On the $p$-adic Beilinson conjecture and the equivariant Tamagawa number conjecture

Let $E/K$ be a finite Galois extension of totally real number fields with Galois group $G$. Let $p$ be an odd prime and let $r>1$ be an odd integer. The $p$-adic Beilinson conjecture relates the values at $s=r$ of $p$-adic Artin $L$-functions attached to the irreducible characters of $G$ to those of corresponding complex Artin $L$-functions. We show that this conjecture, the equivariant Iwasawa main conjecture and a conjecture of Schneider imply the `$p$-part' of the equivariant Tamagawa number conjecture for the pair $(h^0(\mathrm{Spec}(E))(r), \mathbb Z[G])$. If $r>1$ is even we obtain a similar result for Galois CM-extensions after restriction to `minus parts'.


Introduction
Let E/K be a finite Galois extension of number fields with Galois group G and let r be an integer. The equivariant Tamagawa number conjecture (ETNC) for the pair (h 0 (Spec(E))(r), Z[G]) as formulated by Burns and Flach [BF01] asserts that a certain canonical element T Ω(E/K, r) in the relative algebraic K-group K 0 (Z[G], R) vanishes. This element relates the leading terms at s = r of Artin L-functions to natural arithmetic invariants.
If r = 0 this might be seen as a vast generalization of the analytic class number formula for number fields, and refines Stark's conjecture for E/K as discussed by Tate in [Tat84] and the 'Strong Stark conjecture' of Chinburg [Chi83, Conjecture 2.2]. It is known to imply a whole bunch of conjectures such as Chinburg's 'Ω 3 -conjecture' [Chi83,Chi85], the Rubin-Stark conjecture [Rub96], Brumer's conjecture, the Brumer-Stark conjecture (see [Tat84, Chapitre IV, §6]) and generalizations thereof due to Burns [Bur11] and the author [Nic11b]. If r is a negative integer, the ETNC refines a conjecture of Gross [Gro05] and implies (generalizations of) the Coates-Sinnott conjecture [CS74] and a conjecture of Snaith [Sna06] on annihilators of the higher K-theory of rings of integers (see [Nic11a]). If r > 1 the ETNC likewise predicts constraints on the Galois module structure of p-adic wild kernels [Nic19].
The functional equation of Artin L-functions suggests that the ETNC at r and 1 − r are equivalent. This is not known in general, but leads to a further conjecture which is sometimes referred to as the local ETNC. Except for the validity of the local ETNC it therefore suffices to consider the (global) ETNC for either odd or even integers r. Note that the local ETNC is widely believed to be easier to settle. For instance, the 'global epsilon constant conjecture' of Bley and Burns [BB03] measures the compatibility of the closely related 'leading term conjectures' at s = 0 [Bur01] and s = 1 [BB07] and is known to hold for arbitrary tamely ramified extensions [BB03,Corollary 7.7] and also for certain weakly ramified extensions [BC16]. Now suppose that E/K is a Galois extension of totally real number fields and let p be an odd prime. If r < 0 is odd Burns [Bur15] and the author [Nic13] independently have shown that the 'p-part' of the ETNC for the pair (h 0 (Spec(E))(r), Z[G]) holds provided that a certain Iwasawa µ p -invariant vanishes (which conjecturally is always true). The latter condition is mainly present because the equivariant Iwasawa main conjecture (EIMC) for totally real fields then holds by independent work of Ritter and Weiss [RW11] and of Kakde [Kak13].
The case r ≥ 0 is more subtle. Burns and Venjakob [BV06,BV11] (see also [Bur15, Corollary 2.8]) proposed a strategy for proving the p-part of the ETNC for the pair (h 0 (Spec(E))(1), Z[G]). More precisely, this special case of the ETNC is implied by the vanishing of the relevant µ p -invariant, Leopoldt's conjecture for E at p and the 'p-adic Stark conjecture at s = 1'. The latter conjecture relates the leading terms at s = 1 of the complex and p-adic Artin L-functions attached to characters of G by certain comparison periods. Note that Burns and Venjakob actually assume these conjectures for all odd primes p and then deduce the ETNC for the pair (h 0 (Spec(E))(1), Z[ 1 2 ][G]), but their approach has recently been refined by Johnston and the author [JN20a] so that one can indeed work prime-by-prime.
There are similar results on minus parts if L/K is a Galois CM-extension with Galois group G, i.e. K is totally real and L is a totally complex quadratic extension of a totally real field L + . Namely, if r < 0 is even and µ p vanishes, then the minus p-part of the ETNC for the pair (h 0 (Spec(L))(r), Z[G]) holds [Bur15,Nic13]. Burns [Bur20] recently proposed a strategy for proving the minus p-part of the ETNC in the case r = 0. In comparison with the strategy in the case r = 1, Leopoldt's conjecture is replaced with the conjectural non-vanishing of Gross's regulator [Gro81], and the p-adic Stark conjecture is replaced with the 'weak p-adic Gross-Stark conjecture' [Gro81, Conjecture 2.12b] (now a theorem for linear characters by work of Dasgupta, Kakde and Ventullo [DKV18]). For an approach that only relies upon the validity of the EIMC we refer the reader to [Nic11c,Nic16].
The aim of this article is to propose a similar strategy in the remaining cases, i.e. we will consider the ETNC for Tate motives h 0 (Spec(L))(r) where r > 1 and L is a CM-field. Note that we can treat all integers r > 1 simultaneously as the 'plus p-part' of the ETNC for the pair (h 0 (Spec(L))(r), Z[G]) naturally identifies with the corresponding conjecture for the extension L + /K of totally real fields. We show that the p-adic Beilinson conjecture at s = r, a conjecture of Schneider [Sch79] and the EIMC imply the plus (resp. minus) p-part of the ETNC for the pair (h 0 (Spec(L))(r), Z[G]) if r is odd (resp. even).
We follow the formulation of the p-adic Beilinson conjecture in [BBdJR09]. It relates the values at s = r of the complex and p-adic Artin L-functions by certain comparison periods involving Besser's syntomic regulator [Bes00]. For absolutely abelian extensions variants of the p-adic Beilinson conjecture have been formulated and proved by Coleman [Col82], Gros [Gro90,Gro94] and Kolster and Nguyen Quang Do [KNQD98]. Thus the p-adic Beilinson conjecture holds for absolutely abelian characters (see §3.13 for a precise statement).
Let us compare our approach to the earlier work mentioned above. The formulation of both the p-adic Beilinson conjecture and the p-adic Stark conjecture involves the choice of a field isomorphism j : C ≃ C p . We show in §3.12 that the p-adic Beilinson conjecture does not depend upon this choice if and only if a conjecture of Gross [Gro05] holds. The latter is revisited in §3.8 and might be seen as a higher analogue of Stark's conjecture; a similar result in the case r = 1 has recently been established by Johnston and the author in [JN20a]. In both cases the independence of j is therefore equivalent to the rationality part of the appropriate special case of the ETNC. This eventually allows us to establish a prime-by-prime descent result analogous to [JN20a,Theorem 8.1].
