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On the Equality of Three Formulas for Brumer--Stark Units

We prove the equality of three conjectural formulas for the Brumer--Stark units. The first formula has essentially been proven, so the present paper also verifies the validity of the other two formulas.


Introduction
In this paper we prove the equality of three conjectural formulas for Brumer-Stark units made by the first author in [5], [8], and [9] (the last two of these, in collaboration with Spieß).
One significance of this result is that the first formula has essentially been proven in [6], so the present paper also verifies the validity of the other two formulas. Additionally, the third formula, made in [9], relates to a conjecture for the principal minors of the Gross-Regulator matrix. The validation of the third formula here gives a proof of this conjecture for the diagonal entries. Our work generalizes a partial result in this direction established in [12]. See [6] for a discussion of the application of the these formulas toward explicit class field theory.
We now describe our results more precisely. Let F denote a totally real field, and let H denote a finite abelian extension of F . Write G = Gal(H F ). Let R ∞ denote the set of archimedean places of F . Let R be a finite set of places of F containing R ∞ and the places that are ramified in H. Fix a prime ideal p ∈ R that splits completely in H and let S = R ∪ {p}. Finally, we consider an auxiliary finite set T of primes of F , disjoint from R ∪ {p} and satisfying a standard minor condition (see §2.2). The following conjecture was first stated by Tate and called the Brumer-Stark conjecture, [16,Conjecture 5.4]. Conjecture 1.1. Let P be a prime in H above p. There exists an element u T ∈ U p = {u ∈ H * ∶ u v = 1 if v does not divide p} such that u T ≡ 1 (mod T ), and for all σ ∈ G, we have ord P (u σ T ) = ζ R,T (H F, σ, 0).
Here v ranges over all finite and archimedean places of H; in particular, each complex conjugation in H acts as an inversion on U p . The definition of the partial zeta function ζ R,T (H F, σ, 0) is recalled in §2.2. The conjectural element u T ∈ U p satisfying Conjecture 1.1 is called the Brumer-Stark unit for the data (S, T, H, P). Conjecture 1.1 has been recently proved away from 2 in joint work of the first author and Kakde [7]. It is convenient for us to package together u T and its conjugates over F into an element of H * ⊗ Z[G] that we call the Brumer-Stark element: There have been three formulas conjectured for the image of the Brumer-Stark element u p in F * p ⊗ Z [G]. In [5] the first author conjectured a p-adic analytic formula for u p following the methods of Shintani and Cassou-Noguès. We denote this formula by u 1 and state it precisely in §3. The other two formulas, which we denote u 2 and u 3 , were defined in joint work of the first author with Spieß in [8] and [9], respectively. Both of these formulas are cohomological in nature and are defined using the Eisenstein cocycle. They are stated precisely in §5 and §6. The following combines the conjectures of the first author (for i = 1) and the first author with Speiß (for i = 2, 3).
The main result of this paper is the following. Theorem 1.3. The three conjectural formulas for the Brumer-Stark element u p are equal, i.e.
. Recent work of the first author with Kakde has proved that u 1 = u p up to a root of unity under some mild assumptions. Write µ(F * p ) for the group of roots of unity in F * p . Theorem 1.4 (Theorem 1.6, [6]). Suppose that the rational prime p below p is odd and unramified in F . Suppose further that there exists q ∈ S that is unramified in H whose associated Frobenius σ q is the complex conjugation in G. Then Conjecture 1.2 for u 1 holds up to multiplication by a root of unity in F * p : In §7 we prove that u 2 = u 3 via a direct calculation that was foretold in [9]. The proof that u 1 = u 2 , which takes up §8, is more interesting and involves a new idea not present in prior work in this direction. It can be broken into two parts. Suppose that the conductor of the extension H F is f, and denote by E + (f) ⊂ O * F the subgroup of totally positive units congruent to 1 modulo f. We then prove by a direct calculation that u 1 (σ) ≡ u 2 (σ) (mod E + (f)), (1) where u i (σ) denotes the σ component of u i . We remark here that showing the above equation first requires the proof that u 2 = u 3 . Next, let H ′ ⊃ H be another finite abelian extension of F in which p splits completely. Let ff ′ be the conductor of H ′ F where, as above, f is the conductor of H F . For each σ ∈ G, we then show the norm compatibility relation for i = 1, 2, Applying (1) with H replaced by H ′ and combining with (2), we obtain Taking larger and larger conductors ff ′ and passing to a limit, we obtain the desired result u 1 (σ, H) = u 2 (σ, H).
Acknowledgements. The first author was supported by NSF grant DMS 1901939 for the duration of this project. The second author is grateful for the support of the Heilbronn Institute for Mathematical Research and Imperial College London. We thank Mahesh Kakde and Michael Spieß for helpful discussions.

