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

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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.