绿点理论

摘要：

Poizat的建设理论领域的乘法子群的绿点是扩展在几个方向：首先，我们也建设类似的理论，其中的绿点形成一个可分的$\operatorname{End}(E)$ -子模的椭圆曲线$E$。其次，绿点的子群（子模）允许有挠率，而在Poizat的工作中，这种更一般的情况只涉及假设与Tori相交的未证明猜想（CIT）。第三，由Zilber的工作与非交换几何连接的动机，我们构建了一个版本的理论的绿点在乘法群的情况下，其中的杰出的子群是不可分的，但一个$\mathbb{Z}$ -组，我们称之为理论的翡翠点。在随后与Boris Zilber的联合论文中，我们发现了构建理论的自然模型。

英文摘要：

Poizat's construction of theories of fields with a multiplicative subgroup of green points is extended in several directions: First, we also construct similar theories where the green points form a divisible $\operatorname{End}(E)$-submodule of an elliptic curve $E$. Second, the subgroup (submodule) of green points is allowed to have torsion, whereas in Poizat's work this more general case was only dealt with assuming the unproven Conjecture on Intersections with Tori (CIT). Third, motivated by Zilber's work on connections with non-commutative geometry, we construct a version of the theories of green points in the multiplicative group case for which the distinguished subgroup is not divisible but a $\mathbb{Z}$-group, which we call theories of emerald points. In a subsequent joint paper with Boris Zilber we find natural models for the constructed theories.

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