Notation
Recall that we have let F be a totally real field of degree n over Q with ring of integers O = O F . Let E F = O * F denote the group of global units. More generally, for a finite set S of nonarchimedean places of F we denote by E S = E F,S the group of S-units of F . We define S = {q ∶ q q where, for some r ∈ S, r q}.
We also let H F be a totally complex extension containing a CM-subfield. Let f denote the conductor of the extension H F . We write E + (f) for the totally positive units of F that are congruent to 1 (mod f). Write G f for the narrow ray class group of conductor f. Let e be the order of p in G f and suppose that p e = (π) with π ≡ 1 (mod f) and π totally positive. We write O = O p − πO p ⊂ F * p . Define A = A F as the adele ring of F . For a Q-vector space W fix the notation WẐ = W ⊗ ZẐ = W ⊗ Q A Q . For an abelian group A and prime number L, we put A ℓ = A ⊗ Z Q ℓ .
For a place v of F we put For a set S of places of F we let A S denote the adele ring away from S. We also define U S = ∏ v∉S U v , and U S = ∏ v∈S U v . We shall also use the notation F S = (A S F × U S ) ∩ F * . Finally we note that if we have a function f ∶ X → Z and X ⊆ Y then we can extend f to a function f ! ∶ Y → Z by defining we call this the extension of f to Y by 0.

Partial zeta functions
For σ ∈ G = Gal(H F ), we define the partial zeta function Here the sum ranges over all integral ideals a ⊂ O that are relatively prime to the elements of R and whose associated Frobenius element σ a ∈ G is equal to σ. The series (5) converges for Re(s) > 1 and has a meromorphic continuation to C, regular outside s = 1. When the field extension H F is clear from context, we drop it from the notation and simply write ζ R (σ, s). Since p splits completely in H, the zeta functions associated to the sets of primes R and S = R∪{p} are related by the formula ζ S (σ, s) = (1 − Np −s )ζ R (σ, s).
Recall that we have fixed an auxiliary finite set of primes of F , denoted T , that is disjoint from S. The partial zeta function associated to the sets R and T is defined by the group ring equation We assume that the set T contains at least two primes of different residue characteristic or at least one prime η with absolute ramification degree at most ℓ − 2 where η lies above ℓ. With this in place, the values ζ R,T (K F, σ, 0) are rational integers for any finite abelian extension K F unramified outside R and any σ ∈ Gal(K F ). This was shown by Deligne-Ribet [10] and Cassou-Nogués [2].
Our assumption on T implies that there are no nontrivial roots of unity in H that are congruent to 1 modulo T . Thus the p-unit u T in Conjecture 1.1, if it exists, is unique. Note also that our u T is actually the inverse of the u in [11,Conjecture 7.4]. Throughout this paper, for ease of notation, we fix T = {λ} for an appropriate choice of λ.

