Since Dirichlet’s remarkable proof of the analytic class number formula in the first half of the nineteenth century, conjectures on the relationship between the values of \(L\)-functions and invariants of arithmetic objects have motivated a great deal of research. The equivariant Tamagawa number conjecture (ETNC) is a unifying statement concerning the special values of motivic \(L\)-functions which encompasses both the Birch and Swinnerton-Dyer conjecture and the generalized Stark conjectures. This is a deep and challenging assertion which has been proved in very few cases. The Tamagawa number conjecture builds on the conjectures of A. A. Beilinson [Contemp. Math. 55, 1–34 (1986; Zbl 0609.14006)], predicting that the \(L\)-values of smooth projective varieties over \(\mathbb{Q}\) are given by period integrals and regulator maps, up to a rational factor. This conjecture was reformulated by J.-M. Fontaine and B. Perrin-Riou [Proc. Symp. Pure Math. 55, Pt. 1, 599–706 (1994; Zbl 0821.14013)] in a language that was naturally extended to motives with extra symmetries. There are two equivalent formulations of the conjecture. The first is a global formulation that concerns of the vanishing of a certain element in relative \(K\)-theory. The second is a local formulation that concerns the equality of two lattices.
This paper under review is actually an improvement of the main results in the author’s thesis. She proves the local equivariant Tamagawa number conjecture for the motive of an abelian extension of an imaginary quadratic field, with the action of the Galois group ring for all split primes \(p\neq 2, 3\) at all integer values \(s< 0\).
For the entire collection see [Zbl 1279.11002].