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

Behrooz Mosallaei; Farzaneh Ghanbari; Sepideh Farivar; Vahid Nourozi

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

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arxiv 2404.08149 v2 pith:TNNBZFDD submitted 2024-04-11 math.NT math.AG

Behrooz Mosallaei , Sepideh Farivar , Farzaneh Ghanbari , Vahid Nourozi This is my paper · ORCID

classification math.NTmath.AG

keywords numberfiniteformulamathcalactioncartiercertaincharacterized

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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 $\mathcal{X}$.

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