Some effective estimates for André–Oort in Y(1) n
Some effective estimates for André–Oort in Y(1) n
复制标题
Y(1) n 中 André–Oort 的一些有效估计
DOI：
10.1515/crelle-2019-0028
复制
发表时间：
2018
期刊：
影响因子：
--
通讯作者：
E. Kowalski
中科院分区：
文献类型：
--
作者：
Gal Binyamini;E. Kowalski
Abstract Let X ⊂ Y ( 1 ) n {X\subset Y(1)^{n}} be a subvariety defined over a number field ? {{\mathbb{F}}} and let ( P 1 , … , P n ) ∈ X {(P_{1},\ldots,P_{n})\in X} be a special point not contained in a positive-dimensional special subvariety of X. We show that if a coordinate P i {P_{i}} corresponds to an order not contained in a single exceptional Siegel–Tatuzawa imaginary quadratic field K * {K_{*}} , then the associated discriminant | Δ ( P i ) | {|\Delta(P_{i})|} is bounded by an effective constant depending only on deg X {\deg X} and [ ? : ℚ ] {[{\mathbb{F}}:{\mathbb{Q}}]} . We derive analogous effective results for the positive-dimensional maximal special subvarieties. From the main theorem we deduce various effective results of André–Oort type. In particular, we define a genericity condition on the leading homogeneous part of a polynomial, and give a fully effective André–Oort statement for hypersurfaces defined by polynomials satisfying this condition.
Abstract Let X ⊂ Y ( 1 ) n {X\subset Y(1)^{n}} be a subvariety defined over a number field ? {{\mathbb{F}}} and let ( P 1 , … , P n ) ∈ X {(P_{1},\ldots,P_{n})\in X} be a special point not contained in a positive-dimensional special subvariety of X. We show that if a coordinate P i {P_{i}} corresponds to an order not contained in a single exceptional Siegel–Tatuzawa imaginary quadratic field K * {K_{*}} , then the associated discriminant | Δ ( P i ) | {|\Delta(P_{i})|} is bounded by an effective constant depending only on deg X {\deg X} and [ ? : ℚ ] {[{\mathbb{F}}:{\mathbb{Q}}]} . We derive analogous effective results for the positive-dimensional maximal special subvarieties. From the main theorem we deduce various effective results of André–Oort type. In particular, we define a genericity condition on the leading homogeneous part of a polynomial, and give a fully effective André–Oort statement for hypersurfaces defined by polynomials satisfying this condition.