Ash–Pollack–Stevens conjecture on non-rigid finite-slope eigensystems for $\operatorname{GL}_3$

Let $\theta$ be a finite-slope cuspidal Hecke eigensystem of $\operatorname{GL}_3$. An eigensystem is arithmetically rigid if it does not lie in any irreducible component of an eigenvariety containing a Zariski-dense subset of arithmetic points. Ash–Pollack–Stevens conjecture. If $\theta$ is not arithmetically rigid, then $\theta$ is essentially self-dual.

The conjecture concerns when $p$-adic families of overconvergent Hecke eigensystems contain sufficiently many arithmetic points. It predicts that every non-rigid finite-slope cuspidal eigensystem for $\operatorname{GL}_3$ has an essential self-duality, restricting the possible arithmetic behavior of eigenvariety components.