In a little more detail, we formulate conjectural 'higher refined p-adic class number formulae' analogous to [Bur20, Conjecture 3.5] (where r = 0), and show that these follow from the EIMC and Schneider's conjecture in §4.6. Here, as will be shown in §4.5, the latter conjecture ensures that the relevant complexes are semisimple at all Artin characters as Leopoldt's conjecture does in the case r = 1 and the non-vanishing of Gross's regulator does in the case r = 0. This is a necessary condition in order to apply the descent formalism of Burns and Venjakob [BV11]. A second condition is the vanishing of the aforementioned Iwasawa µ p -invariant, but given recent progress of Johnston and the author [JN18,JN20b] on the EIMC without assuming µ p = 0, we wish to circumvent this hypothesis. For this purpose, we develop a different descent argument that makes no use of this assumption, but requires a more delicate analysis of the relevant complexes. The higher refined p-adic class number formula at s = r may then be combined with the p-adic Beilinson conjecture at s = r to deduce the plus, respectively minus, p-part 2. Algebraic Preliminaries 2.1. Derived categories and Galois cohomology. Let Λ be a noetherian ring and let PMod(Λ) be the category of all finitely generated projective Λ-modules. We write D(Λ) for the derived category of Λ-modules and C b (PMod(Λ)) for the category of bounded complexes of finitely generated projective Λ-modules. Recall that a complex of Λ-modules is called perfect if it is isomorphic in D(Λ) to an element of C b (PMod(Λ)). We denote the full triangulated subcategory of D(Λ) comprising perfect complexes by D perf (Λ). If M is a Λ-module and n is an integer, we write M[n] for the complex where M is placed in degree −n. Note that this is compatible with the usual shift operator on cochain complexes. Let L be an algebraic extension of the number field K. For a finite set S of places of K containing the set S ∞ of all archimedean places we let G L,S be the Galois group over L of the maximal extension of L that is unramified outside S(L); here we write S(L) for the set of places of L lying above those in S. We let O L,S be the ring of S(L)-integers in L. For any topological G L,S -module M we write RΓ(O L,S , M) for the complex of continuous cochains of G L,S with coefficients in M. If F is a field and M is a topological G F -module, we likewise define RΓ(F, M) to be the complex of continuous cochains of G F with coefficients in M.
If F is a global or a local field of characteristic zero, and M is a discrete or a compact G F -module, then for r ∈ Z we denote the r-th Tate twist of M by M(r). Now fix a prime p and suppose that S also contains all p-adic places of K. Then for each integer i the cohomology group in degree i of RΓ(O L,S , Z p (r)) naturally identifies with H í et (O L,S , Z p (r)), the i-thétale cohomology group of the affine scheme Spec(O L,S ) with coefficients in thé etale p-adic sheaf Z p (r). We set H í et (O L,S , Q p (r)) := Q p ⊗ Zp H í et (O L,S , Z p (r)). 2.2. Representations and characters of finite groups. Let G be a finite group and let F be a field of characteristic zero. We write R + F (G) for the set of characters attached to finite-dimensional F -valued representations of G, and R F (G) for the ring of virtual characters generated by R + F (G). Moreover, we denote the subset of irreducible characters in R + F (G) and the ring of F -valued virtual characters of G by Irr F (G) and Char F (G), respectively.
For a subgroup H of G and ψ ∈ R + F (H) we write ind G H ψ ∈ R + F (G) for the induced character; for a normal subgroup N of G and χ ∈ R + F (G/N) we write infl G G/N χ ∈ R + F (G) for the inflated character. For σ ∈ Aut(F ) and χ ∈ Char F (G) we set χ σ := σ • χ and note that this defines a group action from the left even though we write exponents on the right of χ. We denote the trivial character of G by ½ G .
2.3. χ-twists. Let G be a finite group and let F be a field of characteristic zero. If M is a Z[G]-module we let M G be the maximal submodule of M upon which G acts trivially. Likewise we write M G for the maximal quotient module with trivial G-action. For any Let p be a prime. For each χ ∈ Irr Cp (G) we fix a subfield F χ of C p which is both Galois and of finite degree over Q p and such that χ can be realized over F χ . We write e χ := |G| −1 χ(1) g∈G χ(g)g −1 for the associated primitive central idempotent in We thereby obtain left, respectively right exact functors M → M (χ) and M → M (χ) from the category of Z p [G]-modules to the category of O χ -modules. Note that there is an Since multiplication by the trace Tr G := g∈G g gives rise to an isomorphism P (χ) ≃ P (χ) for each projective Z p [G]-module P (in fact for each cohomologically trivial Gmodule P ), these functors extend to naturally isomorphic exact functors Proof. Since (finitely generated) Q p [G]-modules are cohomologically trivial, the functors M → M (χ) and M → M (χ) are naturally isomorphic exact functors on the category of finitely generated Q p [G]-modules. The final assertion of the lemma is therefore clear.
We repeat the short argument for convenience. Choose a complex Here A and B are placed in degrees a and a + 1, respectively. Then we obtain a commutative diagram of O χ -modules which implies the claim. If b − a ≥ 2 we choose a complex P • ∈ C b (PMod(Z p [G])) that is isomorphic to C • in D(Z p [G]) and consider the exact sequence of perfect complexes where τ ≥b−1 and τ ≤b−2 denote naive truncation. Note that the complexes τ ≥b−1 P • and τ ≤b−2 P • are acyclic outside [b − 1, b] and [a, b − 2], respectively. It follows by induction . If b − a ≥ 3 then we likewise have that H a (C • ) = H a (τ ≤b−2 P • ) and we may again conclude by induction that we have an isomorphism . If b − a = 2 we may alternatively consider the exact sequence of perfect complexes and deduce as above.
3. The p-adic Beilinson conjecture 3.1. Setup and notation. Let L/K be a finite Galois extension of number fields with Galois group G. For any place v of K we choose a place w of L above v and write G w and I w for the decomposition group and inertia subgroup of L/K at w, respectively. We denote the completions of L and K at w and v by L w and K v , respectively, and identify the Galois group of the extension L w /K v with G w . For each non-archimedean place w we let O w be the ring of integers in L w . We identify G w := G w /I w with the Galois group of the corresponding residue field extension which we denote by L(w)/K(v). Finally, we let φ w ∈ G w be the Frobenius automorphism, and we denote the cardinality of K(v) by N(v). We let S be a finite set of places of K containing the set S ∞ of archimedean places. If a prime p is fixed, we will usually assume that the set S p of all p-adic places is also contained in S.
By a Galois CM-extension of number fields we shall mean a finite Galois extension L/K such that K is totally real and L is a CM-field. Thus complex conjugation induces a unique automorphism τ in the center of G and we denote the maximal totally real subfield of L by L + . Then L + /K is also Galois with group G + := G/ τ .