Shintani zeta functions
Shintani zeta functions are a crucial ingredient in each of the constructions we study. We establish the necessary notation here, following Shintani [13].
For each v ∈ R ∞ we write σ v ∶ F → R and fix the order of these embeddings. We can then embed F into R n by x ↦ (σ v (x)) v∈R∞ . Note that F * acts on R n with x ∈ F * acting by multiplication by σ v (x) on the v-component of any vector in R n . For linearly independent v 1 , . . . , v r ∈ R n + , define the simplicial cone A Shintani set is a subset of R n + that can be written as a finite disjoint union of Shintani cones.
We now recall the definition of Shintani zeta functions. Write f for the conductor of the extension H F . Let b be a fractional ideal of F relatively prime to S and T , and let D be a Shintani set. For each compact open U ⊆ O p , define, for Re(s) > 1, We define ζ R,T (b, D, U, s) in analogy with (6) i.e., by the group ring equation It follows from Shintani's work in [13] that the function ζ R,T (b, D, U, s) has a meromorphic continuation to C. We now want to define conditions on the set of primes T and the Shintani set D to allow our Shintani zeta functions to be integral at 0.
• Nη is a rational prime ℓ; and • the cone C may be written We also say that η is good for a Shintani set D if D can be written as a finite disjoint union of Shintani cones for which η is good.
Definition 2.3. The set T is good for a Shintani set D if D can be written as a finite disjoint union of Shintani cones D = ⊔ C i so that for each cone C i , there are at least two primes in T that are good for C i (necessarily of different residue characteristic by our earlier assumption) or one prime η ∈ T that is good for C i such that Nη ≥ n + 2.
Remark 2.4. Given any Shintani set D, it is possible to choose a set of primes T such that T is good for D. In fact, all but a finite number of prime ideals with prime norm are good for a given Shintani set.
We can now note the required property to allow our Shintani zeta functions to be integral at zero. The proposition below is proved in [5, p.15].
Proposition 2.5. If the set of primes T is good for a Shintani set D, then We are mostly interested in a particular type of Shintani set, one which is a fundamental domain for the action of E + (f). Definition 2.6. We call a Shintani set D a Shintani domain if D is a fundamental domain for the action of E + (f) on R n + . That is, The existence of such domains follows the work of Shintani, in particular from [13,Proposition 4]. We note here some simple equalities that follow from the definitions. More details are given in §3.3 of [5]. Recall we have written G f for the narrow ray class group of conductor f. Let e be the order of p in G f , and write p e = (π) with π ≡ 1 (mod f) and π totally positive. Let D be a Shintani domain and write O = O p − πO p . Then, We now give two technical definitions that are necessary in the definition of u 1 .
Definition 2.7. Let V ⊂ E + (f) be a finite index subgroup (which is necessarily free of rank n − 1). We call a Shintani set D a Colmez domain for V if D is a fundamental domain for the action of V on R n + . That is, We note that in the definition of a Colmez domain we allow ourselves to work with V = E + (f), thus the definition includes Shintani domains. Proposition 2.8. Let V ⊂ E + (f) be a finite index subgroup. Let D and D ′ be Colmez domains for V . We may write D and D ′ as finite disjoint unions of the same number of simplicial cones Proof. [5,Proposition 3.15] proves this result when V = E + (f). The proof of this proposition is analogous.
A decomposition as in (9) is called a simultaneous decomposition of the Colmez domains (D, D ′ ).
there is a simultaneous decomposition as in (9) such that for each cone C i , there are at least two primes in T that are good for C i , or there is one prime η ∈ T that is good for C i such that Nη ≥ n + 2.
Lemma 2.11 (Lemma 3.20, [5]). Let D be a Shintani set and U a compact open subset of O p . Let b be a fractional ideal of F , and let β ∈ F * be totally positive so that β ≡ 1 (mod f) and ord p (β) ≥ 0. Suppose that b and β are relatively prime to R and that b is also relatively prime to T . Let q = (β)p −ordp(β) . Then ζ R,T (bq, D, U, 0) = ζ R,T (b, βD, βU, 0).
We end this section with a lemma of Colmez that allows us to give an explicit Colmez domain. Let α be, up to a sign, one of the standard basis vectors of R n . Note that its ray (αR + ) is preserved by the action of R n + . We define C α (v 1 , . . . , v r ) to be the union of the cone C(v 1 , . . . , v r ) with the boundary cones that are brought into the interior of the cone by a small perturbation by α, i.e., the set whose characteristic function is given by We use the usual bar notation for homogeneous chains Let x 1 , . . . , x n−1 ∈ F . We define the sign map δ ∶ F n → {−1, 0, 1} by the rule where ω(x 1 , . . . , x n ) denotes the n × n matrix whose columns are the images of the x i in R n . We adopt the convention sign(0) = 0.
Then the Shintani set is a Colmez domain for V .
The existence of Colmez domains follows from the work of Shintani in [13]. In Lemma 4.5 we show the existence of units ε 1 , . . . , ε n−1 ∈ E + (f) that satisfy the conditions of Lemma 2.12.
3 The multiplicative integral formula (u 1 ) Definition 3.1. Let I be an abelian topological group that may be written as an inverse limit of discrete groups Denote the group operation on I multiplicatively. For each i ∈ I α , denote by U i the open subset of I consisting of the elements that map to i in I α . Suppose that G is a compact open subset of a quotient of A * F . Let f ∶ G → I be a continuous map, and let µ be a Z-valued measure on G. We define the multiplicative integral, written with a cross through the integration sign, by Let λ be a prime of F such that Nλ = ℓ for a prime number ℓ ∈ Z and ℓ ≥ n + 2. We assume that no primes in S have residue characteristic equal to ℓ. Definition 3.2. Let D be a Shintani domain, and assume that λ is π-good for D. Define the error term ǫ(b, D, π) ∶= By [5,Lemma 3.14], only finitely many of the exponents in (12) are nonzero. [5, Proposition 3.12] and the assumption that λ is π-good for D implies that the exponents are integers. We recall from (8) that the measure is defined as We are now ready to write down the conjectural formula from [5]. We note that for any Shintani domain D we can always choose a prime λ that is π-good for D. In fact, all but a finite number of primes will satisfy this property. Henceforth, we can assume that λ satisfies the property written above and is π-good for D. We now give the main definition of this section. Definition 3.3. Let D be a Shintani domain, and assume that λ is π-good for D. Define As our notation suggests, we have the following proposition.
Proposition 3.4 (Proposition 3.19, [5]). The element u p,λ (b, D) does not depend on the choice of generator π of p e .
The following is conjectured.
Conjecture 3.5 (Conjecture 3.21, [5]). Let e be the order of p in G f , and suppose that p e = (π) with π totally positive and π ≡ 1 (mod f). Let D be a Shintani domain, and let λ be π-good for D. Let b be a fractional ideal of F relatively prime to S and λ. We have the following.
1. The element u p,λ (b, D) ∈ F * p depends only on the class of b ∈ G f ⟨p⟩ and no other choices, including the choice of D, and hence may be denoted u p,λ (σ b ), where σ b ∈ Gal(H F ).
3. Shimura reciprocity law: For any fractional ideal a of F prime to S and to λ, we have As we noted in the introduction, this conjecture has been proved up to a root of unity (Theorem 1.4).
We want to state the formula over F * p ⊗ Z[G] to match with the cohomological constructions.
Definition 3.6. We define

