论文标题
建造三维CM Abelian品种的Hecke字符
Construction of Hecke Characters for Three-dimensional CM Abelian Varieties
论文作者
论文摘要
以复杂的乘法为椭圆形曲线而闻名，$ \ mathbb {q} $ - 理性模型的存在等同于其模构的领域等于$ \ mathbb {q} $，或者其内态态度是9个可能的字段（$ \ ast $）的整数。 Murabayashi和Umegaki证明了Abelian表面的类似结果。对于具有模量有理领域的三维CM Abelian品种，Chun缩小到37个可能的CM字段的列表。在本文中，我们证明他的清单是准确的。通过构建满足Shimura定理的某些Hecke字符，我们证明，这些Abelian品种的28个同种类别具有$ \ Mathbb {q} $ - 型号。因此，与$（\ ast）$的完整类比在这里失败。
It is well-known for an elliptic curve with complex multiplication that the existence of a $\mathbb{Q}$-rational model is equivalent to its field of moduli being equal to $\mathbb{Q}$, or its endomorphism ring being the ring of integers of 9 possible fields ($\ast$). Murabayashi and Umegaki proved analogous results for abelian surfaces. For three dimensional CM abelian varieties with rational fields of moduli, Chun narrowed down to a list of 37 possible CM fields. In this paper, we show that his list is exact. By constructing certain Hecke characters that satisfy a theorem of Shimura, we prove that precisely 28 isogeny classes of these abelian varieties have $\mathbb{Q}$-models. Therefore the complete analogy to $(\ast)$ fails here.
