### Abstract

Let π: Y → X be a branched Z/pZ-cover of smooth, projective, geometrically connected curves over a perfect field of characteristic p > 0. We investigate the relationship between the a-numbers of Y and X and the ramification of the map π. This is analogous to the relationship between the genus (respectively p-rank) of Y and X given the Riemann–Hurwitz (respectively Deuring–Shafarevich) formula. Except in special situations, the a-number of Y is not determined by the a-number of X and the ramification of the cover, so we instead give bounds on the a-number of Y. We provide examples showing our bounds are sharp. The bounds come from a detailed analysis of the kernel of the Cartier operator.

Original language | English (US) |
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Pages (from-to) | 587-641 |

Number of pages | 55 |

Journal | Algebra and Number Theory |

Volume | 14 |

Issue number | 3 |

DOIs | |

State | Published - 2020 |

### Keywords

- A-numbers
- Arithmetic geometry
- Artin–Schreier covers
- Covers of curves
- Invariants of curves

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

*Algebra and Number Theory*,

*14*(3), 587-641. https://doi.org/10.2140/ant.2020.14.587