On the bad reduction of certain U(2, 1) Shimura varieties

Ehud de Shalit, Eyal Z. Goren*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


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 OE, and a principal polarization ϕ. The second surface, S0(p), parametrizes, in addition, a suitably restricted choice of a subgroup H⊂ A[p] of rank p2. The third Picard surface, S~, parametrizes triples (A, ψ, ι) similar to those parametrized by S, but where ψ is a polarization of degree p2. 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 languageAmerican English
Title of host publicationSpringer Proceedings in Mathematics and Statistics
PublisherSpringer New York LLC
Number of pages72
StatePublished - 2018

Publication series

NameSpringer Proceedings in Mathematics and Statistics
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Bibliographical note

Publisher Copyright:
© Springer Nature Switzerland AG 2018.


  • Picard surfaces
  • Shimura varieties
  • Supersingular strata


Dive into the research topics of 'On the bad reduction of certain U(2, 1) Shimura varieties'. Together they form a unique fingerprint.

Cite this