Automorphism groups of algebraic curves with p -rank zero

Abstract

The Hurwitz bound on the size of the K-automorphism group Aut(X ) of an algebraic curve X of genus g 2 defined over a field K of zero characteristic is |Aut(X )| 84(g − 1). For a positive characteristic, algebraic curves can have many more automorphisms than expected from the Hurwitz bound. There even exist algebraic curves of arbitrary high genus g with more than 16g 4 automorphisms. It has been observed on many occasions that the most anomalous examples of algebraic curves with very large automorphism groups invariably have zero p-rank. In this paper, the K-automorphism group Aut(X ) of a zero 2-rank algebraic curve X defined over an algebraically closed field K of characteristic 2 is investigated. The main result is that, if the curve has genus g 2 and |Aut(X )| > 24g(g − 1), then Aut(X ) has a fixed point on X , apart from a few exceptions. In the exceptional cases, the possibilities for Aut(X ) and g are determined.

On some Galois covers of the Suzuki and Ree curves

“…BeingS q an F q 4 -maximal curve, we can apply the results in [15] on zero 2-rank curves. By direct computations |Aut(S q )| ≥ |LS(q)| ≥ 72(g(S q ) − 1), then by [15,Theorem 5.1] we conclude that Aut(S q ) is non-solvable.…”

“…BeingS q an F q 4 -maximal curve, we can apply the results in [15] on zero 2-rank curves. By direct computations |Aut(S q )| ≥ |LS(q)| ≥ 72(g(S q ) − 1), then by [15,Theorem 5.1] we conclude that Aut(S q ) is non-solvable. Applying [15, Theorem 6.1], the commutator Aut(S q ) ′ of Aut(S q ) is one of the following groups: PSL(2, n), PSU(3, n), SU(3, n), S(n) with n = 2 r ≥ 4.…”

Transcendence degree one function fields over a finite field with many automorphisms

“…(iii) The Sylow 2-subgroups of Γ are wreathed, and Γ is isomorphic to either PSL(3, n) with an odd prime power n ≡ 1 (mod 4), or to PSU(3, n), n ≡ −1 (mod 4), or to PSU (3,4).…”

Generic Newton polygons for curves of given p-rank

“…The p-rank stratification of the moduli space of Artin-Schreier curves is discovered in [25]. See also [13,14].…”