
Recent Trends in the Arithmetic of Moduli and Automorphic Forms Date: September 8th to 13th, 2013
Location: Ringhotel Schorfheide, Hubertusstock 2, 16247 Joachimsthal
Main Speakers:
Jochen Heinloth (University of DuisburgEssen)
An introduction to the geometric Langlands program (show abstract [here])
An introduction to the geometric Langlands program (hide abstract [here])
In the first lecture I will try to give some motivation about
the origins of the geometric Langlands program, give some
first examples and then explain which moduli spaces appear
in the formulation. A key result is then a theorem of Satake
and its geometric version, showing how the representation
ring of an algebraic group is encoded in the geometry of the
socalled affine Grassmanninan. Again it is interesting to see
how this works in simple examples. Using this result we will
try to give some more examples of the correspondence,
showing that even 0dimensional moduli spaces can be used
to construct arithmetic objects with interesting properties.
If time permits we will try to end the lectures giving some
ideas about more recent results on the program.
Max Lieblich (University of Washington)
Moduli of sheaves on K3 surfaces (show abstract [here])
Moduli of sheaves on K3 surfaces (hide abstract [here])
I will present some results on the moduli spaces of sheaves on K3
surfaces. One of the crucial properties of K3 surfaces is that one
can construct many moduli spaces of sheaves (or sheaflike objects)
that are themselves K3 surfaces, and one can understand much of
the cohomology of these spaces in terms of the original ambient
surface. This rich collection of spaces yields numerous insights
into the derived categories of K3 surfaces, the geometry of their
moduli, and their arithmetic properties.
Shouwu Zhang (Princeton University)
Diophantische Analysis und Modulfunktionen
(Diophantine analysis and modular functions) (show abstract [here])
Diophantische Analysis und Modulfunktionen
(Diophantine analysis and modular functions) (hide abstract [here])
In his seminar paper in 1952 with the title as above, Kurt Heenger
developed a new method to solve diophantine equations using modular
functions. Using this method, he solved not only the famous Gauss
class number one problem, but also the first important cases of the
congruent number problem. In my lectures, I will explain recent
developments in the line of Heegner's work including the GrossZagier
formula, the Waldspurger formula, and Kolyvagin's theorem, and solutions
of new cases of the congruent number problem.
Participation is restricted to the members of the Research Training Group
Schedule: [here]
Participants: [here]

