Algebraic curves with automorphism groups of large prime order

Abstract

Let X be an algebraic curve of genus g defined over an algebraically closed field K of characteristic p ≥ 0, and q a prime dividing |Aut(X )|. We say that X is a q-curve. Homma proved that either q ≤ g + 1 or q = 2g + 1, and classified (2g + 1)-curves. In this note, we classify (g + 1)-curves, and fully characterize the automorphism groups of q-curves for q = 2g + 1, g + 1. We also give some partial results on q-curves for q = g, g − 1.