Galois covers of arithmetic curves of type (p, ..., p)
Williams, Nicholas John
Thesis or dissertation
University of Exeter
In this thesis we consider the setting where R is a complete discrete valuation ring of mixed characteristic (0, p) where p > 0 is prime. Let (p, ..., p) denote either the group Z/pZ x ... x Z/pZ or the product of rank p group schemes. Given a degree p Galois cover between R-curves, it is understood when the cover has the structure of a torsor under a finite flat group scheme of rank p. We investigate torsors under group schemes of type (p, ..., p) and establish criteria for their existence. We treat the boundary of the formal fibre and extend our knowledge of the conductor and degree of the different in degree p to the (p, p) setting. We also take the opportunity to explain how this can be naturally extended to the general (p, ..., p) case. Finally, we generalise a local vanishing cycles formula for curves known in the degree p case to Galois groups of type (p, p), relating the genus of two points in terms of just the cover's ramification data and the conductors acting at the boundaries.
PhD in Mathematics