Local Andre-Oort conjecture for the universal abelian variety

• 成果类型: Article
• 署名作者: Scanlon, T
• 署名单位: University of California System; University of California Berkeley
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-005-0460-1
• 发表日期: 2006
• 页码: 191-211
• 关键词: model-theory; points

摘要

We prove a p-adic version of the Andre-Oort conjecture for subvarieties of the universal abelian varieties. Let g and n be integers with n >= 3 and p a prime number not dividing n. Let R be a finite extension of W[F-p(alg)], the ring of Witt vectors of the algebraic closure of the field of p elements. The moduli space A = A(g,1,n) of g-dimensional principally polarized abelian varieties with full level n-structure as well as the universal abelian variety p : X --> A over A may be defined over R. We call a point xi is an element of X(R) R-special if X-pi(xi) is a canonical lift and xi is a torsion point of its fibre. Employing the model theory of difference fields and work of Moonen on special subvarieties of A, we show that an irreducible subvariety of X-R containing a dense set of R-special points must be a special subvariety in the sense of mixed Shimura varieties.