## Abstract

Let E be a quadratic imaginary field, and let p be a prime which is inert in E. We study three types of Picard modular surfaces in positive characteristic p and the morphisms between them. The first Picard surface, denoted S, parametrizes triples (A, ϕ, ι) comprised of an abelian threefold A with an action ι of the ring of integers O_{E}, and a principal polarization ϕ. The second surface, S_{0}(p), parametrizes, in addition, a suitably restricted choice of a subgroup H⊂ A[p] of rank p^{2}. The third Picard surface, S~, parametrizes triples (A, ψ, ι) similar to those parametrized by S, but where ψ is a polarization of degree p^{2}. We study the components, singularities and naturally defined stratifications of these surfaces, and their behavior under the morphisms. A particular role is played by a foliation we define on the blowup of S at its superspecial points.

Original language | English |
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Title of host publication | Springer Proceedings in Mathematics and Statistics |

Publisher | Springer New York LLC |

Pages | 81-152 |

Number of pages | 72 |

DOIs | |

State | Published - 2018 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
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Volume | 251 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Bibliographical note

Publisher Copyright:© Springer Nature Switzerland AG 2018.

## Keywords

- Picard surfaces
- Shimura varieties
- Supersingular strata