1·1. Dirichlet L-series at $s = 0$ and annihilation of class groups

Stark’s conjecture predicts a description for the leading term of a general Artin L-series at $s = 0$ up to an unspecified rational factor. Formulating an integral refinement of this conjecture turned out to be a delicate task that Stark himself, in [ Reference Stark54 ], only found a solution to in the case that the order of vanishing of the L-series at $s = 0$ is one. Initial generalisations to higher orders of vanishing, for example the ‘question’ of Stark in [ Reference Grant29 , Reference Tangedal55 ] or a conjecture of Sands [ Reference Sands50 , Conjecture 2·0], were subsequently shown to not hold in general by Rubin [ Reference Rubin48 , Section 4] and Popescu [ Reference Popescu47 ]. Instead, Rubin proposed in loc. cit. what is now commonly referred to as the ‘Rubin–Stark conjecture’.

Going beyond mere integrality, it is expected that this unspecified factor encodes important arithmetic information and, in particular, is linked to the Galois module structure of class groups. The primordial example of this phenomenon is Stickelberger’s theorem from the 19th century, which asserts that the ideal class group of a cyclotomic field is annihilated by a certain element valued in the group ring over the relevant Galois group and constructed from values of Dirichlet L-series at $s = 0$ . The analogous annihilation statement for class groups of finite abelian CM extensions of totally real fields is known as the ‘Brumer–Stark conjecture’ and has very recently been settled by Dasgupta and Kakde [ Reference Dasgupta and Kakde21 ] with additional arguments by Dasgupta, Kakde, Silliman and Wang [ Reference Dasgupta, Kakde, Silliman and Wang23 ]. In certain situations, these results can even be extended to non-abelian CM extensions, see [ Reference Burns and Johnston16 , Reference Ellerbrock and Nickel26 , Reference Johnston and Nickel33 ].

However, outside the setting of totally imaginary extensions of totally real fields the values of the associated Dirichlet L-series at $s = 0$ usually vanish (the only exception being the case considered by Nomura in [ Reference Nomura45 ]) and so naive generalisations of Stickelberger’s theorem become trivial. This led Burns to formulate the question whether in such cases one can instead use higher derivatives of Dirichlet L-series to produce annihilators of class groups (see [ Reference Macías Castillo37 , Question 1·1]) and similar aspects have also been considered by Buckingham [ Reference Buckingham7 , Reference Buckingham8 ].

In this article, we prove new results on the Rubin–Stark conjecture and, moreover, on the annihilation of class groups, in cases of higher orders of vanishing. Indeed, in Theorem 1·1 we extend annihilation results concerning multi-quadratic extensions by Sands [ Reference Sands52 ] and the second author [ Reference Macías Castillo37 ] to general p-elementary abelian extensions $K/k$ of number fields, for arbitrary prime numbers p. This result is conditional on the collection of subextensions $L/k$ of $K/k$ that have degree p validating a conjecture of Burns, Kurihara and Sano (which we recall as Conjecture 1·10). Since we are then also able, in Theorem 1·3, to prove new cases of the Burns–Kurihara–Sano conjecture, we derive a method of systematically producing examples in which the annihilation claim of Theorem 1·1 is valid unconditionally (see Corollary 1·8).

1·2. Statements of the main results

To describe our results in more detail, we fix a finite abelian extension of number fields $K / k$ with Galois group $G \,{:\!=}\, \mathrm{Gal}({K}/{k})$ and, following Rubin [ Reference Rubin48 , Hypothesis 2·1], a triple (S, V, T) of finite sets of places of k with the following properties:

1. (H1) S contains both the set $S_\infty$ of infinite places of k and the places that ramify in K;

2. (H2) $V \subsetneq S$ is a proper subset comprising places which split completely in $K / k$ ;

3. (H3) T is disjoint from S and such that the group $\mathcal{O}^\times_{K, S, T} \,{:\!=}\, \{ a \in K^\times \mid {\mathrm{ord}}_w (a) = 0 \text{ if } w \not \in S_K, {\mathrm{ord}}_w (a - 1) \gt 0 \text{ if } w \in T_K \}$ is ${\mathbb{Z}}$ -torsion free. (Here $S_K$ and $T_K$ denote the sets of places of K that lie above those in S and T, respectively, and ${\mathrm{ord}}_w$ is the normalised valuation attached to w.)

