TY - CHAP

T1 - On the bad reduction of certain U(2, 1) Shimura varieties

AU - de Shalit, Ehud

AU - Goren, Eyal Z.

N1 - Publisher Copyright: © Springer Nature Switzerland AG 2018.

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

KW - Picard surfaces

KW - Shimura varieties

KW - Supersingular strata

UR - http://www.scopus.com/inward/record.url?scp=85054011769&partnerID=8YFLogxK

U2 - https://doi.org/10.1007/978-3-319-97379-1_6

DO - https://doi.org/10.1007/978-3-319-97379-1_6

M3 - فصل

T3 - Springer Proceedings in Mathematics and Statistics

SP - 81

EP - 152

BT - Springer Proceedings in Mathematics and Statistics

ER -