Transferring to a subgroup
In this section we recall the results [12], which allow us to transfer to a subgroup. Let V be a finite index subgroup of E + (f). Recall that π is totally positive, congruent to 1 modulo f and satisfies (π) = p e where e is the order of p in G f . Let D ′ V be a Shintani set which is a fundamental domain for the action of V on R n + and assume that λ is π-good for D ′ V . As before, we shall refer to such Shintani sets as Colmez domains. Let b be a fractional ideal of F relatively prime to S and λ.
We define Proposition 3.7 (Proposition 6.11, [12]). Let K and K ′ be two Colmez domains for V and λ a prime of F such that λ is π-good for K and K ′ . If λ is also good for (K, K ′ ), then u p,λ (b, K) = u p,λ (b, K ′ ).
Let V ⊂ E + (f) be a finite index subgroup. The following proposition shows the relation between u 1 (σ) and u 1 (V, σ). Proposition 3.8 (Proposition 6.12, [12]). Let D be a Shintani domain for E + (f). Let V be a finite index subgroup of E + (f). Write g 1 , . . . , g n−1 for a Z-basis of E + (f) such that g b 1 1 , . . . , g b n−1 n−1 is a Z-basis for V . Define . . , g n−1 ) is some constant that depends on g 1 , . . . , g n−1 and π (up to multiplication by an element of E + (f)), we have 4 Preliminaries for the cohomological formulas

Continuous maps
For topological spaces X and Y let C(X, Y ) denote the set of continuous maps X → Y . If R is a topological ring we let C c (X, R) denote the subset of C(X, R) of continuous maps with compact support. If we consider Y (resp. R) with the discrete topology then we shall also write C 0 (X, Y ) (resp. C 0 c (X, R)) instead of C(X, Y ) (resp. C c (X, Y )). Assume now that X is a totally disconnected topological Hausdorff space and A a locally profinite group. We define subgroups where the sums are taken over all compact open subgroups K of A. So C ◇ c (X, A) is the subgroup of C c (X, A) generated by locally constant maps with compact support X → A and by continuous maps with compact support X → K ⊆ A for some compact open subgroup K ⊆ A. Similarly C ◇ (X, A) is the subgroup of C(X, A) generated by locally constant maps X → A and by continuous maps X → K ⊆ A for some compact open K.
The following notation is used in the formulation of u 2 . Given two arbitary finite, disjoint sets Σ 1 , Σ 2 of places of F and a locally profinite group A we put where ? ∈ {◇, c, 0}. Here, for a set of places S, U S denotes the subgroup of We also introduce a generalisation of the above notation. For S 1 , S 2 disjoint sets of places of F let If S 3 is an additional disjoint set of places we also define

Measures
We now wish to attach to a homomorphism µ ∶ C c (X, Z) → Z[G] an A ⊗ Z[G]-valued measure on X for any abelian group A and finite abelian group G. We write the group operation of A multiplicatively. Firstly, by tensoring µ with the identity we obtain a homomorphism To write this map explicitly we first note that the isomorphism in (13) is given by where g α (x) = 1 if g(x) = α and 0 otherwise.
. Thus the homomorphism µ A is given by and g α is as defined before. If A is profinite we can consider the homomorphism where K ranges over the open subgroups of A.
(which we denote by µ A as well). For a general A (not necessarily profinite) we have seen that µ induces a homomorphism C c (X, for every compact open subgroup K ⊂ A. Combining these maps we see that µ induces a canonical homomorphism In practice, we apply certain specialisations of the general construction above. In the defini- In the definition of u 3 we have a measure on A rather than on

