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 ${𝒳}$.

Consider the moduli space of smooth curves of genus g and p-rank f defined over an algebraically closed field k of characteristic p. It is an open problem to classify which group schemes occur as the p-torsion of the Jacobians of these curves for f < g - 1. We prove that the generic point of every component of this moduli space has a-number 1 when f = g - 2 and f = g - 3. Likewise, we show that a generic hyperelliptic curve with p-rank g - 2 has a-number 1 when p ≥ 3. We also show that the locus of curves with p-rank g - 2 and a-number 2 is non-empty with codimension 3 in when p ≥ 5. We include some other results when f = g - 3. The proofs are by induction on g while fixing g - f. They use computations about certain components of the boundary of .

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 $H^0({𝒳},{\mathrm{\Omega }}^1)$.