周期、指数和潜力股
Period, index and potential sha
摘要：
本文提出了O 'Neil周期指数阻塞映射的理论，并导出了全局域上亏格为1的曲线的算法。我们的第一个结果意味着对每一对正整数（P，I），其中P整除I，I整除P^2，存在数域K和K上的亏格为1的曲线C，其周期为P，指数为I.其次，设E是整体域K上的任意椭圆曲线，P > 1是被K的特征线整除的任意整数。我们在K上构造了无限多个亏格为1的曲线C，其周期为P，指数为P^2，雅可比矩阵为E。我们推导出域扩张下Sharevich-Tate群结构的强结果。
英文摘要：
In this paper we advance the theory of O'Neil's period-index obstruction map and derive consequences for the arithmetic of genus one curves over global fields. Our first result implies that for every pair of positive integers (P,I) with P dividing I and I dividing P^2, there exists a number field K and a genus one curve C over K with period P and index I. Second, let E be any elliptic curve over a global field K, and let P > 1 be any integer indivisible by the characteristic of K. We construct infinitely many genus one curves C over K with period P, index P^2, and Jacobian E. We deduce strong consequences on the structure of Sharevich-Tate groups under field extension.
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