We refer to such a triple (S, V, T) as a ‘Rubin datum’ for $K / k$ . For any Rubin datum and character $\chi$ in $\widehat{G} \,{:\!=}\, \mathrm{Hom}_{\mathbb{Z}} (G, {\mathbb{C}}^\times)$ , the (S-truncated, T-modified) Dirichlet L-series

\[L_{k, S, T} (\chi, s) \,{:\!=}\, \prod_{v \in T} (1 - \chi ({\mathrm{Frob}}_v)\mathrm{N} v^{1 - s} ) \cdot \prod_{v \not \in S} (1 - \chi ({\mathrm{Frob}}_v) \mathrm{N}v^{-s})^{-1}\quad \text{ if } \mathrm{Re}(s) \gt 1\]

is known to admit a meromorphic continuation to ${\mathbb{C}}$ that is holomorphic and of order of vanishing at least $|V|$ at $s = 0$ (cf. [ Reference Tate56 , Chapter I, Proposition 3·4]). We may therefore define the ( $|V|$ -th order) ‘Stickelberger element’

\[\theta_{K / k, S, T}^{(|V|)} (0) \,{:\!=}\, \sum_{\chi \in \widehat{G}} \big( \lim_{s \to 0} s^{-|V|} L_{k, S, T} (\chi^{-1}, s) \big) \cdot e_{\chi},\]

with $e_\chi \,{:\!=}\, |G|^{-1} \sum_{\sigma \in G} \chi (\sigma)^{-1} \sigma$ the usual primitive orthogonal idempotent in ${\mathbb{C}} [G]$ associated with $\chi$ . In addition, we define $X_{K, S} \subseteq Y_{K, S} \,{:\!=}\, \bigoplus_{w \in S_K} {\mathbb{Z}} w$ to be the ${\mathbb{Z}} [G]$ -submodule of elements whose coefficients sum to zero, and denote the Dirichlet regulator isomorphism by

(1·1) \begin{equation} \lambda_{K, S} {\colon} {\mathbb{R}} \otimes_{\mathbb{Z}} \mathcal{O}_{K, S}^\times \stackrel{\simeq}{\longrightarrow} {\mathbb{R}} \otimes_{\mathbb{Z}} X_{K, S},\quad x \otimes a \mapsto - x \sum_{w \in S_K} \log | a|_w \cdot w.\end{equation}

The Rubin–Stark conjecture [ Reference Rubin48 , Conjecture B’] now predicts, via the reinterpretation given in Lemma 2·2 below, that for every homomorphism of $\mathbb{Z}[G]$ -modules $f {\colon} \mathcal{O}_{K, S, T}^\times \to X_{K, S}$ one has

(1·2) \begin{equation}\theta_{K / k, S, T}^{(|V|)} (0) \cdot {\det}_{{\mathbb{R}} [G]} (f_\mathbb{R}\circ \lambda_{K, S}^{-1})\in {\mathbb{Z}} [G]\end{equation}

with $f_\mathbb{R}$ the scalar extension ${\mathbb{R}}\otimes_{\mathbb{Z}} f{\colon}{\mathbb{R}} \otimes_{\mathbb{Z}} \mathcal{O}_{K, S}^\times={\mathbb{R}} \otimes_{\mathbb{Z}} \mathcal{O}_{K, S,T}^\times\longrightarrow\mathbb{R}\otimes_\mathbb{Z} X_{K,S}$ of f. In addition, it is expected that the element in (1·2) annihilates the $S_K$ -class group of K (cf. [ Reference Burns13 , Conjecture 2·4·1] or [ Reference Macías Castillo37 , Question 1·1]). Here we study a refinement of this question that instead considers the $S_K$ -ray class group ${\mathrm{Cl}}_{K, S, T}$ of K mod $T_K$ (defined as the quotient of the group of fractional ideals of ${\mathcal{O}}_{K, S}$ coprime to $T_K$ , by the subgroup of principal ideals with a generator congruent to 1 modulo all $w \in T_K$ ).

To state our first main result in this direction we fix a prime number p, consider a p-elementary abelian extension $K/k$ and write $\Omega$ for the set of degree-p subextensions $L/k$ of $K/k$ .

Theorem 1·1. Let $K / k$ be a p-elementary abelian extension of number fields of degree $p^m$ and fix a Rubin datum (S, V, T) for $K / k$ that satisfies

\[|S| \geq \max \{ |V| + 2, |V| - s_p + (p - 1)(m - 1) + 3\},\]

where $s_p \,{:\!=}\, \dim_{\mathbb{F}_p} ({\mathrm{Cl}}_{k, S, T} \otimes_{\mathbb{Z}} \mathbb{F}_p)$ denotes the p-rank of the S-ray class group mod T of k.

