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

我们将使用理论的组织矩阵最近开发的伯恩斯和Macias卡斯蒂略发展一个普遍的理论精制的几何的伯奇和Swinnerton-Dyer型的阿贝尔品种定义的数域。这将扩展工作的Mazur和鲁宾，他们discribe大部分的算术椭圆曲线的岩泽代数在一个单一的斜埃尔米特矩阵。的基本成分，制定我们的精炼的aesthetures是一定的高度配对，这是与每个组织矩阵。在特殊情况下，我们希望表明这种配对与先前定义的Mazur/Tate，Tan，Schneider和Bertolini/Darmon的高度配对一致（都是一致的），从而表明，我们的精化公式包括Mazur/Tate和Bertolini/Darmon的公式，我们将从另一个方向把我们的公式与等变玉川数猜想（ETNC）的相关情况联系起来。并希望通过这种方式，我们可以使ETNC或其某些明确的后果服从数值和理论验证。

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.