Aim and subject
Shimura varieties,
defined in terms of reductive algebraic groups over the rational numbers, play
an important role in algebraic geometry and number theory. In many cases, their
cohomology provides the link between automorphic and Galois representations as
conjectured by Langlands. But at the same time Shimura varieties are moduli
spaces for rational polarizable Hodge structures. For example, the moduli
spaces of principally polarized abelian varieties are the Shimura varieties
associated to the groups of symplectic similitudes.
A point of a Shimura
variety is called special if its associated Mumford-Tate group is a
torus. In the case of the moduli spaces of abelian varieties, this notion is
the same as that of complex multiplications. In particular, the j-invariants of
special elliptic curves are the values of the classical j-function at the
points of the complex upper half space that are quadratic over the rationals.
These special points are important for the study of the arithmetic of Shimura
varieties.
Another way to think
of special points is to consider the morphisms between Shimura varieties that
are induced by morphisms of the algebraic groups in terms of which they are
defined. The irreducible components of the image of such morphisms are then
called special subvarieties, and the special points are just the
zero-dimensional special subvarieties. Special subvarieties are precisely the
subvarieties defined by Hodge-theoretical conditions, hence they are also
called subvarieties of Hodge type.
At this moment, two
conjectures have been made on the behavior of special points of Shimura
varieties, and partial results have been obtained. These conjectures are the
subject of the workshop.
The André-Oort
conjecture (by Yves André and Frans Oort) says that the special
subvarieties are precisely the subvarieties that contain a dense subset of
special points.
The equidistribution
conjecture says that the Galois orbits of a ``generic'' sequence of
special points in a given Shimura variety are equidistributed for the natural
hyperbolic measure. The word ``generic'' can have two meanings here: a sequence
is generic if it converges to the generic point of the Shimura variety for the
topology whose closed subsets are either all algebraic closed subvarieties, or
all special subvarieties. The André-Oort conjecture says precisely that the two
possible notions of generic coincide.
Both conjectures can
be seen as analogs of known results (formerly known as the Manin-Mumford
conjecture) for torsion points (or points of small height) and subvarieties of
abelian varieties (Raynaud, Ullmo, Zhang (see [1])). The results on the
André-Oort conjecture obtained so-far ([10], [7]) have already been applied in
other fields in mathematics: transcendence theory (Wolfahrt, see [7]),
Birch and Swinnerton-Dyer conjecture (Vatsal, Cornut, see [6])). As the
André-Oort conjecture characterizes subvarieties of Hodge type, it may have
implications for the Hodge conjecture. See [4], [5] and [14] for the most
recent work on the equidistribution conjecture.
The aim of the
workshop is to bring together the people working on these conjectures, and so
to combine the various approaches:
·
reduction modulo p (Moonen, Oort);
·
Galois orbits and Hecke correspondences
(Edixhoven, Yafaev);
·
equidistribution (Clozel, Cohen, Oh, Ullmo,
Zhang);
- Diophantine approximation (André).
A combination of these approaches could lead
to important progress. A question of particular interest is to see if
Edixhoven's method ([9]) can be used to derive the existence of a dense set of
positive dimensional special subvarieties from the existence of a dense set of
special points, and if this can then be used as input for results of Clozel and
Ullmo ([5]).
A second aim of the
workshop is to provide a good introduction into this subject for a somewhat
larger public.
Invited Keynote
Participants
·
Yves André (Paris)
·
Pascal Autissier (Paris)
·
Daniel Bertrand (Paris)
·
Florian Breuer
·
Laurent Clozel (Paris)
·
Christophe Cornut (Paris)
·
Marc Hindry (Paris)
·
Chandrashekhar Khare (Salt Lake City)
·
Philippe Michel (Montpellier)
·
Ben Moonen (Amsterdam)
·
Rutger Noot (Rennes)
·
Hee Oh (Princeton)
·
Richard Pink (Zürich)
·
Jacques Tilouine (Paris)
·
Emmanuel Ullmo (Paris)
·
Nike Vatsal (Seattle)
·
Gisbert Wüstholz (Zürich)
·
Andrei Yafaev (Londen)
·
Shouwu Zhang (New York)
Program
The workshop will be spread out over five
days. The first two days will serve as an introduction to the subject. This is
necessary because the experts working in this subject have their different
approaches. These first two days can also be considered as a short
instructional conference, accessible to a larger public of mathematicians
working in algebraic geometry, algebraic or analytic number theory, or ergodic
theory. The first two days we plan to have 4 lectures each day. The remaining 3
days we plan to have only two lectures per day, thus leaving ample time for
interaction between the participants. The participants are encouraged to work
together, to organise discussions and informal lectures in smaller groups.
References
[1]
A. Abbes. Hauteurs et
discrétude (d'après L. Szpiro, E. Ullmo et S. Zhang).
Séminaire Bourbaki, Vol. 1996/97. Astérisque No. 245, (1997),
Exp. No. 825, 4, 141-166.
[2] Y. André. G-functions and
geometry. Aspects Math. E.13, Vieweg 1989.
[3] Y. André. Distribution des
points CM sur les sous-variétés des variétés de modules de variétés abéliennes.
Manuscript, April 1997.
[4] L. Clozel, H. Oh, and
E. Ullmo. Hecke operators and equidistribution of Hecke points.
Invent. Math. 144 (2001), no. 2, 327-351.
[5] L. Clozel and E. Ullmo. Equidistribution
de sous-variétés spéciales. Preprint, October 2002.
[6]
C. Cornut. Non-trivialité
des points de Heegner. C. R. Acad. Sci. Paris,
Ser. I 334 (2002) 1039-1042.
[7]
S.J. Edixhoven and
A. Yafaev. Subvarieties of Shimura varieties. To appear in Annals
of Mathematics. Available on Edixhoven's home page.
[8]
S.J. Edixhoven. On the
André-Oort conjecture for Hilbert modular surfaces. Moduli of abelian
varieties, Progress in Mathematics 195 (2001), 133-155, Birkhäuser.
Available on the author's home page.
[9] S.J. Edixhoven. Special points
on products of modular curves. Preprint, January 2003. Available on the
author's home page.
[10] B.J.J. Moonen. Linearity properties
of Shimura varieties, II. Compositio Math. 114 (1998), no. 1, 3-35.
[11] F. Oort. Canonical liftings and
dense sets of CM-points. In: Arithmetic Geometry, Cortona Italy, October
1994 (ed. F. Catanese). Ist. Naz. Alta Mat. F. Severi 1997, Cambridge
Univ. Press; pp. 228-234.
[12]
A. Yafaev. A conjecture of
Yves André. Preprint, January 2003. Submitted for publication.
[13] A. Yafaev. On a result of Ben
Moonen. Preprint, January 2003.
[14]
S. Zhang. Elliptic curves,
L-functions, and CM-points. Preprint, July 2002.
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