$a$-numbers of covers in characteristic two: semigroups and Riemann-Roch spaces

Abstract

We compute the $a$-numbers of the Garcia-Güneri-Stichtenoth curves, the Beelen-Montanucci curves, and the Skabelund covers of Suzuki curves in characteristic two. The proofs use the identity $a(X)=\ell(\tfrac12\operatorname{div}(dt))$ and reduce the calculations to Riemann-Roch spaces on the base curves. We obtain transfer formulas for odd-degree radical covers, with one or several points at infinity, and evaluate them using Weierstrass semigroups and plane sections of the Hermitian curve. For the first two families we also give cyclic models over the projective line and relate the formulas to Elkin's rank formula. We then treat two classes of Artin-Schreier fibre products. A supersingular genus-six family has a-number $3$ at the zero parameter and $2$ elsewhere, although the genera and $a$-numbers of its quadratic quotients remain fixed. We determine its Ekedahl--Oort types as $(0,0,0,1,2,3)$ and $(0,1,1,2,3,4)$,