Special Points in Shimura Varieties

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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