The local-global principle for symmetric determinantal representations of smooth plane curves
The local-global principle for symmetric determinantal representations of smooth plane curves
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光滑平面曲线对称行列式表示的局部全局原理
DOI：
10.1007/s11139-016-9775-3
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Tetsushi Ito
中科院分区：
文献类型：
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作者：
Yasuhiro Ishitsuka;Tetsushi Ito
一条光滑的平面曲线被称为允许一个对称行列式表示，如果它可以由一个元素为三元线性形式的对称矩阵的行列式来定义。本文研究了特征不等于二的整体域上光滑平面曲线的对称行列式表示的存在性的局部-整体原理。当平面曲线的度小于或等于3时，我们将对称行列式表示的问题与Severi-Brauer簇和mod 2 Galois表示上的丢番图问题联系起来，并证明了局部-整体原理对二次曲线和三次曲线成立.我们还构造反例的本地全球的原则，四次使用的结果芒福德，哈里斯，和盐田的θ特性。
A smooth plane curve is said to admit a symmetric determinantal representation if it can be defined by the determinant of a symmetric matrix with entries in linear forms in three variables. We study the local–global principle for the existence of symmetric determinantal representations of smooth plane curves over a global field of characteristic different from two. When the degree of the plane curve is less than or equal to three, we relate the problem of finding symmetric determinantal representations to more familiar Diophantine problems on the Severi–Brauer varieties and mod 2 Galois representations, and prove that the local–global principle holds for conics and cubics. We also construct counterexamples to the local–global principle for quartics using the results of Mumford, Harris, and Shioda on theta characteristics.