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The Rank of the Cartier operator on Picard Curves

Vahid Nourozi, Farhad Rahmati

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 $H^0(\mathcal{X},Ω^1)$.

The Rank of the Cartier operator on Picard Curves

Abstract

For an algebraic curve defined over an algebraically closed field of characteristic , the -number is the dimension of the space of exact holomorphic differentials on . We compute the -number for a family of certain Picard curves, using the action of the Cartier operator on .
Paper Structure (3 sections, 4 theorems, 17 equations)

This paper contains 3 sections, 4 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. The Cartier operator
  3. The $a$-number of Picard Curves

Key Result

Proposition 2.2

(Global Properties of $\mathfrak{C}$). For all $\omega \in \Omega_{K/K_q}$ and all $f \in K$,

Theorems & Definitions (8)

  • Definition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 3.1
  • proof
  • Example 3.2