Narrow Results

This paper presents a formula for the $a$-number of certain maximal curves characterized by the equation $y^{\frac{q+1}{2}}=x^m+x$ over the finite field ${𝔽}_{q^2}$. $a$-number serves as an invariant for the isomorphism class of the $p$-torsion group scheme. By utilizing the action of the Cartier operator on $H^0({𝒳},{\mathrm{\Omega }}^1)$, we establish a closed formula for the $a$-number of ${𝒳}$.

For a curve in positive characteristic, the Cartier operator acts on the vector space of its regular differentials. The a-number is defined to be the dimension of the kernel of the Cartier operator. In [a-numbers of curves in Artin–Schreier covers, Algebra Number Theory 14(3) (2020) 587–641], Booher and Cais use a sheaf-theoretic approach to give bounds on the a-numbers of Artin–Schreier covers. In this paper, I generalize that approach to arbitrary powers of the Cartier operator, yielding bounds for the dimension of the kernel. These bounds give new restrictions on the Ekedahl-Oort type of Artin–Schreier covers.

For an algebraic curve ${𝒳}$ defined over an algebraically closed field of characteristic $p>0$, $a$-number $a({𝒳})$ is the dimension of the space of exact holomorphic differentials on ${𝒳}$. We computed the $a$-number for a family of certain Picard curves using the action of the Cartier operator on $H^0({𝒳},{\mathrm{\Omega }}^1)$.