Eisenstein cocycles
We now define the Eisenstein cocycle. The cohomological constructions for u 2 and u 3 require different variations. Write S p for the primes of F above p that split completely in H. Let O F,Sp denote the ring of Here rec H F denotes the Artin reciprocity map for the extension H F . It follows from work of Shintani that the L-function in (14) has a meromorphic continuation to C. Furthermore, for D, b and s fixed, the values Let λ be a prime of F such that Nλ = ℓ for a prime number ℓ ∈ Z and ℓ ≥ n + 2. We assume that no primes in S have residue characteristic equal to ℓ. We then define the smoothed Shintani L-function Proof. The result follows from [9, Page 7].
Let F * + denote the group of totally positive elements of F . Let E Sp,+ denote the group of totally positive units in O Sp which we view as a subgroup of F * + . Let x 1 , . . . , x n−1 ∈ F * + . Recall the definition of C e 1 (x 1 , . . . , x n ) from (10) and the definition of δ(x 1 , . . . , x n ) from (11). The following proposition follows directly from [3, Theorem 1.6].
Then µ b,λ is an E Sp,+ -invariant homogeneous (n − 1)-cocycle yielding a class where the limit is over increasingly finer covers V of the support of f by compact open subgroups V ⊆ F Sp and t V ∈ V is any element of V .
We define the Eisenstein cocycle associated to λ by Here {b 1 , . . . , b h } is a set of integral ideals representing the narrow class group of O F,Sp (i.e., the group of fractional ideals of O F,Sp modulo the group of fractional principal ideals generated by totally positive elements of F ).
For more details on this construction, see §2 of [9]. Note that we use rec −1 H F rather then rec H F to make our formulation of u 3 consistent with u p , u 1 and u 2 . We expand further on this at the start of §6.
We now give a variation on the Eisenstein cocycle.
Here, the Shintani zeta function is defined in (7), δ is defined in (11) and C e 1 (x 1 , . . . , x n ) is defined in (10). Then ν p b,λ is a homogeneous (n − 1) on E + (f) p with values in the space of Z-distribution on F p . This follows from [3, Theorem 2.6]. We obtain a class Here ν p b,λ is viewed as an element of Hom(C c (F p , Z), Z) via the integration pairing from Remark 4.3. We also define where the sum ranges over a system of representatives of G f ⟨p⟩. This construction is adapted from the construction of ω p f,λ in §3.3 of [9]. We write W for F considered as a Q-vector space, and W ∞ = W ⊗ Q R. As before, let λ be a prime of F such that Nλ = ℓ for a prime number ℓ ∈ Z and ℓ ≥ n + 2. We assume that no primes in S have residue characteristic equal to ℓ.
By fixing an ordering of the infinite places, v ∈ R ∞ , we fix an identification W ∞ ≅ R n . Let v ∈ R ∞ be the infinite place corresponding to the standard basis element e 1 ∈ R n . We define F ℓ,v to be F viewed as a (diagonally embedded) subset of A S ℓ ∪{v} F . If D is a Shintani set and Φ ∈ C c (WẐ, Z) then, following [8], we define the Dirichlet series It is known to converge for Re(s) > 1 and extend to the whole complex plane except for possibly a simple pole at s = 0. Moreover, if D and Φ are as given in the following proposition then L(D, Φ; s) is holomorphic. We remark that the set S does not appear in the definition of this Dirichlet series.
In the following proposition we will decorate the L-function with λ since the choice of Φ incorporates Proof. This proposition follows the combination of [

