Lorentz Center

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## Heights and Moduli Spaces |

The
aim of the workshop is to discuss recent developments around heights and moduli
spaces. Moduli spaces belong to the most basic and intensively studied objects
in mathematics. They are geometric objects parametrizing
other geometric objects of a specific kind, such as curves, abelian
varieties or vector bundles. Traditionally
moduli spaces have been studied using topological or analytical methods. This
has led to fruitful connections with complex geometry and mathematical physics.
That moduli spaces also possess deep and interesting arithmetic aspects was
already discovered in the 19th century, but Grothendieck's
theory of schemes, developed in the early 1960s, made it clear that moduli
spaces can actually be defined over the ring of integers. The
arithmetic viewpoint on moduli spaces came into full bloom in the 1980s with G.
Faltings's proof of the Mordell
conjecture on finiteness of rational points on curves, and the discovery by B.
Gross and D. Zagier of a formula establishing a
precise connection between the derivative of the L-series of a modular elliptic
curve evaluated at its critical point, and the height of a CM point. In both
results heights play a crucial role. After
these results the arithmetic of moduli spaces, and especially modular heights,
became a hot topic of research, and results in this area continue to attract
the attention of many. For example in the direction of Faltings's
proof one could mention ongoing work on isogeny estimates, or the development
of higher dimensional Arakelov theory, especially
related to logarithmically singular hermitian
bundles; in the direction of Gross-Zagier there are
ongoing generalizations of the key formulas to other arithmetic groups. The
central notion in all recent developments is that of a Shimura variety. Shimura
varieties provide a geometrical bridge between the world of Galois
representations, on the one hand, and the world of automorphic
representations, on the other. Their geometric and arithmetic structure is thus
very rich and the full picture is only gradually getting unfolded. During
the workshop we wish to emphasize examples and actual computations rather than
general theory. Often in mathematics, deeper knowledge is steered by a
challenging set of guiding examples. There are already many of such examples
around to work on, but there seems to be a need to work out more in order to
enhance further developments on the theoretical side. Topics that we want to
focus on include: Shimura varieties and special cycles, interactions with automorphic forms, isogeny estimates, Bogomolov
type conjectures, effective height bounds, adelic
heights. [Back] |