Organising matrices, height pairings and refined conjectures ofthe Birch and Swinnerton-Dyer type

我们将使用Burns和Macias Castillo最近发展的组织矩阵理论来发展关于数域上定义的阿贝尔变种的Birch和Swinnerton-Dyer类型的精化猜想的普遍理论。这将扩展Mazur和Rubin的工作，他们用单个斜厄米特矩阵描述了岩泽代数上椭圆曲线的大部分算术。形成我们精炼猜想的基本成分是与每个组织矩阵相关的特定高度配对。在特殊情况下，我们希望证明这个高度对与先前定义的Mazur/Tate、Tan、Schneider和Bertolini/Darmon的高度对重合，从而表明我们改进的猜想包括Mazur/Tate和Bertolini/Darmon.在不同的方向上，我们的猜想将与等变Tamagawa数猜想(ETNC)的相关情况相联系，并希望通过这种方式，我们可以使ETNC或其某些显式结果服从于数值和理论验证.

We will use the theory of organising matrices recently developed by Burns and Macias Castillo to develop a universal theory of refined conjectures of the Birch and Swinnerton-Dyer type for abelian varieties defined over number fields. This will extend work of Mazur and Rubin where they discribe much of the arithmetic of an elliptic curve over an Iwasawa algebra in terms of a single skew-Hermitian matrix.The essential ingredient to formulate our refined conjectures is a certain height pairing which is associated to each organising matrix. In special cases we hope to show that this pairing coincides with previously defined height pairings of Mazur/Tate, Tan, Schneider and Bertolini/Darmon (which all coincide), and thus show that our refined conjectures include those of Mazur/Tate and Bertolini/Darmon.In a different direction we will relate our conjectures to the relevant case of the Equivariant Tamagawa Number Conjecture (ETNC) and hope that in this way we can make the ETNC or certain explicit consequences thereof amenable to numerical and theoretical verifications.