Andrew Obus

Wild tame-by-cyclic extensions

Suppose G is a semi-direct product of the form Z/p⋊Z/m where p is prime and m is relatively prime to p. Suppose K is a complete local field of characteristic p > 0 with algebraically closed residue field. The main result states necessary and sufficient conditions on the ramification filtrations that occur for wildly ramified G-Galois extensions of K. In addition, we prove that there exists a parameter space for G-Galois extensions of K with given ramification filtration, and we calculate its dimension in terms of the ramification filtration. We provide explicit equations for wild cyclic extensions of K of degree p.

The local lifting problem for A4

We solve the local lifting problem for the alternating group A_4, thus showing that it is a local Oort group. Specifically, if k is an algebraically closed field of characteristic 2, we prove that every A_4-extension of k[[s]] lifts to characteristic zero. As a consequence, every A_4-branched cover of smooth projective curves in characteristic 2 lifts to characteristic zero.

Unramified Brauer Classes on Cyclic Covers of the Projective Plane

Let \( {X} \rightarrow \mathbb{P}^{2}\) be a p-cyclic cover branched over a smooth, connected curve C of degree divisible by p, defined over a separably closed field of characteristic diffierent from p. We show that all (unramified) p-torsion Brauer classes on X that are fixed by Aut\( ({X}/\mathbb{P}^{2})\) arise as pull-backs of certain Brauer classes on \( {\rm{k}}(\mathbb{P}^{2})\) that are unramified away from C and a fixed line L. We completely characterize these Brauer classes on \( {\rm{k}}(\mathbb{P}^{2})\) and relate the kernel of the pullback map to the Picard group of X.

Reduction of dynatomic curves

The dynatomic modular curves parametrize polynomial maps together with a point of period

Good reduction of three-point Galois covers

n

Wild ramification kinks

. It is known that the dynatomic curves

Conductors of wild extensions of local fields, especially in mixed characteristic (0,2)

Y_1(n)

Cyclic extensions and the local lifting problem

are smooth and irreducible in characteristic 0 for families of polynomial maps of the form

On Colmez’s product formula for periods of CM-abelian varieties

f_c(z) = z^m +c

A generalization of the Oort conjecture

where