The Rank of the Cartier operator on Picard Curves

Author:

Vahid Nourozi, Farhad Rahmati

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), Number Theory (math.NT)

journal:

date:

2023-06-12 16:00:00

Abstract

For an algebraic curve $\mathcal{X}$ defined over an algebraically closed field of characteristic $p > 0$, the $a$-number $a(\mathcal{X})$ is the dimension of the space of exact holomorphic differentials on $\mathcal{X}$. We compute the $a$-number for a family of certain Picard curves, using the action of the Cartier operator on $H^0(\mathcal{X},\Omega^1)$.

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