Presentation
Autour des conjectures de Zilber-Pink / Around the
Zilber-Pink conjectures
The Zilber-Pink conjectures encompass a whole array of problems,
originating in the Mordell conjecture, and including the conjectures
of Manin-Mumford and Mordell-Lang, the André-Oort conjecture, and
questions on unlikely intersections raised by Bombieri-Masser-Zannier
and by Zilber. Given a semi-abelian scheme or more generally, a mixed
Shimura variety S, one studies how an algebraic
subvariety intersects the special subvarieties of
S. Following the standard motto of diophantine geometry, the
conjectures express that the geometry of S should govern its
arithmetic. For instance, if a curve meets the special subvarieties
of codimension at least 2 infinitely often, then it should lie in a strict special subvariety of S.
Important progress has been made on these problems in the last decade,
based on a variety of approaches which has turned the Zilber-Pink
conjectures into a meeting point of classical diophantine methods
(heights, Lehmer-type problems), arithmetic geometry (density of Galois
orbits), Hodge theory (Mumford-Tate and monodromy groups), and model
theory (o-minimality).
The objective of the summer school is two-fold. On the one hand, three
courses will be given during the first week, providing the audience
with an introduction to the theories of Shimura varieties, of heights
on algebraic groups, and of o-minimality. On the other hand, shorter
courses, lectures and problem sessions,
will be held during both weeks, giving the state of the art on the
conjectures, and listing open (and hopefully accessible) research
problems connected to these topics.
In addition, a satellite day will take
place on Saturday 30 June 2012.
|