2025 Theses Doctoral

Convolution Monodromy Groups of Tori and the Shafarevich Conjecture

Ji, Caleb

Sheaf convolution endows the category of perverse sheaves (with suitable modifications) on a connected commutative algebraic group over 𝔽̄_𝑞 with the structure of a Tannakian category. We establish an explicit fiber functor in the case of the 𝑛-dimensional torus and use it to compute the convolution monodromy groups corresponding to certain perverse sheaves defined by hypersurfaces.

Using the approach of Lawrence-Sawin, we apply these monodromy computations to prove some instances of the Shafarevich conjecture for hypersurfaces in tori. We also improve the result of Lawrence-Venkatesh on the Shafarevich conjecture for hypersurfaces in projective space by making it effective. We end by discussing a question of Katz on the specialization of these convolution monodromy groups.