3.2.
Higher K-theory. For an integer n ≥ 0 and a ring R we write K n (R) for the Quillen K-theory of R. In the cases R = O L,S and R = L the groups K n (O L,S ) and K n (L) are equipped with a natural G-action and for every integer r > 1 the inclusion O L,S ⊆ L induces an isomorphism of Z[G]-modules Moreover, if S ′ is a second finite set of places of K containing S, then for every r > 1 there is a natural exact sequence of Z[G]-modules Both results (3.1) and (3.2) are due to Soulé [Sou79]; see [Wei13, Chapter V, Theorem 6.8]. We also note that sequence (3.2) remains left-exact in the case r = 1. The structure of the finite Z[G w ]-modules K 2r−1 (L(w)) has been determined by Quillen [Qui72] (see also [Wei13, Chapter IV, Theorem 1.12 and Corollary 1.13]) to be 3.3. The regulators of Borel and Beilinson. Let Σ(L) be the set of embeddings of L into the complex numbers; we then have |Σ(L)| = r 1 +2r 2 , where r 1 and r 2 are the number of real embeddings and the number of pairs of complex embeddings of L, respectively. For an integer k ∈ Z we define a finitely generated Z-module which is endowed with a natural Gal(C/R)-action, diagonally on Σ(L) and on (2πi) −k . The invariants of H k (L) under this action will be denoted by H + k (L), and it is easily seen that we have The action of G on Σ(L) endows H + k (L) with a natural G-module structure. Let r > 1 be an integer. Borel [Bor74] has shown that the even K-groups K 2r−2 (O L ) (and thus K 2r−2 (O L,S ) for any S as above by (3.2) and (3.3)) are finite, and that the odd K-groups K 2r−1 (O L ) are finitely generated abelian groups of rank d r . More precisely, for each r > 1 Borel constructed an equivariant regulator map The covolume of this lattice is called the Borel regulator and will be denoted by R Bor r (L). Moreover, Borel showed that where ζ * L (1 − r) denotes the leading term at s = 1 − r of the Dedekind zeta function ζ L (s) attached to the number field L.
In the context of the ETNC, however, it is more natural to work with Beilinson's regulator map [Beȋ84]. By a result of Burgos Gil [BG02] Borel's regulator map is twice the regulator map of Beilinson. Hence we will work with ρ r := 1 2 ρ Bor r in the following.
Remark 3.1. We will sometimes refer to [Nic19] where we have worked with Borel's regulator map. However, if we are interested in rationality questions or in verifying the p-part of the ETNC for an odd prime p, the factor 2 essentially plays no role. In contrast, the p-adic Beilinson conjecture below predicts an equality of two numbers in C p so that this factor indeed matters.
3.4. The Quillen-Lichtenbaum Conjecture. Fix an odd prime p and assume that S contains S ∞ and the set S p of all p-adic places of K. Then for any integer r > 1 and i = 1, 2 Soulé [Sou79] has constructed canonical G-equivariant p-adic Chern class maps . We need the following deep result.
Theorem 3.2 (Quillen-Lichtenbaum Conjecture). Let p be an odd prime. Then for every integer r > 1 and i = 1, 2 the p-adic Chern class maps ch (p) r,i are isomorphisms. Proof. Soulé [Sou79] proved surjectivity. Building on work of Rost and Voevodsky, Weibel [Wei09] completed the proof of the Quillen-Lichtenbaum Conjecture.
Let p be a prime. For an integer n ≥ 0 and a ring R we write K n (R; Z p ) for the Ktheory of R with coefficients in Z p . The following result is due to Hesselholt and Madsen [HM03].
Theorem 3.3. Let p be an odd prime and let w be a finite place of L. Then for every integer r > 1 and i = 1, 2 there are canonical isomorphisms of Z p [G w ]-modules ). 3.5. Local Galois cohomology. We keep the notation of §3.1. In particular, L/K is a Galois extension of number fields with Galois group G. Let p be an odd prime. We denote the (finite) set of places of K that ramify in L/K by S ram and let S be a finite set of places of K containing S ram and all archimedean and p-adic places (i.e. S ∞ ∪ S p ∪ S ram ⊆ S).
Let M be a topological G L,S -module. Then M becomes a topological G Lw -module for every w ∈ S(L) by restriction. For any i ∈ Z we put For any integers r and i we define Lemma 3.4. Let r > 1 be an integer. Then we have isomorphisms of Q p [G]-modules Proof. This is [Nic19, Lemma 3.3] (see also [Bar09, Lemma 5.2.4]). The case i = 1 will be crucial in the following so that we briefly recall its proof. Let w ∈ S p (L) and put D Lw , where B dR denotes Fontaine's de Rham period ring. Then the Bloch-Kato exponential map ) is an isomorphism for every w ∈ S p (L) as follows from [BK90, Corollary 3.8.4 and Example 3.9]. Since the groups H 1 et (L w , Z p (r)) are finite for w ∈ S p (L), we obtain isomorphisms of Q p [G]-modules 3.6. Schneider's conjecture. Let M be a topological G L,S -module. For any integer i we denote the kernel of the natural localization map We recall the following conjecture of Schneider [Sch79,p. 192].
Conjecture 3.5 (Sch(L, p, r)). Let r = 0 be an integer. Then the Tate-Shafarevich group Remark 3.6. It is not hard to show that Conjecture 3.5 does not depend on the choice of the set S. Remark 3.8. It can be shown that Schneider's conjecture for r = 1 is equivalent to Leopoldt's conjecture (see [NSW08, Chapter X, §3]).
Lemma 3.11. Let r = 0 be an integer and let p be an odd prime. Then the Tate- Proof. The first claim is [Nic19, Proposition 3.8 (i)]. The second claim is immediate.
3.7. Artin L-series. Let L/K be a finite Galois extension of number fields with Galois group G and let S be a finite set of places of K containing all archimedean places. For any irreducible complex-valued character χ of G we denote the S-truncated Artin L-series by L S (s, χ), and the leading coefficient of L S (s, χ) at an integer r by L * S (r, χ). We shall sometimes use this notion even if L * S (r, χ) = L S (r, χ) (which will happen frequently in the following).
Recall that there is a canonical isomorphism For any r ∈ Z we also put Let r > 1 be an integer. Since Borel's regulator map (3.5) induces an isomorphism of R[G]-modules, the Noether-Deuring Theorem (see [NSW08,Lemma 8.7.1] for instance) implies the existence of Q[G]-isomorphisms Let χ be a complex character of G and let V χ be a C[G]-module with character χ. Composition with ρ r • φ 1−r induces an automorphism of Hom G (Vχ, .11] conjectured the following higher analogue of Stark's conjecture. . Remark 3.13. It is not hard to see that Gross's conjecture does not depend on S and the choice of φ 1−r (see also [Nic11a,Remark 6]). A straightforward substitution shows that if it is true for χ then it is true for χ τ for every choice of τ ∈ Aut(C).
