The a-number of the Serre-type maximal curves of Abdón-Bezerra-Quoos

Abstract

For every prime power $q$, we determine the $a$-number of the Serre-type maximal curves $X(q,n)$: $y^{q^2}-y=x^{(q^n+1)/(q+1)}$ over $\mathbb{F}_{q^{2n}}$, $n\ge 3$ odd, introduced by Abdón, Bezerra and Quoos as a generalization of Serre's curve $y^4+y=x^3$ over $\mathbb{F}_{64}$. The Weierstrass semigroup at the unique point at infinity gives an explicit basis of holomorphic differentials, on which the Cartier operator acts by monomials; this reduces the $a$-number to a lattice-point count, which we evaluate in closed form.