Lifting Problem on Automorphism Groups of Cyclic Curves

TL;DR

This paper investigates conditions under which automorphism groups of hyperelliptic and cyclic curves over algebraically closed fields can be lifted from positive characteristic to characteristic zero, generalizing previous results.

Contribution

It establishes necessary and sufficient conditions for lifting automorphism groups of hyperelliptic and cyclic curves from characteristic p to zero.

Findings

Derived criteria for automorphism group liftability in hyperelliptic curves

Extended results to a broader family called cyclic curves

Provided a framework for understanding automorphism group behavior across characteristics

Abstract

Let X be a smooth projective hyperelliptic curve over an algeraically closed field k of prime characteristic p. The aim of this note is to find necessary and sufficient conditions on the automorphism group of the curve X to be lifted to characteristic zero. The results will be generalised for a certain family of curves that we call cyclic curves.