Let \(K/k\) be a finite Galois extension of number fields and let \(G=\mathrm{Gal}(K/k)\). The equivariant Tamagawa number conjecture (eTNC) for the pair \((h^{0}(\mathrm{Spec}(K)),\mathbb{Z}[G])\) relates the leading terms at \(s=0\) of Artin \(L\)-functions attached to \(K/k\) to certain natural arithmetic invariants, and may be viewed as an equivariant refinement of the analytic class number formula at \(s=0\). It implies a number of other conjectures involving values or leading terms of Artin \(L\)-functions at \(s=0\). We abbreviate the eTNC for the pair \((h^{0}(\mathrm{Spec}(K)),\mathbb{Z}[G])\) to \(\mathrm{eTNC}(K/k)\), and for a rational prime \(p\), we write \(\mathrm{eTNC}(K/k)_{p}\) for its so-called \(p\)-part. Then \(\mathrm{eTNC}(K/k)\) holds if and only if \(\mathrm{eTNC}(K/k)_{p}\) holds for all \(p\).
We henceforth specialise to the case in which \(G\) is abelian. D. Burns and C. Greither [Invent. Math. 153, No. 2, 303–359 (2003; Zbl 1142.11076)], together with additional arguments for the \(2\)-part by M. Flach, [J. Reine Angew. Math. 661, 1–36 (2011; Zbl 1242.11083)], proved that \(\mathrm{eTNC}(K/k)\) holds in the case that \(k=\mathbb{Q}\). W. Bley, [Doc. Math. 11, 73–118 (2006; Zbl 1178.11070)] adapted this strategy to prove that if \(k\) is an imaginary quadratic field and \(p\) is an odd prime that splits in \(k\) and does not divide the class number of \(k\), then \(\mathrm{eTNC}(K/k)_{p}\) holds. The main result of the article under review improves this last result by significantly weakening the assumptions on \(p\), as described below.
In D. Burns et al., [Algebra Number Theory 11, No. 7, 1527–1571 (2017; Zbl 1455.11158)], the authors extended the aforementioned methods of Burns and Greither to arbitrary abelian extensions. Roughly speaking, their strategy is as follows (we suppress some technical details and hypotheses). Fix a prime \(p\) and a finite abelian extension \(K/k\). Let \(k_{\infty}/k\) be a \(\mathbb{Z}_{p}\)-extension and let \(K_{\infty} = Kk_{\infty}\). Suppose that each of the following conjectures (all of which are stated in loc.cit.) holds for \(K_{\infty}/k\).

(i)

the higher-rank Iwasawa main conjecture;

(ii)

the Iwasawa-theoretic Mazur-Rubin-Sano conjecture; and

(iii)

Gross’ finiteness conjecture (also called condition (F)).

Then \(\mathrm{eTNC}(K/k)_{p}\) holds.
In the article under review, the authors apply this strategy to prove the following result. Let \(p\) be a prime number, let \(k\) be an imaginary quadratic field, and let \(K/k\) be a finite abelian extension. Then \(\mathrm{eTNC}(K/k)_{p}\) holds whenever

(\(\star\))

either \(p \nmid [K:k]\) or the \(\mu\)-invariant of a certain Iwasawa module vanishes.

We remark that the \(\mu\)-invariant is known to vanish whenever \(p\) splits in \(k\) or \([K:k]\) is a power of \(p\), and so the result is unconditional in these cases.
The idea of the proof is as follows. The condition (\(\star\)) is needed to show that (i) the higher-rank Iwasawa main conjecture holds for an appropriate extension \(K_{\infty}/k\). Both (ii) the Iwasawa-theoretic Mazur-Rubin-Sano conjecture and (iii) condition (F) are proven for wide classes of extensions \(K_{\infty}/k\). From these results, the authors show that there exists a choice of \(k_{\infty}\) such that the all three conjectures (i), (ii) and (iii) hold for \(Kk_{\infty}/k\), and then apply the above strategy to show that \(\mathrm{eTNC}(K/k)_{p}\) holds. We remark that once this is established, they are then able to deduce that the Iwasawa-theoretic Mazur-Rubin-Sano conjecture in fact holds unconditionally for all extensions \(K_{\infty}/k\) when \(k\) is imaginary quadratic.