Convolution monodromy groups and the Shafarevich conjecture for hypersurfaces in tori

2024/08/16 by Caleb Ji, Ji, Caleb
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2408.08482

openalex publication_date 2024/08/16 · openalex created_date 2024/09/13 · openalex updated_date 2026/07/28

Abstract

The Shafarevich conjecture for a class of varieties over a number field posits the finitude of those with good reduction outside a finite set of primes. In the case of hypersurfaces in the torus \mathbbG_mⁿ, a natural class to consider are those with a fixed Newton polyhedron that are nondegenerate with respect to it. Using an approach similar to that of Lawrence-Sawin for abelian varieties (arXiv:2004.09046), we prove the Shafarevich conjecture for certain classes of Newton polyhedra. In the course of our proof we construct a fiber functor for the Tannakian category of perverse sheaves on the torus in positive characteristic and compute the weights of the Frobenius on it, which leads to the big monodromy results which are key to this method.

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