We record some cases where Gross's conjecture is known and deduce a few new cases. If q = ℓ n is a prime power, we let Aff(q) be the group of affine transformations on F q . Thus we may write Aff(q) Theorem 3.14. Let L/K be a finite Galois extension of number fields with Galois group G and let χ ∈ R(G) be a virtual character. Let r > 1 be an integer. Then Gross's conjecture (Conjecture 3.12) holds in each of the following cases.
(i) χ is absolutely abelian, i.e. there is a normal subgroup N of G such that χ factors through G/N ≃ Gal(L N /K) and L N /Q is abelian; is totally real and r is even; For any integer k we write for the canonical C[G×Gal(C/R)]-equivariant isomorphism which is induced by mapping l ⊗ z to (σ(l)z) σ∈Σ(L) for l ∈ L and z ∈ C. Now fix an integer r > 1. We define a C[G]isomorphism Here, the first isomorphism is induced by Proposition 3.15. Fix an integer r > 1 and a character χ. Then Gross's conjecture 3.12 holds if and only if we have A S φr (χ σ ) = A S φr (χ) σ for all σ ∈ Aut(C). Proof. This is [Nic19, Proposition 5.5].
3.9. The comparison period. We henceforth assume that p is an odd prime and that L/K is a Galois CM-extension. Recall that τ ∈ G is the unique automorphism induced by complex conjugation. For each n ∈ Z we define a central idempotent e n : . Now let r > 1 be an integer. Since L is CM, the idempotent e r acts trivially on be the i-th syntomic cohomology group as considered by Besser [Bes00]. We let reg w r : The map µ p (r) shows up in the formulation of the p-adic Beilinson conjecture. However, the following map will be more suitable for the relation to the ETNC. We define a C p [G]homomorphismμ Here, the first map is induced by the p-adic Chern class map ch (p) r,1 which is an isomorphism by Theorem 3.2; the arrow is the natural localization map, and the last isomorphism is induced by the Bloch-Kato exponential maps (see Lemma 3.4).
The following result will be crucial for relating the p-adic Beilinson conjecture to the ETNC.
Proof. For any abelian group A we write A for its p-completion, that is A := lim ← −n A/p n A. The localization maps (3.12) induce a map For each w ∈ S p (L) the Universal Coefficient Theorem [Wei13, Chapter IV, Theorem 2.5] implies that there is a natural (injective) map Moreover, by [Bes00, Corollary 9.10] there is a natural map H 1 Here, the left-hand vertical arrow is induced by the syntomic regulator, and the map on the right by the isomorphism in Theorem 3.3. Moreover, the composite map ) is the Bloch-Kato exponential map (3.7) by [Bes00, Proposition 9.11]. Unravelling the definitions we now see that the maps µ p (r) andμ p (r) coincide.
We will henceforth often not distinguish between the maps µ p (r) andμ p (r). Since the Tate-Shafarevich group X 1 (O L,S , Z p (r)) is torsion-free by Lemma 3.11, the following result is now immediate. Definition 3.18. Let j : C ≃ C p be a field isomorphism and let ρ ∈ R + Cp (G). Let r > 1 be an integer. We define the comparison period attached to j, ρ and r to be We record some basic properties of Ω j (r, −).
Remark 3.20. Since µ ∞ (r) is an isomorphism, for any two choices of field isomorphism j, j ′ : C ≃ C p we have that Ω j (r, ρ) = 0 if and only if Ω j ′ (r, ρ) = 0.
Here ε r ∈ Z p [G] are certain idempotents such that the ε r -parts of both C p [G]-modules in the exterior products are free of the same rank r ε j . If r > 1 and Sch(L, p, r) holds, then one may take ε r = e r and r ǫ j = 0. In this case the diagram [BKS20, p. 125] gives an exact sequence of C p [G]-modules The non-trivial map is (up to sign) the dual of µ p (r) • ρ −1 r . Hence the exterior product on the left of (3.13) canonically identifies with The authors then use the isomorphism (3.13) to define generalized Stark elements and to state [BKS20, Conjecture 3.6] which might be seen as an analogue and refinement of a conjecture of Rubin [Rub96] in the case r = 0. It is then shown in [BKS20, §4] that their conjecture is implied by the appropriate special case of the ETNC. As the formulation of the latter involves the Bloch-Kato exponential map rather than the syntomic regulator, a variant of Proposition 3.16 is already implicit in their work (for instance, see [BKS20, Remark 2.7]) 3.10. p-adic Artin L-functions. Let E/K be a finite Galois extension of totally real number fields and let G = Gal(E/K). Let p be an odd prime and let S be a finite set of places of K containing S p ∪ S ∞ . For each ρ ∈ R Cp (G) the S-truncated p-adic Artin L-function attached to ρ is the unique p-adic meromorphic function L p,S (s, ρ) : Z p → C p with the property that for each strictly negative integer n and each field isomorphism j : C ≃ C p we have where ω : G K → Z × p is the Teichmüller character and we view ρ ⊗ ω n−1 as a character of Gal(E(ζ p )/K). By a result of Siegel [Sie70] the right-hand side does indeed not depend on the choice of j. In the case that ρ is linear, L p,S (s, ρ) was constructed independently by Deligne and Ribet [DR80], Barsky [Bar78] and Cassou-Nogués [CN79]. Greenberg [Gre83] then extended the construction to the general case using Brauer induction.
3.11. Statement of the p-adic Beilinson conjecture. We now formulate our variant of the p-adic Beilinson conjecture.
Conjecture 3.23 (The p-adic Beilinson conjecture). Let E/K be a finite Galois extension of totally real number fields and let G = Gal(E/K). Let p be an odd prime and let S be a finite set of places of K containing S p ∪ S ∞ . Let ρ ∈ R + Cp (G) and let r > 1 be an integer. Then for every choice of field isomorphism j : C ∼ = C p we have (3.14) L p,S (r, ρ) = Ω j (r, ρ ⊗ ω r−1 ) · j L S (r, (ρ ⊗ ω r−1 ) j −1 ) . Theorem 3.28. Let E/K be a finite Galois extension of totally real number fields and let G = Gal(E/K). Let p be an odd prime and let S be a finite set of places of K containing S p ∪ S ∞ . Let ρ ∈ R + Cp (G) and let r > 1 be an integer. We put ψ := ρ ⊗ ω r−1 . If Ω j (r, ψ) = 0 for some (and hence every) choice of field isomorphism j : C ∼ = C p then the following statements are equivalent.
Proof. The first and second occurrence of 'and hence every' in the statement of the theorem follow from Remark 3.13 and Remark 3.20, respectively.
