论文标题
曲线雅各布人的均等
Parity of ranks of Jacobians of curves
论文作者
论文摘要
我们调查了雅各布人的Selmer曲线群体，这些曲线群体接受了非平凡的自动形态群体的作用，并将其应用于Selmer等级的奇偶校验的研究。在Shafarevich-Tate的猜想下，我们就纯粹的局部不变式的任意雅各布的摩尔德尔（Mordell）的平等表达了。后者可以看作是局部根数的算术类似物，在桦木 - swinnerton-dyer的猜想下，同样控制阿贝利亚品种等级的奇偶。作为应用程序，我们给出了椭圆曲线的奇偶校验猜想的新证明。本文的核心致力于开发雅各布人的算术理论，用于曲线的Galois封面，包括其L功能的分解以及Brauer Relative与Selmer群体之间的相互作用。
We investigate Selmer groups of Jacobians of curves that admit an action of a non-trivial group of automorphisms, and give applications to the study of the parity of Selmer ranks. Under the Shafarevich--Tate conjecture, we give an expression for the parity of the Mordell--Weil rank of an arbitrary Jacobian in terms of purely local invariants; the latter can be seen as an arithmetic analogue of local root numbers, which, under the Birch--Swinnerton-Dyer conjecture, similarly control parities of ranks of abelian varieties. As an application, we give a new proof of the parity conjecture for elliptic curves. The core of the paper is devoted to developing the arithmetic theory of Jacobians for Galois covers of curves, including decomposition of their L-functions, and the interplay between Brauer relations and Selmer groups.