Colmez subgroups
In the definitions for the Eisenstein cocycle and its variants the sign map δ appears. For the explicit calculations we want to perform later it is convenient if we can work with a finite index subgroup V ⊆ E + such that V = ⟨g 1 , . . . , g n−1 ⟩ and that we are able to choose π such that, after writing g n = π, • for τ ∈ S n we have δ([g τ (1) . . . g τ (n−1) ]) = sign(τ ).
We refer to such subgroups as Colmez subgroups. We define Let H ⊂ R n be the hyperplane defined by Tr(z) = 0. Then, Log(E + ) is a lattice in H. If z = (z 1 , . . . , z n ) ∈ R n + and Log(z) ∈ R n is not an element of H, then we define the projection We have Log(z H ) ∈ H. Note that z and z H lie on the same ray in R n + . For any M > 0 and i = 0, 1, . . . , n − 1, write l i (M ) for the element of H which has value M in the (i + 1) place and −M (n − 1) in the other places. We endow R n with the sup-norm. We denote by B(x, r) the ball centred at x of radius r.
The following lemma, which builds on [4, Lemma 2.1], allows us to find a collection of possible subsets V = ⟨g 1 , . . . , g n−1 ⟩ such that we get a nice sign property that allows us to more easily explicitly calculate the Eisenstein cocycle.
is some constant we define that depends only on R) we have the following: For i = 1, . . . , n − 1 let g i ∈ E + and g n = g π ∈ π H E + such that Log(g i ) ∈ B(l i (M ), R) and Log(g π ) ∈ B(l 0 (M ), R). Then • ⟨g 1 , . . . , g n−1 ⟩ ⊆ E + is a finite index subgroup, and Proof. This proof largely follows the ideas of Colmez in his proof of [4, Lemma 2.1]. First, note that both Log(E + ) and Log(π H E + ) are lattices inside H. There exists a constant R 1 ∶= R(E + , π) such that for all M > 0 and any r > R(E + , π) there exist g 1 , . . . , g n−1 ∈ E + and g π ∈ π H E + such that Log(g i ) ∈ B(l i (M ), r) for i = 1, . . . , n − 1 and Log(g π ) ∈ B(l 0 (M ), r). The existence of R 1 follows from Dirichlet's Unit Theorem and, in particular, the non-vanishing of the regulator of a number field. Since the l i (M ) form a basis of H, the Log(g i ) form a free family of finite index in Log(E + ), if M is large enough relative to r, say M > k(r). Now take M satisfying: ). Hence, the matrix given by [g 1 . . . g n−1 ] is written Expand ∆ and isolate the diagonal term; using the bounds we defined previously we obtain ) > 0 according to ii). We then show the other required sign properties in the same way.
It is required in our later calculations to make the following sign calculation.
Lemma 4.6. For i = 1, . . . , n − 1 let g i ∈ E + be chosen as in Lemma 4.5. Write S for the n × n matrix with rows Log(g 1 ), . . . , We now subtract the first column from each of the other columns and expand the determinant along the bottom row. This gives, after letting Write S ′ for the matrix in (17) and note that −2R < B i,j < 2R, for all i, j = 1, . . . , n − 1. When expanding the determinant of S ′ and isolating the diagonal terms using the bounds from before, we observe: Thus, Since we have assumed M > 4(n! − 1)R we have It thus remains to show that for n ≥ 2 the following holds Firstly, one can see by calculating that the inequality holds for n = 2. Remarking that as n increases the difference between the two terms in brackets decreases, gives that the value of the left hand side of (18) must increase with n. Thus (18) holds.
We now let K 2 (R) = max(K 1 (R), 4(n! − 1)R) so that both Lemma 4.5 and Lemma 4.6 hold if M > K 2 (R). Corollary 4.7. Let r > 0 be an integer, D + an r × r diagonal matrix with positive entries, A ∈ M n×r (R) and S as in Lemma 4.6. Then the block matrix Proof. Write d 1 , . . . , d r ∈ R >0 for the diagonal entries of D + . Using cofactor expansion with the last r columns of B one can see that the determinant of B is equal to Using Lemma 4.6 and the fact that the entries of D + are positive, the result follows.
We recall the definition of k(r) from the proof of Lemma 4.5 and note the following lemma.
Proof. Since the l i (M ) form a free family we need to only choose M big enough so that for i, j = 1, . . . , n − 1 and i ≠ j we have It is easy to calculate that the distance between l i (M ) and Thus we let K = √ 2(n − 1) n.
It follows from the above lemma that if M > K 2 (R) then for any λ > 0 we have that λM > K 2 (λR). Lemma 4.9. There exists such that we have the following. Firstly, for i = 1, . . . , n − 1 we can choose Proof of Lemma 4.9. We firstly choose the f i ∈ E + via Lemma 4.5 and Lemma 4.6, and let V f = ⟨f 1 , . . . , f n−1 ⟩. I.e., we have Log By writing the matrix representing the generators we have chosen for V f in an upper triangular form, we can make the following choice of generators of E + . Let ⟨δ 1 , . . . , δ n−1 ⟩ = E + such that for some τ ∈ S n−1 we have, for i = 1, . . . , n − 1, By changing the sign if necessary we choose a 1 > 0. Furthermore, we note that changing the values of the b i,j in the choice of V f does not change the index of the subgroup.
For ease of notation, let a = ∏ n−1 i=1 a i . For i = 2, . . . , n − 1 there exists R g,i > 0 and M g,i > 0 such that for all M > M g,i , there exists α ∈ E + with Log(α) ∈ B(l τ (i) (M ), R g,i ) and with q i a nonzero integer with absolute value coprime to a. We note that this is only possible for i > 2 since we require the freedom of having at least one additional component we can vary. We . . . , R g,n−1 ) and M ′ g = max(M g,2 , . . . , M g,n−1 ). We now find q 1 > a 1 which is coprime to a and such that q 1 a 1 M f > M ′ g and q 1 a 1 R f > R ′ g . We now fix R g = q 1 a 1 R f and M g = q 1 a 1 M f . Clearly R g > R 1 and it follows from Lemma 4.8 that M g > K 2 (R g ). We then choose g τ (1) = δ q 1 1 , it is immediate that Log(g τ (1) ) ∈ B(l 1 (M g ), R g ). For i = 2, . . . , n − 1, we have shown that there exist g τ (i) ∈ E + with Log(g τ (i) ) ∈ B(l τ (i) (M g ), R g ) and with q i a nonzero integer with absolute value coprime to a. Let V g = ⟨g 1 , . . . , g n−1 ⟩, the result follows.

