieties" by Joe Harris, notes by Joachim Kock.
Lecture 3: Arithmetic over function fields of surfaces
Tsen and Lang proved the function field of a surface over an algebraically closed field is C2. I will discuss a conjectural generalization analogous to the generalization of Tsen's theorem in the previous lecture. I will discuss a proof under additional hypotheses. One corollary is a new proof of a theorem of de Jong about Brauer groups of function fields of surfaces.
References: "Algèbres simples centrales sur les corps de fonctions de deux variables (d'aprés A. J. de Jong)" by Jean-Louis Colliot-Thélène. "Almost proper GIT-stacks and discriminant avoidance" by A. J. de Jong and J. Starr Preprint
The schedule will run approximately as follows:
- Geometry and arithmetic of curves: closed and open.
- Kodaira dimension
a. Iitaka's program: Viehweg's additivity theorem
b. Rationally Connected and Iitaka fibrations
c. Lang's conjectures and implications
d. Logarithmic Kodaira dimension
e. The Lang-Vojta conjecture
- Campana's program
a. C-folds, Kodaira dimension and Campana's core
b. Integral points and the category of C-folds
c. Campana's arithmetic conjectures
- The minimal model program
a. Cone of curves
b. Bend and break
c. Cone theorem
d. The Minimal Model Conjecture
e. Flip conjectures
Vojta, Campana and ABC
b. Vojta's conjectures
c. Vojta and ABC
d. Campana and ABC
- M. Hindry and J. Silverman, Diophantine geometry. An introduction, Graduate Texts in Mathematics, 201, Springer-Verlag, New York, 2000
- O. Debarre, Higher-dimensional algebraic geometry, Universitext. Springer-Verlag, New York, 2001
- F. Campana, Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 3, 499-630
- F. Campana, Fibres multiples sur les surfaces: aspects geométriques, hyperboliques et arithmétiques, Manuscripta Math. 117 (2005), no. 4, 429-461
- P. Vojta, A more general abc-conjecture, Internat. Math. Res. Notices 1998, no. 21, 1103-1116
Consider a projective variety defined over a number field together with a height function on its algebraic points. There are many situations where Galois orbits of algebraic points of small height exhibit remarkable equidistribution properties : the projective space itself with its usual height (max of coordinates), abelian varieties with the Néron-Tate height, or heights on the projective line normalized by an algebraic dynamical system of degree at least 2.
Such an equidistribution holds not only for the complex topology but also for the p-adic one, provided one works in the framework of p-adic analytic spaces as defined by V. Berkovich.
The schedule will run approximately as follows:
1. Equidistribution on the projective line (elementary methods)
2. Arakelov geometry and equidistribution
3. Equidistribution on Berkovich spaces
- L. Szpiro, E. Ullmo, S. Zhang, Équirépartition des petits points. Invent. Math. 127 (1997), no. 2, 337-347.
- A. Chambert-Loir, Mesures et édistribution sur les espaces de Berkovich. J. f. die reine und angewandte Mathematik (2006), to appear.
Computational Aspects of Arithmetic Algebraic Geometry
The schedule will run as follows:
Mo. G.-M. Greuel
Th. M. Stoll: Computations with genus 2 curves
Fr: M. Stoll
The schedule will run as follows:
Mo. J. Jahnel
Tue. J. Jahnel
Thu. S. Wiedmann
Fr: S. Wiedmann
The schedule will run as follows:
Mo. O. Labs: Construction of Hypersurfaces with Singularities via Finite Field Experiments
Tue. O. Labs
Thu. J. Schicho: How to use Lie Algebras for Solving Diophantine Equations
Fr. J. Schicho
We discuss geometric and arithmetic properties of the moduli space
of polarized abelian varieties in positive characteristic. In
characteristic zero we have strong tools at our disposal: besides
algebraic-geometric theories we can use analytic and topological
methods. It seems that we are at a loss in positive characteristic.
However the opposite is true. Phenomena, only occurring in positive
characteristic provide us with strong tools to study moduli spaces.
We discuss two theorems recently proved. The main focus will be several
basic techniques, which will be used to prove these theorems, but which
have a much more general scope.
