SEMISTABLE MODELS OF HYPERELLIPTIC CURVES IN THE WILD CASE & DIFFERENTIAL OPERATORS ON P-ADIC MODULAR FORMS

This PhD thesis is divided into two parts, collecting two independent pieces of work I completed during my PhD, both in the realm of number theory and arithmetic geometry. The first part presents a joint work with J. Yelton, regarding semistable models of hyperelliptic curves in the wild case. To this subject, which was already the topic of my MSc. thesis, I devoted part of my research work during my PhD, until completing it during summer 2022. The second part of this thesis discusses some research findings related to a completely unrelated topic proposed to me by my PhD advisor Fabrizio Andreatta, namely the one of geometric constructions of differential operators on sheaves of p-adic modular forms.