The geometric torsion conjecture for Hilbert modular varieties
-- Benjamin Bakker, October 3, 2014

<p>


A celebrated theorem of Mazur asserts that the order of
the torsion part of the Mordell-Weil group of an elliptic
curve over Q is absolutely bounded; it is conjectured that the
same is true for abelian varieties over number fields, though
very little progress has been made in proving it. The natural
geometric analog where Q is replaced by the function field of a
complex curve -- dubbed the geometric torsion conjecture -- is
equivalent to the nonexistence of low genus curves in
congruence towers of Siegel modular varieties. In joint work
with J. Tsimerman, we prove the geometric torsion conjecture
for abelian varieties with real multiplication. The proof
uses the hyperbolic geometry of Hilbert modular varieties to
produce new bounds on Seshadri constants of the canonical
bundle along the boundary.