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.