Congruences for critical values of higher derivatives of twisted Hasse–Weil L-functions

Let be an abelian variety over a number field and let be a finite cyclic extension of of -power degree for an odd prime . Under certain technical hypotheses, we obtain a reinterpretation of the equivariant Tamagawa number conjecture (‘eTNC’) for , and as an explicit family of -adic congruences involving values of derivatives of the Hasse–Weil -functions of twists of , normalised by completely explicit twisted regulators. This reinterpretation makes the eTNC amenable to numerical verification and furthermore leads to explicit predictions which refine well-known conjectures of Mazur and Tate.