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arXiv · 2401.13707

The $a$-number of $y^{q^2+q+1} = x^{q^2+1} + x^q $ over finite field

Vahid Nourozi·Behrooz Mosallaei
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Abstract

In this paper, we compute a formula for the $a$-number of curve $\mathbb{X}$ given by the equation $y^{q^2 + q + 1} = x^{q^2 + 1} - x^q$ over the finite field $\mathbb{F}_{q^2}$. The $a$-number is an invariant of the isomorphism class of the $p$-torsion group scheme. In this paper, we compute a closed formula for the $a$-number of $\mathbb{X}$ using the action of the Cartier operator on $H^0(\mathbb{X}, \Omega^1)$.

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Vahid Nourozi, Behrooz Mosallaei. 2024-01-23. The $a$-number of $y^{q^2+q+1} = x^{q^2+1} + x^q $ over finite field. https://arxiv.org/abs/2401.13707

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