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

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 $H^0(\mathcal{X},{\mathrm{\Omega }}^1)$, we establish a closed formula for $a$-number of $\mathcal{X}$.