This is an investigation into the possible existence and consequences of a Birch–Swinnerton-Dyer-type formula for L -functions of elliptic curves twisted by Artin representations. We translate expected properties of L -functions into purely arithmetic predictions for elliptic curves, and show that these force some peculiar properties of the Tate–Shafarevich group, which do not appear to be tractable by traditional Selmer group techniques. In particular, we exhibit settings where the different p -primary components of the Tate–Shafarevich group do not behave independently of one another. We also give examples of “arithmetically identical” settings for elliptic curves twisted by Artin representations, where the associated L -values can nonetheless differ, in contrast to the classical Birch–Swinnerton-Dyer conjecture.

中文翻译：

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椭圆曲线Artin扭曲L值的BSD型公式

这是对由Artin表示扭曲的椭圆曲线的L函数的Birch-Swinnerton-Dyer型公式的可能存在和后果的调查。我们将L函数的期望性质转换为椭圆曲线的纯算术预测，并表明它们迫使Tate–Shafarevich组具有某些特殊的属性，而这些属性似乎无法通过传统的Selmer组技术来处理。特别是，我们展示了Tate–Shafarevich组的不同p-主成分彼此之间行为不独立的环境。与经典的Birch-Swinnerton-Dyer猜想相反，我们还给出了由Artin表示扭曲的椭圆曲线的“算术相同”设置的示例，其中相关的L值仍然可以不同。