Let j, j ′ : C ∼ = C p be field isomorphisms and let χ := ψ j −1 . Then j = j ′ • σ for some σ ∈ Aut(C) and so ψ j ′−1 = χ σ . For every Q[G]-isomorphism φ r as in (3.11) we have which does not depend on j. Hence we have By Proposition 3.15 the last expression is equal to 1 if and only if Gross's conjecture at s = 1 − r (Conjecture 3.12) holds for the characterχ.
Corollary 3.29. Let E/K be a finite Galois extension of totally real number fields and let G = Gal(E/K). Fix a prime p and let r > 1 be an integer. Assume that Sch(E(ζ p ), p, r) holds. If the p-adic Beilinson conjecture at s = r holds for all ρ ∈ R + Cp (G) then Gross's conjecture at s = 1 − r holds for χ ⊗ ω r−1 for all χ ∈ R + C (G).

Absolutely abelian characters. Since our conjecture is compatible with that of [BBdJR09] by Remark 3.26 and invariant under induction and inflation of characters by
Remark 3.27, we deduce the following result from work of Coleman [Col82] (see [BBdJR09,Proposition 4.17]).
Theorem 3.30. Let E/K be a finite Galois extension of totally real number fields and let G = Gal(E/K). Let p be a prime and let r > 1 be an integer. Suppose that ρ ∈ R + Cp (G) is an absolutely abelian character, i.e., there exists a normal subgroup N of G such that ρ factors through G/N ∼ = Gal(E N /K) and E N /Q is abelian. Then the p-adic Beilinson conjecture (Conjecture 3.23) holds for ρ.
Let G be a compact p-adic Lie group that contains a closed normal subgroup H such that Γ := G/H is isomorphic to Z p . We fix a topological generator γ of Γ. The Iwasawa where T π is a finitely generated free O-module. For g ∈ G we denote its image in Γ under the canonical projection by g. for λ ∈ Λ O (Γ), t ∈ T π and g ∈ G.
For each complex C • ∈ D perf (Λ(G)) we define a complex Given an open normal subgroup U of G we set C • U := Z p [G/U] ⊗ L Λ(G) C • and, if U is contained in the kernel of π, we furthermore obtain a complex The Bockstein homomorphism in degree i is defined to be the composite homomorphism , where the middle arrow is the tautological map. We obtain a bounded complex of Omodules ∆ C • ,π : . . .
where the term H i (C • U,(π) ) is placed in degree i. Note that the Bockstein homomorphisms and the complex ∆ C • ,π actually depend on the choice of γ, though our notation does not reflect this. The complex C • is called semisimple at π if Q p ⊗ Zp ∆ C • ,π is acyclic (for any and hence every choice of γ).
For any Λ(G)-module M we put M π : Lemma 4.1. Assume that G is a compact p-adic Lie group of dimension 1 and let C • ∈ D perf (Λ(G)) be acyclic outside degree a for some a ∈ Z. Further assume that the Λ(G)-module H a (C • ) has projective dimension at most 1. Then for every continuous homomorphism π : G → Aut O (T π ) the complex C • π is acyclic outside degree a and there is a canonical isomorphism of Λ O (Γ)-modules H a (C • π ) ≃ H a (C • ) π . Proof. This has been shown in the course of the proof of [Bur20, Lemma 5.6]. We repeat the short argument for convenience.
We may assume that a = 0. Then C • may be represented by a complex P −1 d → P 0 , where P −1 and P 0 are projective Λ(G)-modules placed in degrees −1 and 0, respectively, and the homomorphism d is injective. Then C • π is represented by where d ′ is injective since d is. The result follows.

Algebraic K-theory. Let R be a noetherian integral domain with field of fractions E. Let A be a finite-dimensional semisimple E-algebra and let A be an R-order in A.
For any field extension F of E we set A F := F ⊗ E A. Let K 0 (A, F ) = K 0 (A, A F ) denote the relative algebraic K-group associated to the ring homomorphism A ֒→ A F . We recall that K 0 (A, A F ) is an abelian group with generators [X, g, Y ] where X and Y are finitely generated projective A-modules and g : F ⊗ R X → F ⊗ R Y is an isomorphism of A Fmodules; for a full description in terms of generators and relations, we refer the reader to [Swa68,p. 215]. Moreover, there is a long exact sequence of relative K-theory (see [Swa68, Chapter 15]) The reduced norm map Nrd = Nrd A F : A F → ζ(A F ) is defined componentwise on the Wedderburn decomposition of A F and extends to matrix rings over A F (see [CR81,§7D]); thus it induces a map K 1 (A F ) −→ ζ(A F ) × , which we also denote by Nrd.
In the case E = F the relative K-group K 0 (A, A) identifies with the Grothendieck group whose generators are [C • ], where C • is an object of the category C b tors (PMod(A)) of bounded complexes of finitely generated projective A-modules whose cohomology modules are R-torsion, and the relations are as follows: Chapter 2] or [Suj13, §2], for example). We denote the full triangulated subcategory of D perf (A) comprising perfect complexes whose cohomology modules are R-torsion by D perf tors (A). Then every object C • of D perf tors (A) defines a class [C • ] in K 0 (A, A).
Let p be an odd prime and let G be a one-dimensional compact p-adic Lie group that surjects onto Z p . Then G may be written as G = H ⋊ Γ with a finite normal subgroup H of G and a subgroup Γ ≃ Z p . Let O be the ring of integers in some finite extension F of , we say that ξ is a characteristic element for x. We also set Q(G) := Q Qp (G).
We include the following consequence of (4.4) for later use.
Its kernel is a projective Λ O (G)-module P by assumption. Since M is R-torsion, we see that the classes of P and Λ O (G) m in K 0 (Λ O (G)) have the same image in K 0 (Q F (G)). Hence they coincide by (4.4). In other words, P and Λ O (G) m are stably isomorphic. By enlarging m if necessary, we may assume that P is free of rank m and we have established the existence of (4.5). By [NSW08,Lemma 5.3.11] this sequence induces an exact sequence of Z p [G]-modules It follows that M Γ ′ is a free Z p -module of the same rank as M Γ ′ . This proves the remaining claims. Now let π : G → Aut O (T π ) be a continuous homomorphism as in (4.1) and set n := rank O (T π ). There is a ring homomorphism Φ π : Λ(G) → M n×n (Λ O (Γ)) induced by the continuous group homomorphism where g denotes the image of g in G/H ≃ Γ. By [CFK + 05, Lemma 3.3] this homomorphism extends to a ring homomorphism Q(G) → M n×n (Q F (Γ)) and this in turn induces a homomorphism For ξ ∈ K 1 (Q(G)) we set ξ(π) := Φ π (ξ). If π = π ρ is an Artin representation with character ρ, we also write Φ ρ and ξ(ρ) for Φ πρ and ξ(π ρ ), respectively, and let be the map defined by Ritter and Weiss in [RW04]. By [Nic13, Lemma 2.3] (choose r = 0) we have a commutative triangle We shall also write ξ * (ρ) for the leading term at T = 0 of the power series Φ ρ (ξ). We choose a maximal Λ(Γ 0 )-order M(G) in Q(G) such that Λ(G) is contained in M(G).

