Abstract

For a smooth projective curve \(\mathcal{C}\) defined over an algebraic number field k, we investigate the finiteness of the set of generalized Jacobians \({{J}_{\mathfrak{m}}}\) of \(\mathcal{C}\) associated with modules \(\mathfrak{m}\) defined over \(k\) such that a fixed divisor representing a class of finite order in the Jacobian J of \(\mathcal{C}\) provides the torsion class in the generalized Jacobian \({{J}_{\mathfrak{m}}}\). Various results on the finiteness and infiniteness of the set of generalized Jacobians with the above property are obtained depending on the geometric conditions on the support of \(\mathfrak{m}\), as well as on the conditions on the field k. These results are applied to the problem of the periodicity of a continued fraction expansion constructed in the field of formal power series \(k((1{\text{/}}x))\) for special elements of the field of functions \(k(\tilde {\mathcal{C}})\) of the hyperelliptic curve \(\tilde {\mathcal{C}}:{{y}^{2}} = f(x)\).

中文翻译：

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论代数数域上具有非平凡挠点的广义雅可比行列式集合的有限性

摘要

对于在代数数域k上定义的平滑射影曲线\(\mathcal{C}\)，我们研究了广义雅克比矩阵\({{J}_{\mathfrak{m}}}\)的有限性\(\mathcal{C}\)与在\(k\)上定义的模块\(\mathfrak{m}\)关联，使得表示\(\mathcal{C的雅可比J中的一类有限阶的固定除数) }\)提供广义雅可比行列式\({{J}_{\mathfrak{m}}}\)中的扭转类。根据\(\mathfrak{m}\)支持的几何条件以及域k上的条件，可以得到具有上述性质的广义雅可比行列式集合的有限性和无限性的各种结果。这些结果适用于在函数域的特殊元素的形式幂级数\(k((1{\text{/}}x))\)域中构造的连分数展开式的周期性问题\超椭圆曲线\ (\tilde {\mathcal{C}} :{{y}^{2}} = f(x)\) 的 (k(\tilde {\mathcal{C}})\)。