On the existence of curves with prescribed \(a\)-number (Q2032831)

Let \(k\) be an algebraically closed field of characteristic \(p>0\), let \(X\) be a curve defined over \(k\), and let \(\text{Jac}(X)\) be its Jacobian. One of the most important invariants of \(X\) is its \(a\)-number \(a_{X}\), which is defined by \[a_{X}=\text{dim}_{k}(\text{Hom}(\alpha_{p}, \text{Jac}(X))),\] where \(\alpha_{p}\) is the group scheme which is the kernel of Frobenius on the additive group scheme \(\mathbb{G}_{a}\). It is well known that the \(a\)-number of \(X\) is equal to \(g-r\), where \(g\) is the genus of \(X\) and \(r\) is the rank of the Cartier-Manin matrix, that is, the matrix for the Cartier operator defined on \(H^{0}(X,\Omega^{1}_{X})\). In this paper, the author studies the existence of Artin-Schreier curves with large \(a\)-number, namely \(a_{X}=g-1\) and \(a_{X}=g-2\), proving among other important results that such curves can be written in some particular forms. Also, by computing the rank of the Hasse-Witt matrix of the curve, bounds on the \(a\)-number of trigonal curves of genus \(5\) in small characteristic are also given.