4.3.
Cohomology with compact support. Let p be an odd prime. We denote the cyclotomic Z p -extension of a number field K by K ∞ and set Γ K := Gal(K ∞ /K). Let S be a finite set of places of K containing the set S ∞ ∪ S p . Let M be a topological G K,S -module. Following Burns and Flach [BF01] we define the compactly supported cohomology complex to be Let E/K be a finite Galois extension of totally real fields and set L := E(ζ p ). Then L is a CM-field and we denote its maximal totally real subfield by L + as in §3.9. Set G := Gal(L ∞ /K) and let χ cyc : G −→ Z × p , be the p-adic cyclotomic character defined by σ(ζ) = ζ χcyc(σ) for any σ ∈ G and any p-power root of unity ζ. The composition of χ cyc with the projections onto the first and second factors of the canonical decomposition Z × p = ζ p−1 × (1 + pZ p ) are given by the Teichmüller character ω and a map that we denote by κ.
Assume in addition that S contains all places that ramify in L ∞ /K. For each integer r we define a complex of Λ(G)-modules C • r,S := RΓ c (O K,S , e r Λ(G) ♯ (r)), where Λ(G) ♯ (r) denotes the Λ(G)-module Λ(G) upon which σ ∈ G K acts on the right via multiplication by the element χ r cyc (σ)σ −1 ; here σ denotes the image of σ in G. Note that the complexes C • r,S are perfect by [FK06, Proposition 1.6.5] and we have natural isomorphisms (4.6) C • r,S ≃ C • 1,S ⊗ L Zp Z p (r − 1) for every integer r.
Each Λ(G)-module M naturally decomposes as a direct sum M + ⊕ M − with M ± = 1±τ 2 M. Similarly, each complex C • of D perf (Λ(G)) gives rise to subcomplexes (C • ) + and (C • ) − . Moreover, we let (C • ) ∨ := RHom(C • , Q p /Z p ) be the Pontryagin dual of C • . By a Shapiro Lemma argument and Artin-Verdier duality we then have isomorphisms in D(e r Λ(G)). We let M S be the maximal abelian pro-p-extension of L + ∞ unramified outside S. Then X S := Gal(M S /L + ∞ ) is a finitely generated Λ(G + )-module, where we put G + := G/ τ = Gal(L + ∞ /K). Iwasawa [Iwa73] has shown that X S is in fact torsion as a Λ(Γ L + )-module. We let µ p (L + ) denote the Iwasawa µ-invariant of X S and note that this does not depend on the choice of S (see [NSW08, Corollary 11.3.6]). Hence µ p (L + ) vanishes if and only if X S is finitely generated as a Z p -module. It is conjectured that we always have µ p (L + ) = 0 and as explained in [JN18,Remark 4.3], this is closely related to the classical Iwasawa 'µ = 0' conjecture for L at p. Thus a result of Ferrero and Washington [FW79] on this latter conjecture implies that µ p (L + ) = 0 whenever E/Q and thus L/Q is abelian.
The only non-trivial cohomology groups of RΓ(O L + ∞ ,S , Q p /Z p ) ∨ occur in degrees −1 and 0 and canonically identify with X S and Z p , respectively. Hence (4.7) with r = 1 and (4.6) imply that for each integer r the cohomology of C • r,S is concentrated in degrees 2 and 3 and we have 4.4. The main conjecture. The following is an obvious reformulation of the equivariant Iwasawa main conjecture for the extension L + ∞ /K (without its uniqueness statement). Conjecture 4.4 (equivariant Iwasawa main conjecture). Let L/K be a Galois CMextension such that L contains a primitive p-th root of unity. Let S be a finite set of places of K containing S ∞ and all places that ramify in L + ∞ /K. Then there exists an element ζ S ∈ K 1 (Q(G + )) such that ∂(ζ S ) = [C • 1,S ] and for every irreducible Artin representation π ρ of G + with character ρ and for each integer n ≥ 1 divisible by p − 1 we have for every field isomorphism j : C ∼ = C p .
Part (i) of the following theorem has been shown by Ritter and Weiss [RW11] and by Kakde [Kak13] independently. Part (ii) is due to Johnston and the present author [JN20b]. By starting out from the work of Deligne and Ribet [DR80], Greenberg [Gre83] has shown that for each topological generator γ of Γ there is a unique element f ρ,S (T ) in the quotient field of Q c p ⊗ Zp Z p T (which we will identify with Q c p ⊗ Zp Λ(Γ) via the usual map that sends γ to 1 + T ) such that L p,S (1 − s, ρ) = f ρ,S (u s − 1), where u := κ(γ). For each integer a we let x → t a cyc (x) be the automorphisms on Q c (G) induced by g → χ cyc (g) a g for g ∈ G. We use the same notation for the induced group homomorphisms on K 1 (Q(G)) and K 1 (e r Q(G)), r ∈ Z.
Corollary 4.7. Let r ∈ Z be an integer and let π ρ be an irreducible Artin representation of G + with character ρ. Then f ρ,S (u 1−r (1+T )−1) is a characteristic element of C • r,ρ⊗ω r−1 . Proof. Since the main conjecture (Conjecture 4.4) holds 'over the maximal order' by [JN18,Theorem 4.9] (this result is essentially due to Ritter and Weiss [RW04]), the equality (4.9) holds unconditionally up to a factor j ρ (x) for some x ∈ ζ(M(G)) × . Thus the claim follows from Lemma 4.3.
) equals d r+1 if and only if Schneider's conjecture Sch(L, p, r) holds.
