Organising matrices, height pairings and refined conjectures ofthe Birch and Swinnerton-Dyer type
Final Report Abstract
Let F/k be a cyclic extension of number fields of odd prime power degree. Let A/k denote an abelian variety. We write A F for the base change of A and let M F = h1(A F)(1) be the associated motive. The relevant case of the Equivariant Tamagawa Number Conjecture (short ETNC) is then an equivariant refinement of the Birch and Swinnerton-Dyer conjecture. In joint work with D. Macias Castillo, "Congruences for critical values of higher derivatives of twisted Hasse-Weil L-functions", Crelle Journal 722 (2017), 105-136, we are able, under suitable technical hypothesis including triviality of certain Tate-Shafarevic groups, to define explicit twisted regulators and thereby produce explicit reformulations of the ETNC. Based on this we obtain consequences of the conjectured validity of the ETNC in the spirit of 'refined conjectures of the BSD type' as initiated by Mazur and Tate.