If for all subextensions $L / k$ in $\Omega$ the Burns–Kurihara–Sano conjecture [ Reference Burns, Kurihara and Sano17 , Conjecture 7·3] is valid for $(L/k,S,V,T)$ , then

\[\big \{ \theta_{K / k, S, T}^{(|V|)} (0) \cdot {\det}_{{\mathbb{R}} [G]} ( f_\mathbb{R}\circ \lambda_{K, S}^{-1} ) \mid f \in \mathrm{Hom}_{{\mathbb{Z}} [G]} ( \mathcal{O}_{K, S, T}^\times, X_{K, S}) \big \}\,\,\subseteq\,\, \mathrm{Ann}_{{\mathbb{Z}} [G]} ({\mathrm{Cl}}_{K, S, T}).\]

In particular, the Rubin–Stark conjecture is valid for $(K/k,S,V,T)$ .

To prove Theorem 1·1 (in Section 4) we first deduce in Lemma 4·2 the validity of the Rubin–Stark conjecture for $K / k$ from the assumed validity of conjecture [ Reference Burns, Kurihara and Sano17 , Conjecture 7·3] for all subextensions in $\Omega$ . The annihilation statement in Theorem 1·1 is then deduced from this by varying the Rubin datum in combination with Cebotarev’s density theorem, as in the theory of ‘Stark systems’ (see, for example, [ Reference Burns, Sakamoto and Sano18 , Section 4]). Although this latter aspect of the argument is of a general nature, we prefer to focus on the concrete situation of Theorem 1·1 in this article and to discuss the general formalism elsewhere.

As our second main result, we prove new cases of the Burns–Kurihara–Sano conjecture.

Results on the Rubin–Stark conjecture in the literature outside the classical cases where at most one archimedean place of k splits in K or the degree $[K : k]$ is at most two are sparse (see Remark 1·11 for a list of known cases). By combining Theorems 1·3 and 1·1 with Example 1·5, we now obtain the following method to systematically produce new examples in which the conjecture is valid.

Corollary 1·8. Let p be a prime number, let $\zeta_p$ be a primitive $p$ -th root of unity, and let $\kappa$ be a number field with the property that $\kappa \cap {\mathbb{Q}} (\mu_p) = {\mathbb{Q}}$ . Let $a_1, \dots, a_m$ be elements of $\kappa$ that are $\mathbb{F}_p$ -linearly independent in $\kappa^\times / (\kappa^\times)^p$ , and set $k \,{:\!=}\, \kappa (\mu_p)$ and $K \,{:\!=}\, k (\sqrt[p]{a_1}, \dots, \sqrt[p]{a_m})$ .

If (S, V, T) is a Rubin datum for $K / k$ with

\[|S| \geq \max \{ |V| + 2, |V| - s_p + (p - 1)(m - 1) + 3\},\]

then

\[\big \{ \theta_{K / k, S, T}^{(|V|)} (0) \cdot {\det}_{{\mathbb{R}} [G]} ( f_\mathbb{R}\circ \lambda_{K, S}^{-1} ) \mid f \in \mathrm{Hom}_{{\mathbb{Z}} [G]} ( \mathcal{O}_{K, S, T}^\times, X_{K, S}) \big \}\,\,\subseteq\,\, \mathrm{Ann}_{{\mathbb{Z}} [G]} ({\mathrm{Cl}}_{K, S, T}).\]

In particular, the Rubin–Stark conjecture is valid for $(K/k,S,V,T)$ .

1·3. The conjectures of Rubin–Stark and Burns–Kurihara–Sano

In this section we state the Rubin–Stark conjecture and the conjecture [ Reference Burns, Kurihara and Sano17 , Conjecture 7·3] of Burns, Kurihara and Sano, and we discuss the list of cases in which either conjecture is known to be valid. The formulations given here, in terms of the products of the form (1·2), are equivalent to the original versions of the conjectures by Lemma 2·2 below.

We fix a finite abelian extension of number fields $K/k$ with Galois group $G \,{:\!=}\, \mathrm{Gal}({K}/{k})$ and Rubin datum (S,V,T).