We now return to the situation considered in §4.3. Before proving the next result we recall that for a finitely generated Λ O (Γ)-torsion module M the following are equivalent: Proposition 4.9. Let r > 1 be an integer and let ρ be an irreducible Artin character of G + such that ker(ρ) contains Γ L + . Set ψ := ρ ⊗ ω r−1 . Then we have that H i (C • r,S,ψ ) = 0 for i = 2, 3 and natural isomorphisms of O ψ -modules (4.10) H 2 (C • r,S,ψ ) Γ ≃ H 1 c (O L,S , Z p (r)) (ψ) ≃ X 1 (O L,S , Z p (r)) (ψ) and (4.11) and a short exact sequence of O ψ -modules r,S,ψ ) Γ and H 3 (C • r,S,ψ ) Γ are finite. Proof. We put O := O ψ for brevity. Since the complex C • r,S is acyclic outside degrees 2 and 3 and the functor r,S,ψ ) vanishes for i ≥ 4. We let U = Γ L = Gal(L ∞ /L). Then U is contained in the kernel of ψ, and by [FK06, Proposition 1.6.5] we have an isomorphism . We now consider the exact sequence (4.2) for the case at hand and various integers i. We will repeatedly apply Lemma 2.2. In particular, the complex C • r,S,U,(ψ) is acyclic outside degrees 1, 2 and 3 by Proposition 4.8 (i). For i ≤ 0 we find that H i (C • r,S,ψ ) Γ and H i+1 (C • r,S,ψ ) Γ vanish. Thus H i (C • r,S,ψ ) vanishes for i ≤ 0 and even for i = 1 once we show that the O-module H 1 (C • r,S,ψ ) Γ is trivial. We already know that it is finite. Sequence (4.2) in the case i = 1 and Lemma 2.2 now give rise to a short exact sequence Since the central idempotent e r annihilates H + −r (L) ⊗ Z p and e r e ψ = e ψ we have an isomorphism H 1 c (O L,S , Z p (r)) (ψ) ≃ X 1 (O L,S , Z p (r)) (ψ) by Proposition 4.8 (ii). Since the Tate-Shafarevich group X 1 (O L,S , Z p (r)) is torsion-free by Lemma 3.11, the O-module X 1 (O L,S , Z p (r)) (ψ) is free. Thus H 1 (C • r,S,ψ ) Γ vanishes as desired and we have established (4.10). Proposition 4.8 (iii), Lemma 2.2 and the case i = 3 of sequence (4.2) imply (4.11). Finally, sequence (4.12) is the case i = 2 of sequence (4.2).
Lemma 4.10. Let r be an arbitrary integer. Then there are finitely generated e r Λ(G)modules Y r,S and Z r with all of the following properties: (i) The projective dimension of both Y r,S and Z r is at most 1; (ii) both Y r,S and Z r are torsion as R-modules; (iii) there is an exact triangle Proof. We first consider the case r = 0. It is shown in [JN19, Proposition 8.5] that the complex where Y 0,S is placed in degree 2. More precisely, in the notation of [JN19] we have Here T is a finite set of places of K disjoint from S with certain properties, and G w∞ denotes the decomposition group at a chosen place w ∞ of L ∞ above v for each v ∈ T ; moreover, we write ind G U M := Λ(G) ⊗ Λ(U ) M for any open subgroup U of G and any Λ(U)module M. By [JN19, Lemmas 8.4 and 8.5] the modules Z 0 and Y 0,S are R-torsion and of projective dimension at most 1. Thus (i) and (ii) also hold for Y r,S := Y 0,S (r) = Y T S (r − 1) and Z r := Z 0 (r) = e r v∈T ind G Gw ∞ Z p (r − 1). It is now clear that (iv) holds and that C • r,S is isomorphic to the complex in D(e r Λ(G)). Hence (iii) also holds. Moreover, the coinvariants (Z r ) Γ L are clearly finite for r = 1.
Remark 4.11. We point out that similar constructions repeatedly appear in the literature. In fact, by [Nic13,Theorem 2.4] the complex C • S (L + ∞ /K) naturally identifies with the complex constructed by Ritter and Weiss [RW02]. Choosing their maps Ψ andψ in [RW04,p.562 f.] suitably one can take Y 1,S = coker(Ψ) and Z 1 = coker(ψ). Moreover, Burns [Bur20, §5.3.1] constructed an exact triangle in D(e 0 Λ(G)) of the form where the set T is as in the proof of Lemma 4.10. The complexes RΓ(K(v), e 0 Λ(G) ♯ (1)) are acyclic outside degree 1 and we have e 0 Λ(G)-isomorphisms Theorem 4.12. Let r > 1 be an integer and let ρ be an irreducible Artin character of G + such that ker(ρ) contains Γ L + . Set ψ := ρ ⊗ ω r−1 . Then the following conditions are equivalent.
(iii) We have that H 2 c (O L,S , Q p (r)) (ψ) vanishes. (iv) The period Ω j (r, ψ) is non-zero for any (and hence every) choice of j : C ≃ C p .
If these equivalent conditions hold (in particular if Sch(L, p, r) holds), then the complex C • r,S is semisimple at ψ.
Proof. We have already observed in the proof of Proposition 4.9 that X 1 (O L,S , Z p (r)) (ψ) is a free O-module, where we again set O := O ψ . The equivalence of (i) and (ii) is therefore clear. We have an exact sequence of Z p [G]-modules Since the e r Q p [G]-modules e r H 1 et (O L,S , Q p (r)) ≃ e r K 2r−1 (O L )⊗Q p and e r P 1 (O L,S , Q p (r)) ≃ e r (L⊗ Q Q p ) are (non-canonically) isomorphic and X 2 (O L,S , Z p (r)) is finite by Proposition 4.8 (iv), there is a (non-canonical) isomorphism of e r Q p [G]-modules Thus also X 1 (O L,S , Q p (r)) (ψ) and H 2 c (O L,S , Q p (r)) (ψ) are (non-canonically) isomorphic and so (ii) and (iii) are indeed equivalent. The equivalence of (ii) and (iv) is easy (see Remark 3.21).
Since (Z r,ψ ) Γ is finite by Lemma 4.10 (v) we have that h r,ψ (0) = 0. Thus L p,S (r, ρ) = f ρ,S (u 1−r − 1) is non-zero if and only if g r,S,ψ (0) = 0 if and only if Y Γ r,S,ψ vanishes, where the latter equivalence uses Lemma 4.2 (with M = Y r,S,ψ and G = Γ) and Lemma 4.10 (i) and (ii). By (4.13) we have an exact sequence of Λ O (Γ)-modules Since taking Γ-invariants is left exact and Z Γ r,ψ vanishes by another application of Lemma 4.2, it follows that there is an isomorphism Y Γ r,S,ψ ≃ H 2 (C • r,S,ψ ) Γ . The latter identifies with X 1 (O L,S , Z p (r)) (ψ) by Proposition 4.9 which proves the claim.
Finally, if these equivalent conditions all hold, then Proposition 4.9 implies that the O-modules H i (C • r,S,ψ ) Γ and H i (C • r,S,ψ ) Γ are finite for all i ∈ Z. It follows from the short exact sequences (4.2) that H i (C • r,S,U ) is finite for all i ∈ Z, where we set U = Γ L as before. Thus the whole complex Q p ⊗ Zp ∆ C • r,S ,ψ vanishes. In particular, the complex C • r,S is semisimple at ψ.