1. Introduction: Density of a Hecke orbit, and a conjecture by Grothendieck.
2. Serre-Tate theory.
3. The Tate-conjecture: l-adic and p-adic.
4. Dieudonné modules and Cartier modules.
5. A conjecture by Manin and the weak Grothendieck conjecture.
6. Hilbert modular varieties.
7. Purity of Newton polygon stratifications and deformations of p-divisible groups.
8. Proof of the density of ordinary Hecke orbits.
9. Proof of the Grothendieck conjecture.
Introduction and discussion of the basic theorems (1 hour), extensive discussions of the basic tools (6 hours, every hour can be followed independently of the others), the proofs of the two theorems discussed (2 hours, here we use the tools exposed before). We will provide notes with ample references and precise formulations of all results discussed. We expect that the audience is familiar with the idea of a moduli space. We will use the concepts of: an abelian variety, isogenies between abelian varieties, and the notion of a polarization. Other tools will be introduced and discussed in our course.
Dieudonné theory arose during the 1950's after it was realized
through the work of Chevalley that the Lie theory dictionary between
algebraic groups and their Lie algebras breaks down if the field over which
one works has positive characteristic. Dieudonné, succeeded by Barsotti,
Cartier, Gabriel, Manin, ... developed a theory to describe formal groups in
terms of associated modules over appropriate rings. Barsotti and then Tate
independently introduced p-divisible groups, studied their relations with
abelian varieties (or more generally abelian schemes) and gave geometric and
arithmetic applications. Grothendieck in 1966, in the course of his
reflections which led to his conception of crystalline cohomology,
formulated a program for crystalline Dieudonné theory. This program was
initially developed by Grothendieck himself, then in part by Messing,
Mazur-Messing and more systematically by Berthelot, Breen, Messing who gave
a homological definition of the Dieudonne crystal associated to a
p-divisible group or more generally associated to any finite, locally-free
group scheme defined over a p-adic base scheme and annihilated by a power of
p. This work led to a well greased machine and established the
full-faithfulness of the crystalline Dieudonné functor over any locally
Noetherian regular scheme of characteristic p. Subsequent work of de Jong
and independently Kato established that the functor from p-divisible groups
to locally-free Dieudonné crystals is an equivalence of categories when the
base scheme is smooth over a field possessing a finite p-basis. de Jong
also established the functor from p-divisible groups up to isogeny to
Dieudonné iso-crystals is fully-faithful over any base scheme of finite type
over a field admitting a finite p-basis. In addition he gave a spectacular
application of the theory by establishing Tate's conjecture on homomorphisms
between p-divisible groups over normal irreducible locally Noetherian
schemes of characteristic p. In the 1990's Zink developed a quite different
approach to the theory, via his concept of display. Displays are quite
elementary and concrete objects living in the domain of (semi)-linear
algebra and it is remarkable that using them one can recapture the Dieudonne
crystals. The talks will be devoted to explaining Zink's beautiful ideas
and results and to discussing their relation with the crystalline Dieudonné
theory, which will be summarized in a much more cursory manner.
- Th. Zink, The display of a formal p-divisible group, Asterisque, 278, 127 - 248, 2002
- Th. Zink, A Dieudonné theory of p-divisible groups, in Class Field Theory - Its Centenary and Prospects, 139 - 160, Tokyo, 2001
- Th. Zink, Windows for displays of p-divisible groups, in Moduli of Abelian Varieties, Progress in Mathematics vol. 195, 491 - 518, Birkhäuser Verlag, 2001
- P. Berthelot, L. Breen, W. Messing, Theorie de Dieudonné cristalline II, Lecture Notes in Mathematics, volume 930, Springer Verlag, 1982
- P. Berthelot, W. Messing, Theorie de Dieudonné cristalline III: theoremes d'equivalence et pleine fidelite, in Grothendieck Festschrift, vol I, 173 - 247, Progress in Mathematics, vol. 86, Birkhäuser Verlag, 1990
- A.J. de Jong, Crystalline Dieudonné module theory via formal and rigid geometry, Publ. Math. IHES, no. 82, 5 - 96, 1995
- A.J. de Jong, W. Messing, Crystalline Dieudonné theory over excellent schemes, Bull. Soc, Math. France, t. 127, 333 - 348, 1999
- J.-M. Fontaine, Groupes p-divisible sur les corps locaux, Asterisque, 47-48, 1977
- A. Grothendieck, Groupes de Barsotti-Tate et cristaux de Dieudonné, Seminaire de Mathematiques Superieures, no. 45 (Ete 1970), Les Presses de l'Universite de Montreal, 1974
We will explain how to generalize the Frobenius action on the de Rham cohomology and the Cartier isomorphism to the non-commutative setting. As an application, we will prove the degeneration, under appropriate assumptions, of the non-commutative Hodge-to-de Rham spectral sequence, using the method of Deligne-Illusie.