1-cocycles attached to homomorphisms
Let g ∶ F * p → A be a continuous homomorphism, where A is a locally profinite group. We want to define a cohomology class c g ∈ H 1 (F * p , C c (F p , A)) attached to g. We define an F * p -action on C c (F * p , Z) by (xf )(y) = f (x −1 y). The following definition is due to Spieß and first appears in [14,Lemma 2.11]. This definition is crucial in making the constructions of the first author and Spieß's cohomological formulas work and we also remark that the definition is unusual in that it appears as though the cocycle z g should be a coboundary. However, it may not be a coboundary since g does not necessarily extend to a continuous function on F p . Definition 4.10. Let g ∶ F * p → A be a continuous homomorphism, where A is a locally profinite group. Let f ∈ C c (F p , Z) such that f (0) = 1. We define c g to be the class of the cocycle for x ∈ F * p and y ∈ F p .
The second term in (19) is allowed to be evaluated at 0 ∈ F p since we can extend continuously the function from F * p to F p as In particular, we can consider the class

Homology of a group of units
Recall that S p is the set of primes of F above p that split completely in H. We label the elements of S p by p 1 , . . . , p r , i.e., we let r = #S p . Now let 0 ≤ k ≤ r be an integer and write Here we have S 0 = ∅. By Dirichlet's unit theorem, the group of totally positive S k -units E S k ,+ is free abelian of rank n + k − 1. Thus the homology group H n+k−1 (E S k ,+ , Z) is free abelian of rank 1.
In the comological formulas u 2 and u 3 we are required to choose a generator of the homology group with the correct choice of sign. Let E + = ⟨ε 1 , . . . , ε n−1 ⟩. Recall we have written G f for the narrow ray class group of conductor f. For i = 1, . . . , r, let e i be the order of p i in G f , and write p e i i = (π i ) with π i ≡ 1 (mod f) and π i totally positive. For ease of notation, write π i = ε n−1+i . We then choose the following generator for the group Here µ = ±1 and is equal to the sign of the determinant of a specific matrix. For k = 0, . . . , r, let σ 1 (x)), . . . , log(σ n (x)), ord p 1 (x), . . . , ord p k (x)).
Here L(1) ∈ R n+k−1 is the vector with 1 in the first n − 1 components and 0 in the last r components. This choice is as given in [15,Remark 2.1] in the case k = 0.
Let V ⊂ E + be a finite index subgroup. Let V = ⟨ε 1 , . . . , ε n−1 ⟩ and write Similarly to (20) we choose a generator 5 Cohomological formula I (u 2 ) This section follows the construction given in [8, §3.1]. Throughout this section we use the notation established in §4.1. Let η p be the generator of H n (E p,+ , Z) as defined in (20). Let F be a fundamental domain for the action of F * E p,+ on (A p F ) * U p . Then ½ F is an element of H 0 (E p,+ , C(F, Z)) ≅ (C(F, Z)) Ep,+ . Taking the cap product gives ½ F ∩ η p ∈ H n (E p,+ , C(F, Z)). We now define ϑ p ∈ H n (F * , C c (∅, Z) p ) as the homology class corresponding to ½ F ∩ η p under the . We now follow the construction of [8, §6]. Since the local norm residue symbol for H F at p is trivial we omit it from the reciprocity map, i.e. we consider the homomorphism Let R ′ = R − R ∞ . We can view rec p H F as an element of H 0 (F * , C(R ′ , Z[G]) p ) and denote by its image under the map Here the cap product is induced by the map For a locally profinite abelian group A we have a canonical map which induces a cap-product pairing In particular we can consider Here c id is as defined in Definition 4.10. Now choose v ∈ R ∞ to be the infinite place corresponding to the standard basis element e 1 ∈ R n . Write R v ∞ = R ∞ − {v}. Recall that we write W for F considered as a Q-vector space. In [8, §5.3], the following map is defined.
We postpone giving the definition of ∆ * until the next section. Now consider the canonical pairing, where we recall the definition of µ F * p from §4.2, Noting that F ℓ,v is acting trivially on F * p ⊗Z[G] we see that (24) induces, via cap-product, a pairing The first author and Spieß then conjecture that the element u 2 (σ) is equal to the image of the Gross-Stark unit in F * p under σ. We end this section by stating some known properties of this construction. c) Let r be a nonarchimedean place of F with r ∉ S ∪ T where T is the set of places of F which lie above a rational prime q . Then we have d) Assume that H has a real archimedean place w ∤ v. Then u 2 (σ) = 1 for all σ ∈ G.
e) Let L F be a finite abelian extension of F containing H and unramified outside S. Then we have Remark 5.2. In the proposition above we correct a small typo in [8, Proposition 6.3, c)] by replacing σ r with σ −1 r .

