Pure Math colloquium

Jacob Tsimmerman, University of Toronto

“Bounding Torsion in Geometric Families of Abelian Varieties”

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 towards a proof. We explain how 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 B. Bakker we prove the geometric torsion conjecture in the special case of abelian varieties with real multiplication. In fact, we present a general method for ruling out low genus curves in quotients of locally symmetric spaces, by using hyperbolic geometry to produce bounds on Seshadri constant. This method moreover allows us to prove a geometric analogue of another famous problem, namely the Frey-Mazur conjecture, asserting that isogeny classes of elliptic curves E over function fields of complex curves are classified by the associated torsion group schemes E[n] for a fixed n.

Refreshments will be served in MC 5046 at 3:00 p.m. Everyone is welcome to attend.