The $a$-number of $y^n=x^m+x$ over finite fields

TL;DR

This paper derives a formula for the $a$-number, an invariant of the $p$-torsion group scheme, for certain maximal curves over finite fields defined by the equation $y^{(q+1)/2} = x^m + x$, using the Cartier operator.

Contribution

It provides a new explicit formula for the $a$-number of specific maximal curves characterized by a particular algebraic equation over finite fields.

Findings

Derived a closed-form formula for the $a$-number.

Applied the Cartier operator to compute the $a$-number.

Enhanced understanding of the $p$-torsion structure of these curves.

Abstract

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