Speaker: Xinyao Zhang (The University of Tokyo)

Title: Zariski density of modular points in Eisenstein deformation spaces

Time: 10:30-11:30 July 9, 2026 (Thursday)

Place: MCM110

Abstract: In this talk, I will discuss Zariski density problems for modular points in universal pseudo-deformation spaces attached to residually reducible two-dimensional Galois representations. For residually irreducible representations, related pro-modularity results were previously obtained by Böckle, Emerton, and others. In the Eisenstein case, however, the geometry of pseudo-deformation spaces is more subtle. I will explain a method based on deformation-theoretic dimension estimates, the infinite fern, and Bloch-Kato Selmer vanishing. As an application, one obtains new cases of the Fontaine-Mazur conjecture in the irregular setting.