Semistable reduction of hyperelliptic curves over finite extensions of the 2-adic numbers

Constructing semistable models for hyperelliptic curves serves as an important ingredient in many interesting problems in mathematics such as solving generalized Fermat equations and generalizing the famous Tate's algorithm for hyperelliptic curves. In recent years, explicit methods for constructing semistable models for hyperelliptic curves defined over local field having characteristics not equal to 2 has been examined thoroughly by Dokchitser-Dokchitser-Maistret-Morgan (2017). Their method, however, relies heavily on the fact that the residue characteristics of the local fields are not 2 and does not apply for the characteristic 2 case. In this thesis, we take a different approach to construct semistable models for a specific class (double root clusters) of hyperelliptic curves defined over finite extensions of the 2-adic numbers. We then demonstrate our methods by constructing an explicit semistable model for a given hyperelliptic curve as a proof of concept. Our result serves as a small step towards a general method for computing semistable models of hyperelliptic curves defined over local fields with residue characteristic 2 for the specific class of curves that we are interested in.