Convolution monodromy groups and the Shafarevich conjecture for hypersurfaces in tori

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 Gₘⁿ, 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.