The Tate Conjecture for a family of surfaces of general type with p g = q = 1 and K 2 = 3

2015/04/01 by Christopher Lyons
Mathematics · #Advanced Algebra and Geometry #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic number #Algebraic surface #Cohomology #Combinatorics #Computer science #Conjecture #Discrete mathematics #Extension (predicate logic) #Galois module #Geometry #Homotopy and Cohomology in Algebraic Topology #Mathematical analysis #Mathematics #Monodromy #Programming language #Pure mathematics #Representation (politics) #Surface (topology) #Type (biology) #math.AG #math.NT #msc:11G99 #msc:14D05 #msc:14G99 #msc:14J29

paper · pdf · doi:10.1353/ajm.2015.0011

published as Amer. J. Math. 137 (2015), no. 2, 281-311 · 26 pages. Final version

openalex publication_date 2015/04/01 · arxiv created 2015/08/08 · arxiv updated 2015/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We prove a big monodromy result for a smooth family of complex algebraic surfaces of general type, with invariants p_g=q=1 and K²=3, that has been introduced by Catanese and Ciliberto. This is accomplished via a careful study of degenerations. As corollaries, when a surface in this family is defined over a finitely generated extension of ℚ, we verify the semisimplicity and Tate conjectures for the Galois representation on the middle ℓ-adic cohomology of the surface.

Citations

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