Conjecture 1·9 (Rubin–Stark, [ Reference Rubin48 , Conjecture B’]). One has

\[\big \{ \theta_{K / k, S, T}^{(|V|)} (0) \cdot {\det}_{{\mathbb{R}} [G]} ( f_\mathbb{R}\circ \lambda_{K, S}^{-1} ) \mid f \in \mathrm{Hom}_{{\mathbb{Z}} [G]} ( \mathcal{O}_{K, S, T}^\times, X_{K, S}) \big \}\,\,\subseteq\,\, {\mathbb{Z}} [G].\]

We set $K_T^\times \,{:\!=}\, \{ a \in K^\times \mid {\mathrm{ord}}_w (a - 1) \gt 0 \text{ if } w \in T_K\}$ . Then the ‘integral dual Selmer group’ $\mathrm{Sel}_{K, S, T}$ is defined by Burns–Kurihara–Sano [ Reference Burns, Kurihara and Sano17 , Definition 2·1] as the cokernel of the map

\[ \prod_{w \not \in S_K \cup T_K} {\mathbb{Z}} \to \mathrm{Hom}_{\mathbb{Z}} ( K_T^\times, {\mathbb{Z}} ),\quad (x_w)_w \mapsto \big \{ a \mapsto \sum_{w} x_w {\mathrm{ord}}_w (a) \big \}.\]

It fits into a canonical exact sequence of G-modules

\begin{align*} 0 \longrightarrow \mathrm{Hom}_{\mathbb{Z}} ( {\mathrm{Cl}}_{K, S, T}, {\mathbb{Q}} / {\mathbb{Z}}) \longrightarrow \mathrm{Sel}_{K, S, T} \longrightarrow \mathrm{Hom}_{\mathbb{Z}} ( \mathcal{O}_{K, S, T}^\times, {\mathbb{Z}} ) \longrightarrow 0,\end{align*}

with all duals endowed with the contragredient G-action.

In the sequel, for $n\geq 0$ , we write ${\operatorname{Fitt}}^n_{{\mathbb{Z}} [G]} (M)$ for the $n$ - th Fitting ideal in $\mathbb{Z}[G]$ of a finitely presented ${\mathbb{Z}} [G]$ -module M (see, for example, [ Reference Northcott46 , Section 3·1] or [ Reference Nickel43 ]). Given a subset I of ${\mathbb{C}} [G]$ , we denote by $I^\#$ the image of I under the involution of ${\mathbb{C}} [G]$ that inverts elements of G.

Burns–Kurihara–Sano use the Selmer group to refine Conjecture 1·9 as follows.

Conjecture 1·10 (Burns–Kurihara–Sano [ Reference Burns, Kurihara and Sano17 , Conjecture 7·3]). One has

(1·3) \begin{equation} \big \{ \theta_{K / k, S, T}^{(|V|)} (0) \cdot {\det}_{{\mathbb{R}} [G]} ( f_\mathbb{R}\circ \lambda_{K, S}^{-1} ) \mid f \in \mathrm{Hom}_{{\mathbb{Z}} [G]} ( \mathcal{O}_{K, S, T}^\times, X_{K, S}) \big \} = {\operatorname{Fitt}}^{|V|}_{{\mathbb{Z}} [G]} ( \mathrm{Sel}_{K, S, T})^\#.\end{equation}

1. (a) One can directly verify the conjecture for the following general classes of extensions $K / k$ :

   1. (i) if $[K\,{:}\,k] \leq 2$ , then it follows from the analytic class number formula (see [ Reference Rubin48 , Corollary 3·2 and Theorem 3·5]);

   2. (ii) if $k = {\mathbb{Q}}$ and $V = S_\infty$ is the singleton comprising the unique infinite place of ${\mathbb{Q}}$ , then it follows by means of a direct computation that shows that the relevant Rubin–Stark element (as defined in section 2·1) can be expressed in terms of a cyclotomic unit (cf. [ Reference Tate56 , Chapter III, Section 5]);

   3. (iii) if k is an imaginary quadratic field and $V = S_\infty$ is the singleton comprising the unique infinite place of k, then it follows from Kronecker’s Second Limit Formula for elliptic units (cf. [ Reference Tate56 , Chapter IV, Proposition 3·9]);

   4. (iv) if $V = {\varnothing}$ , then it is a consequence of work of Cassou-Noguès [ Reference Cassou-Noguès19 ] and, independently, Deligne and Ribet [ Reference Deligne and Ribet24 ] (cf. [ Reference Gross30 , Proposition 3·7]).

1. (b) In addition, the conjecture has been directly verified in the following particular cases.

   1. (i) Grant [ Reference Grant29 ] has verified it for $k = {\mathbb{Q}} ( \zeta_5)$ and $K = k ( \sqrt[5]{\epsilon})$ with $\zeta$ a primitive 5th root of unity and $\epsilon \,{:\!=}\, - \zeta^2 - \zeta^3$ .

   2. (ii) If $K / k$ is multi-quadratic, then Dummit, Sands and Tangedal [ Reference Dummit, Sands and Tangedal25 ], Sands [ Reference Sands51 ], and the second author [ Reference Macías Castillo37 ] have verified it in special cases.

