论文标题
具有理性产品的单数模量三元
Triples of singular moduli with rational product
论文作者
论文摘要
我们表明，所有三重三元$（x_1，x_2，x_3）singular moduli满足$ x_1 x_1 x_2 x_3 \ in \ mathbb {q}^{\ times} $是“琐事”。也就是说，要么$ x_1，x_2，x_3 \ in \ mathbb {q} $; \ Mathbb {q} $和其余$ x_j，x_k $的$ x_i \ in \ mathbb {q} $不同，度$ 2 $，而在$ \ mathbb {q} $上进行了共轭;或$ x_1，x_2，x_3 $是成对的，程度为$ 3 $，并且在$ \ mathbb {q} $上结合。该定理是最好的，并且是Bilu，Luca和Pizarro-Madariaga在二维中的自然三维类似物。它建立了André-subvarieties家族的André-猜想$v_α\ subset \ mathbb {c}^3 $由公式定义的$ x_1 x_1 x_2 x_2 x_3 =α\ in \ mathbb {q} $。
We show that all triples $(x_1,x_2,x_3)$ of singular moduli satisfying $x_1 x_2 x_3 \in \mathbb{Q}^{\times}$ are "trivial". That is, either $x_1, x_2, x_3 \in \mathbb{Q}$; some $x_i \in \mathbb{Q}$ and the remaining $x_j, x_k$ are distinct, of degree $2$, and conjugate over $\mathbb{Q}$; or $x_1, x_2, x_3$ are pairwise distinct, of degree $3$, and conjugate over $\mathbb{Q}$. This theorem is best possible and is the natural three dimensional analogue of a result of Bilu, Luca, and Pizarro-Madariaga in two dimensions. It establishes an explicit version of the André--Oort conjecture for the family of subvarieties $V_α \subset \mathbb{C}^3$ defined by an equation $x_1 x_2 x_3 = α\in \mathbb{Q}$.
