2022/10/19 by Valentín Hernández, Benjamin Schraen, Hernandez, Valentin +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2210.10564
openalex publication_date 2022/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
If ρ is an automorphic modulo p Galois representation, it is natural to wonder if automorphic points are Zariski dense in the deformation space of ρ. We prove new results in this direction in the case of a unitary group split (and unramified) at p. Namely, if ρ is associated to an automorphic form for a unitary group (which contributes to coherent cohomology), we prove that the "infinite fern" (i.e. the image of an appropriate Eigenvariety) in the polarised deformation space of ρ is Zariski dense in a non-empty union of irreducible components. This generalises in particular results of Gouvêa-Mazur for GL2/\mathbb Q, Chenevier for U(3) and recently Hellmann-Margerin-Schraen. The novelty is that we use the local model of Breuil-Hellmann-Schraen to control tangent spaces in the local deformation rings, and a geometric argument on the Eigenvariety originally due to Bellaïche-Chenevier and Taïbi to reduce to points with enormous image. At those points, we can use a recent result of Newton-Thorne to control the vanishing of a Selmer group. In particular, we do not need to assume any "Taylor-Wiles" hypothesis on ρ, which can in particular be irreducible. If we moreover add Taylor-Wiles hypothesis on ρ and an extra hypothesis at p, we have by a result of Allen the Zariski density everywhere.