Convolution monodromy groups and the Shafarevich conjecture for hypersurfaces in tori

TL;DR

This paper proves the Shafarevich conjecture for certain hypersurfaces in tori with fixed Newton polyhedra by employing monodromy group techniques and Tannakian category methods, extending previous approaches to new classes.

Contribution

It introduces a novel approach using monodromy groups and Tannakian categories to prove the Shafarevich conjecture for hypersurfaces in tori with fixed Newton polyhedra.

Findings

Proved finiteness of hypersurfaces with good reduction outside a finite set of primes.

Constructed a fiber functor for perverse sheaves on the torus in positive characteristic.

Computed Frobenius weights leading to big monodromy results.

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 $\mathbb{G}_m^n$, 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.