- P. Deligne, L. Illusie, Relèvements modulo p2 et décomposition du complexe de de Rham, Inventiones Math. 89 (1987), 247-270
- J.-L. Loday, Cyclic homology, Springer-Verlag, Berlin, 1998
Classical modular symbols are elements of the 1-homology groups of modular curves represented by the classes of geodesics joining two cusps. Studying integrals of cusp forms of weight two along them, one gets beautiful formulas both for coefficients of these forms and for values of their Mellin transforms at integer points of the critical strip. This technique can be generalized to cusp forms of higher weight (this is also classical). I will briefly report about these resulta, and then turn to a recent development where integrals are replaced by iterated integrals in the sense of Chen. Many unsolved problems will be discussed.
- Yu. Manin, Parabolic points and zeta-functions of modular curves, Math. USSR Izvestija, publ. by AMS, vol. 6, No. 1 (1972), 19 - 64
- Yu. Manin, Periods of parabolic forms and $p$--adic Hecke series Math. USSR Sbornik, 21:3 (1973), 371 - 393
- Yu. Manin, Iterated integrals of modular forms and noncommutative modular symbols 37 pp., e--Print math.NT/0502576
- L. Merel, Quelques aspects arithmétiques et géométriques de la théorie des symboles modulaires Thése de doctorat, Université Paris VI, 1993.
- L. Merel, Universal Fourier expansions of modular forms, Springer Lecture Notes in Math., vol. 1585 (1994), 59 - 94
We will discuss rational points and rational curves on algebraic varieties over
finite fields. In addition to effects of positive characteristic (e.g., unirationality
of certain surfaces of intermediate and general type), there are unexpected
geometric and arithmetic properties provided by Galois symmetries
over finite fields.
- F. Bogomolov, Y. Tschinkel, Rational curves and points on K3 surfaces, Amer. J. Math. 127, (2005), 825-835
- F. Bogomolov, Y. Tschinkel, Curves in abelian varieties over finite fields, Int. Math. Res. Not. 4, (2005), 233-238
This course concerns equidistribution properties of special points and
The schedule will run approximately as follows;
1. The André-Oort conjecture and the Manin-Mumford conjecture.
In this lecture the statement and results concerning the André-Oort conjecture are explained. A proof of the Manin-Mumford conjecture obtained by translating the strategy for the André-Oort conjecture in the abelian case will be given. (The proof is quite close to the one given by Hindry.)
2. Equidistribution of special subvarieties of Shimura varieties.
I will explain the paper with Clozel with some preliminaries on ergodic theory (Ratner, Margulis) and some generalisations.
- L. Clozel, E Ullmo, Equidistribution de sous-variétés spéciales, Annals of Maths 161, (2005), 1571-1588
- P. Deligne, Variétés de Shimura: interprétation modulaire et techniques de construction de modèles canoniques, in Automorphic Forms, Representations, and L-functions part 2, A. Borel, W. Casselman (eds), Proc. of Symp. in Pure Math. 33, AMS, (1979), 247-290
- B. Edixhoven, A. Yafaev, Subvarieties of Shimura varieties, Ann. Math. (2) 157, (2003), 621-645
- B. Klingler, A. Yafev, On the André-Oort conjecture, preprint 2006
- E. Ullmo, Equidistribution de sous-variétés spéciales II, preprint 2005
- E. Ullmo, A. Yafev, Galois orbits of special subvarieties of Shimura varieties, preprint 2006
- A. Yafaev, A conjecture of Yves André, to appear in Duke Maths. Journal
- A. Yafaev, On a result of Moonen on the moduli space of principally polarized abelian varieties, Compositio Math. 141, (2005) no 5, 1103-1108
In this course we discuss finite field analogs of classical geometric results for algebraic varieties.
- B. Poonen, Bertini theorems over finite fields, Ann. of Math. (2) 160 (2004), no. 3, 1099-1127