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Conference "Global Fields"October 25 - 28, 2011Moscow, Russia |
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Organisers: Philippe Lebacque (Laboratoire de Mathématiques de Besançon ), Michael Tsfasman (CNRS, Laboratoire Poncelet, Institute for Information Transmission Problems), Alexey Zykin (Laboratoire Poncelet, State University Higher School of Economics, IITP)
Thursday 27 October, 15:40 - 17:40
Teichmuller curves are rather exotic objects that have received a lot
of attention in recent years but are still only very partially understood.
A Teichmuller curve is an algebraic curve in the moduli space of curves
of genus g which is totally geodesic for the Teichmuller metric.
For large g it is not even known whether there are infinitely many such curves,
but for g=2 a great deal is known, thanks to the work of McMullen and others:
there are countably many such curves, each lying on some (unique) Hilbert
modular surface, and conversely one or two Teichmuller curves on each Hilbert
modular surface. In recent work with M. Moller we have found much more
explicit descriptions of these curves than were previously known,
as the zero-loci of explicit Hilbert modular forms; this casts new light
on the number theory and algebraic geometry of the curves and also on
the properties of the associated Picard-Fuchs differential equations.