Potential theory, ultrafilters, and uniform boundedness of rational torsion on abelian varieties

Abstract: In 1996, building off of prior work by Mazur and Kamienny, Merel proved his famous theorem on the uniform boundedness of the $K$-rational torsion of an elliptic curve over a number field $K$. The (strong) Torsion Conjecture generalizes this theorem to abelian varieties of arbitrary dimension: it states that given any positive integers $g$ and $N$, there is a $B=B(g,N)$ such that for any $g$-dimensional abelian variety $A$ over a number field $K$ satisfying $[K:\mathbb{Q}]\le N$, the set of $K$-rational torsion points of $A$ has order at most $B$. In this talk, I will present a function field analogue of this conjecture as well as a sketch of its proof. This is joint work with Jit Wu Yap.

algebraic geometrynumber theory

Audience: researchers in the topic

MIT number theory seminar

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