Number Theory Seminar
Vlad Serban
Northwestern
A p-adic strengthening of the Manin-Mumford conjecture
Abstract: Let $G$ be an abelian variety or a product of multiplicative groups
$\mathbb{G}_m^n$ and let $C$ be an embedded curve. The Manin-Mumford
conjecture (a theorem by work of Lang, Raynaud et al.) states that only
finitely many torsion points of $G$ can lie on $C$ unless $C$ is in
fact
a subgroup of $G$. I will show how these purely algebraic statements
extend to suitable analytic functions on open $p$-adic unit poly-disks.
These disks occur naturally as weight spaces parametrizing families of
$p$-adic automorphic forms for $GL(2)$ over a number field $F$. When
$F=\mathbb{Q}$, the "Hida families" in question play a crucial role in
the study of modular forms. When $F$ is imaginary quadratic, I will
explain how our results imply that Bianchi modular forms are sparse in
these $p$-adic families.
Friday May 6, 2016 at 11:00 AM in SEO 427