   3. (iii) McGown, Sands and Vallières [ Reference McGown, Sands and Vallières38 ] have numerically verified it for $V = S_\infty$ in the 19197 examples of k a real quadratic field and K a totally real cubic extensions of k of discriminant less than $10^{12}$ and $V = S_\infty$

1. (c) It holds if $S \setminus V$ contains a place that splits completely in K (cf. [ Reference Rubin48 , Proposition 3·1]).

2. (d) The examples listed in (a) are by now sufficiently well understood to allow for a proof of $\mathrm{eTNC}(K/k)$ [ Reference Burns, Kurihara and Sano17 , Conjecture 3·6]. By [ Reference Burns, Kurihara and Sano17 , Theorem 7·5], for any given Rubin datum for $K/k$ , the conjecture [ Reference Burns, Kurihara and Sano17 , Conjecture 7·3], and hence also the Rubin–Stark conjecture, is a consequence of $\mathrm{eTNC}(K/k)$ . Using functoriality properties of the eTNC, one thus obtains the validity of both the Burns–Kurihara–Sano and Rubin–Stark conjectures for any $K / k$ (and Rubin datum) such that $F \subseteq k \subseteq K \subseteq H$ , with $H / F$ a finite Galois extension for which $\mathrm{eTNC} (H / F)$ holds. The same conclusion is true if the ‘minus part’ $\mathrm{eTNC}^- (H / F)$ of $\mathrm{eTNC} (H / F)$ holds and k is totally real and K is totally imaginary.

   In this direction, the following is currently known:

1. (i) $\mathrm{eTNC} (H / F)$ holds if $[H:F]=2$ ; this case is proved by Kim [ Reference Kim36 , Section 2·4];

2. (ii) $\mathrm{eTNC} (H / F)$ holds if $F = {\mathbb{Q}}$ ; this is work of Burns and Greither [ Reference Burns and Greither15 ] with additional arguments for the 2-component by Flach [ Reference Flach28 ];

3. (iii) $\mathrm{eTNC} (H / F)$ holds if F an imaginary quadratic field such that all prime divisors of $[H : F]$ split in k or validate Iwasawa’s $\mu$ -vanishing conjecture; this case is proved by Hofer and the first author [ Reference Bullach and Hofer11 , Theorem B] and extends previous work of Bley [ Reference Bley2 , Reference Bley3 , Reference Bley4 ].

4. (iv) $\mathrm{eTNC} (H / F)^-$ holds if F is a totally real field and H is CM; this is work of the first author, Burns, Daoud and Seo [ Reference Bullach, Burns, Daoud and Seo10 ] with additional arguments for the 2-component by Dasgupta, Kakde and Silliman [ Reference Dasgupta, Kakde and Silliman22 ]. Earlier work in this direction includes [ Reference Atsuta and Kataoka1 , Reference Nickel41 , Reference Nickel42 , Reference Nickel44 ]. (The results in [ Reference Bullach, Burns, Daoud and Seo10 ] crucially rely on work of Dasgupta and Kakde [ Reference Dasgupta and Kakde21 ] on the Strong Brumer–Stark conjecture, and we remark that the Rubin–Stark conjecture can alternatively be directly deduced from the Strong Brumer–Stark conjecture, see [ Reference Dasgupta and Kakde21 , Theorem 1·6]).

1. Further examples of, not necessarily abelian, extensions $H / F$ for which $\mathrm{eTNC}(H/F)$ is known at present include the following:

   1. (i) H is a totally real Galois extension of $F = {\mathbb{Q}}$ such that either $\mathrm{Gal}({K}/{{\mathbb{Q}}}) \cong S_3$ and H has discriminant less than $10^{20}$ or $\mathrm{Gal}({H}/{{\mathbb{Q}}}) \cong D_{12}$ and H has discriminant less than $10^{30}$ ; by Hofmann, Johnston, and Nickel [ Reference Johnston and Nickel34 , Corollary A·3].

   2. (ii) a particular family of Quaternionic extensions H of $F=\mathbb{Q}$ ; by Burns and Flach [ Reference Burns and Flach14 , Theorem 4·1].

   3. (iii) one example of a Galois extension H of $F=\mathbb{Q}$ with $\mathrm{Gal}({H}/{{\mathbb{Q}}}) \cong A_4$ ; numerical verification by Navilarekallu [ Reference Navilarekallu40 ].

   4. (iv) when a number of standard conjectures are known to be valid, further results can be deduced from the examples above, see [ Reference Johnston and Nickel32 , Section 4] and [ Reference Johnston and Nickel34 , Section 10].