Models of Curves and Wild Ramification
Models of Curves and Wild Ramification
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曲线模型和狂野分支
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发表时间：
2010
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通讯作者：
Dino J. Lorenzini
中科院分区：
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作者：
Dino J. Lorenzini
设K是一个完整的离散估值域，其环为整数OK，残差域K特征为p≥0，假设为代数闭域。设X/K表示g属≥1的光滑固有几何连通曲线，设X/ OK表示其最小正则模型。当g≥2或g = 1且X(K) 6=∅时，存在有限伽罗瓦扩展L/K极小值，其性质为XL/L具有半稳定约简。令X /OL表示XL/L的最小正则模型。从X的特殊纤维的组合性质的知识中可以推断出X '的特殊纤维或扩展L/K的哪些性质？让我们首先考虑椭圆曲线E/K的情况。Tate在其著名算法[26]的总结中指出：设p 6= 2,3。设E/K是一条加性约简的椭圆曲线。那么E/K有潜在的乘法约简，当且仅当它在n ~ 0 ~ 0有In型约简。换句话说，当且仅当其在OK上的还原类型为II， II， III， III， IV， IV或I0时，E/K具有潜在的良好还原性。
Let K be a complete discrete valuation field with ring of integers OK and residue field k of characteristic p ≥ 0, assumed to be algebraically closed. Let X/K denote a smooth proper geometrically connected curve of genus g ≥ 1, and let X /OK denote its minimal regular model. When g ≥ 2, or g = 1 and X(K) 6= ∅, there exists a finite Galois extension L/K minimal with the property that XL/L has semi-stable reduction. Let X /OL denote the minimal regular model of XL/L. Which properties of the special fiber of X ′ or of the extension L/K can be inferred from the knowledge of the combinatorial properties of the special fiber of X ? Let us consider first the case of an elliptic curve E/K. Tate noted the following in the summary of his famous algorithm [26]: Let p 6= 2, 3. Let E/K be an elliptic curve with additive reduction over OK . Then E/K has potentially multiplicative reduction if and only if it has reduction of type In for some n > 0. In other words, E/K has potentially good reduction if and only if its reduction type over OK is either II, II, III, III, IV, IV or I0.