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Thursday, March 10, 2016

**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 show that Bianchi modular forms are sparse in these $p$-adic families.