Remark 4.13. By specializing K = L + in the above argument, we see that Sch(L, p, r) implies that Y Γ L r,S vanishes and that (Y r,S ) Γ L is finite. 4.6. Higher refined p-adic class number formulae. We keep the notation of §4.3. We let ∂ p : , C p ) be the composition of the inverse of the reduced norm and the connecting homomorphism to relative K-theory. By abuse of notation we shall use the same symbol for the induced maps on 'e r -parts'. We recall that G denotes Gal(L/K) and that G + = G/ τ = Gal(L + /K). We define Let us assume that Schneider's conjecture Sch(L, p, r) holds. Then we actually have that L p,S (r) ∈ ζ(e r Q p [G]) × and the cohomology groups of the complex e r RΓ c (O L,S , Z p (r)) are finite by Proposition 4.8 and Theorem 4.12. This complex therefore is an object in D perf tors (e r Z p [G]). We now state our conjectural higher refined p-adic class number formula. Conjecture 4.14. Let r > 1 be an integer and assume that Sch(L, p, r) holds. Then in Remark 4.15. In the case r = 1 Burns [Bur20, Conjecture 3.5] has formulated a conjectural refined p-adic class number formula. Conjecture 4.14 might be seen as a higher analogue of his conjecture. Accordingly, Theorem 4.17 below is the higher analogue of [Bur20, Theorem 3.6].
Lemma 4.16. Let r > 1 be an integer and assume that Sch(L, p, r) holds. Then does not depend on the set S.
Proof. Let S ′ be a second sufficiently large finite set of places of K. By embedding S and S ′ into the union S ∪ S ′ we may and do assume that S ⊆ S ′ . By induction we may additionally assume that S ′ = S ∪ {v}, where v is not in S. In particular, v is unramified in L/K and v ∤ p. By [BF01, (30)] we have an exact triangle (4.14) where RΓ f (L w , Z p (r)) is a perfect complex of Z p [G w ]-modules which is naturally quasiisomorphic to with terms in degree 0 and 1. We set where the first and second equality follow from (4.14) and (4.15), respectively. This implies the claim.
Our main evidence for Conjecture 4.14 is provided by the following result which, crucially, does not depend upon the vanishing of µ p (L + ).
Theorem 4.17. Let r > 1 be an integer and assume that Sch(L, p, r) holds. If the equivariant Iwasawa main conjecture (Conjecture 4.4) holds for L + ∞ /K (and so in particular if µ p (L + ) = 0 or if G + has an abelian Sylow p-subgroup), then Conjecture 4.14 holds.
Proof. We first observe that the complex C • r,S is semisimple at ρ ⊗ ω r−1 for all ρ ∈ Irr Cp (G + ) by Theorem 4.12. Moreover, if we put U = Γ L as above, then we have an isomorphism in D(e r Z p [G]). If Conjecture 4.4 holds, then by Proposition 4.6 there is a characteristic element ζ r,S of [C • r,S ] such that ζ r,S (ρ⊗ω r−1 ) = f ρ,S (u 1−r (1+T )−1). Since f ρ,S (u 1−r −1) = L p,S (r, ρ) = 0 we have that ζ * r,S (ρ ⊗ ω r−1 ) = L p,S (r, ρ). If µ p (L + ) vanishes or p does not divide the cardinality of G + , then [BV11, Theorem 2.2] implies the claim (as noted above the complexes Q p ⊗ Zp ∆ C • r,S ,ρ⊗ω r−1 all vanish). In order to avoid these assumptions, we proceed as follows. Observe that both (Z r ) Γ L and (Y r,s ) Γ L are finite by Lemma 4.10 (v) and Theorem 4.12 (or rather Remark 4.13), respectively, since Schneider's conjecture holds by assumption. Recall from Lemma 4.10 (iii) that we have an exact triangle in D(e r Λ(G)). It now follows from (4.16) and Lemma 4.2 for both Z r and Y r,S that we likewise have an exact triangle in D(e r Z p [G]). Let H r and G r,S in ζ(e r Q(G)) × be the reduced norms of characteristic elements of Z r and Y r,S , respectively. Note that both H r and G r,S are actually reduced norms of matrices with coefficients in e r Λ(G). Since Conjecture 4.4 holds by assumption, we may assume that Nrd(ζ r,S ) = G r,S /H r , where ζ r,S is the characteristic element of [C • r,S ] that occurs in Proposition 4.6. Now by (the proof of) [Nic10, Theorem 6.4] one has and H r is defined similarly. Hence we obtain aug Γ (j ρ⊗ω r−1 (Nrd(ζ r,S )))e ρ⊗ω r−1   ( * ) = ∂ p (L p,S (r)).
Here the first and second equality follow from [Nic11d, Lemma 2.3] and (4.9), respectively. This finishes the proof of ( * ).
Let us write K 0 (e r Z p [G], Q p ) tors for the torsion subgroup of K 0 (e r Z p [G], Q p ).
Corollary 4.18. Let r > 1 be an integer and assume that Sch(L, p, r) holds. Then we have that HR p (L/K, r) ∈ K 0 (e r Z p [G], Q p ) tors .
Proof. Let K ′ ⊆ L + be a totally real field containing K. Denote the Galois group Gal(L/K ′ ) by G ′ . Then HR p (L/K, r) maps to HR p (L/K ′ , r) under the natural restriction map K 0 (e r Z p [G], Q p ) → K 0 (e r Z p [G ′ ], Q p ). If L ′ ⊆ L is a Galois CM-extension of K, then likewise HR p (L/K, r) maps to HR p (L ′ /K, r) under the natural quotient map K 0 (e r Z p [G], Q p ) → K 0 (e r Z p [Gal(L ′ /K)], Q p ). Since HR(L ′ /K ′ , r) vanishes for each intermediate Galois CM-extension of degree prime to p by Theorem 4.17, we deduce the result by a slight modification of the argument given in [ .1], we may and do assume thatL = L. Since both Schneider's conjecture and the p-adic Beilinson conjecture hold, Corollary 3.29 implies that Gross's conjecture at s = 1−r holds for all even (odd) irreducible characters of G if r is even (odd). By [Nic19, Proposition 5.5 and Theorem 6.5] (or rather the 'e r -parts' of these results) this is indeed equivalent to the rationality of T Ω(Q(r) L , e r Z[ 1 2 ][G]). Let us define Ω j (r) := ρ∈Irr Cp (G + ) Ω j (r, ρ ⊗ ω r−1 )e ρ⊗ω r−1 ∈ ζ(e r C p [G]) × .
Remark 4.20. Fix an integer r > 1. We have assumed throughout thatL contains a p-th root of unity. What was actually needed in the above considerations, however, is that ω r−1 restricted to GL is trivial. Let E ⊆L r ⊆L be the smallest intermediate field such that ω r−1 restricted to GL r is trivial. Then we can replaceL byL r throughout. Note thatL r is totally real and thusL + r =L r whenever r is odd, whereas we haveL r =L otherwise. In particular, we can replaceL by E whenever r ≡ 1 mod (p − 1).
Corollary 4.21. Let p be an odd prime and let r > 1 be an integer such that r ≡ 1 mod (p − 1). Let E/K be a Galois extension of totally real fields with Galois group G. Assume that Schneider's conjecture Sch(E, p, r), the p-adic Beilinson conjecture (Conjecture 3.23) for E/K and the equivariant Iwasawa   ) holds for all but finitely many r > 1.