Date of Degree
Algebraic Geometry | Number Theory
Lifting Problem, Local Lifting Problem, Quaternion Actions, Lifting Curves, Hurwitz Trees
The lifting problem asks whether one can lift Galois covers of curves defined over positive characteristic to Galois covers of curves over characteristic zero. The lifting problem has an equivalent local variant, which asks if a Galois extension of complete discrete valuation rings over positive characteristic, with algebraically closed residue field, can be lifted to characteristic zero. In this dissertation, we content ourselves with the study of the local lifting problem when the prime is 2, and the Galois group of the extension is the group of quaternions. In this case, it is known that certain quaternion extensions cannot be lifted, but it is unknown whether there is any quaternion extension which can be lifted. To this end, we construct Hurwitz trees for certain local quaternion extensions, showing that all known obstructions to local lifting vanish. We also classify Hurwitz trees appearing as quotients of the quaternion extension, and prove several lifting results which provides heuristic evidence that some quaternion extension lifts.
Mitchell, George, "The Local Lifting Problem for Curves with Quaternion Actions" (2022). CUNY Academic Works.