On derivatives of Artin \(L\)-series
Stark conjectures on units in global fields predict the rationality of certain quotients ``leading term of an \(L\)-function divided by a regulator''. So-called refined Stark conjectures make integrality statements about these quotients. Whilst the original conjectures were formulated for any Galois extension \(K/k\), the refined versions mostly assumed the extensions to be abelian (for example in the Rubin-Stark conjectures). The article under review is a breakthrough in several respects. It formulates a ``refined non-abelian Stark conjecture'' (Conj. 2.4.1), relates it to the earlier versions, and it provides rather strong evidence for the truth of this conjecture. To wit, the author proves that the Leading Term Conjecture for L-functions (also called ETNC for an appropriate motive) implies this Conjecture 2.4.1. Moreover, the article gives plenty of unconditional results on the conjecture and its consequences, and provides very detailed information concerning relations to other work. In this review we have to be selective, because of the length and encompassing character of this article. Let us begin by trying to sketch the statement of the pivotal Conjecture 2.4.1, also called the Central Conjecture (CC). Some notation is necessary. Always, \(S\) is a finite set of places of \(k\) containing the infinite and the ramified places. Let \(U_{K,S}\) denote the \(S\)-units of \(K\) and \(X_{K,S}\) the kernel of augmentation on the free \(\mathbb Z\)-module with basis \(S(K)\). There is also a finite auxiliary set \(T\), disjoint from \(S\), and an equivariant \(S\)-truncated \(T\)-modified \(L\)-function, whose definition we will not recall here. These lead to equivariant leading terms \(\theta^{(r)}(0)\) depending on all the data. Then for any \(G\)-linear map \(\phi: U_{K,S} \to X_{K,S}\), one has the product \(h:=\theta^{(r)}(0) R(\phi)\), where \(R(\phi)\) is the ``determinant'' (more precisely: the reduced norm) of \(\phi\) followed by \textit{the inverse of} the usual regulator map. Because of this convention concerning the regulator map, which differs from Stark's papers, the product \(h\) just mentioned is the correct analogue of the quotient mentioned at the beginning of this review. Conjecture 2.4.1 now assumes the existence of \(r\) totally split places in \(K/k\) and postulates a strong integrality property for \(h\): it should be the ``determinant'' (reduced norm) of a quadratic matrix over \({\mathbb Z}[G]\) fulfilling some more conditions. It is also part of the conjecture that \(\gamma h\) annihilates \(Cl({\mathcal O}_{K,S})\), with \(\gamma\) an explicit, easy-to-describe extra factor in \({\mathbb Z}[G]\). (We glossed over several fine points; to mention one, the statement has to be changed a little if \(G\) admits symplectic irreducible characters.) The overall structure of the paper is as follows: \S1 mentions, among many other things, three theorems A, B, C which will be consequences of the validity of (CC) in specific situations. We cannot spell them out here; they give rather precise and novel annihilation statements for class groups. \S2 describes (CC) and a related conjecture. \S3 provides links with earlier conjectures, demonstrating in particular that (CC) implies the conjectures of Rubin-Stark and of Brumer (both in the abelian setting). In fact Burns obtains a non-abelian version of the Brumer conjecture. In \S4 the main results are formulated; first and foremost the fact that the Leading Term Conjecture (LTC) implies (CC). From cases where LTC is proved (e.g. \(K\) absolutely abelian) one obtains immediately some interesting unconditional consequences, and proofs for Theorems A, B and C. In \S5 we see some important reduction steps (e.g. change of the set \(S\)). Section 6 discusses the Leading Term Conjecture; starting from its formulation, a much more explicit version involving concrete complexes is given. In \S\S7-8, the difficult technical details of the implication (LTC) \(\Rightarrow\) (CC) are given. \S9 proves two results left over from \S4. In \S10-\S12, one finds proofs either of (CC) or a variant (2.6.1) of it, under special hypotheses; let us just mention that (CC) is proved for an infinite class of quaternionic extensions of \({\mathbb Q}\), thus keeping up the nice tradition of using these extensions as an interesting testing ground. Some further comments: At the end of \S3 the author discusses a very interesting family of dihedral extensions in order to illustrate his conjectures. In \S4 we even find an example \(K/\mathbb Q\) with Galois group \(A_4\). This is taken from work of Navilarekallu. (Apparently a minor slip happened: the description of the field diverges from the original one, and the fields seem to be different in fact. But taking the original description instead seems to produce exactly what Burns wished to demonstrate at this point.) \S8 is devoted to the proof of the annihilation statement in the Central Conjecture. As an additional complication, one needs Tate sequences with small sets \(S\), but the author is able to build on previous work of Ritter-Weiss, Nickel, and the reviewer. His setting is certainly more general. Let us conclude by repeating that this important paper contains plenty of interesting results, and this review certainly does not cover all of them.
scientific article; zbMATH DE number 5976095
scientific article