The map ∆ *
We now define the map ∆ * . For more information and the more general construction we refer to [8, §5.3]. Throughout this section we let A = F * p ⊗ Z[G] to ease notation. For sets X 1 , X 2 and a map ψ ∶ X 1 × X 2 → A, we write Where supp(ψ) is the support of ψ. Proposition 5.3. Let X 1 , X 2 be totally disconnected topological Hausdorff spaces, with X 1 discrete. Let A be a locally profinite group. The map is an isomorphism.
Proof. We calculate the inverse map as follows. Let Y 1 = Supp(X 1 , X 2 , ψ) ⊆ X 1 . Note that Y 1 is finite since ψ has compact support. Then provides an inverse to (27).
Proof. Since we have and F * S 3 U S 3 is finite, we are able to apply Proposition 5.3.
For every w ∈ R ∞ we write δ w ∶ F * w → {±1} for the sign map. We also put

Recall that for a subgroup
We now calculate this map explicitly.
. The image of ψ under (30) is given by Here ψ(y, Proof. The result follows from Corollary 5.4 and (29).
Proposition 5.6. We have Here δ is as defined in (11). Furthermore, if we write then the image of Ψ under the isomorphism in (31) is represented by Here f λ∪v is the image of f in ∏ w∈λ∪v F * w and ψ(f λ∪v , ⋅) ∈ C ◇ c ({p}, R ′ , A) λ,∞ (δ).
Proof. It is easy to see that Thus, by weak approximation we have The result follows by Shapiro's lemma. The explicit description of the map follows simply by tracing through the definitions.
The last map we need to construct before giving the definition of ∆ * is the Recall that we have written S ′ = R ′ ∪ {p} and A λ,∞ F ≅ WẐ λ . There exist canonical homomorphisms By Proposition 5.3 the map (33) is an isomorphism. Let I S ′ ∪λ denote the group of fractional ideals of F that are coprime to S ′ ∪ λ. Since (A S ′ ∪λ,∞ F ) * U S ′ ∪λ,∞ is isomorphic to I S ′ ∪λ , the ring C 0 c (∅, Z) S ′ ∪λ,∞ can be identified with the group ring Z[I S ′ ∪λ ]. We define (32) as the tensor product is the inclusion map induced by extension by 0 and I S ′ ∪λ ∶ Z[I S ′ ∪λ ] → C c (A S ′ ∪λ,∞ F , Z) maps a fractional ideal a ∈ I S ′ ∪λ to the characteristic function ofâ S ′ ∪λ = a(∏ p∉S ′ ∪λ O p ), which we denote by char(a(∏ p∉S ′ ∪λ O p )). For where the sum ranges over Proof. Let ψ ∈ C ◇ c ({p}, R ′ , A) λ,∞ , and define Z as above. Then the image of ψ under the inverse map of the isomorphism (33) is To calculate the effect of I S ′ ∪λ , first note that the isomorphism where S m is the set of places that divide m and m(q) is the integer such that the fractional ideal q −m(q) m is coprime to q and π q is a uniformiser associated to the prime ideal q. We then view the image as an element of (A S ′ ∪λ,∞ F ) * U S ′ ∪λ,∞ by imposing that at the places away from S m the value is 1. Thus when we identify C 0 Applying I S ′ ∪λ we have Returning to ψ we have under the map ∆ λ Lastly, the image of the above under the map (34) is: We are now able to define ∆ * via the composition

Transferring to a subgroup
Let V be a finite index subgroup of E + . Let η p,V be the generator of H n (V ⊕ ⟨π⟩, Z) as defined in (21). Let F V be a fundamental domain for the action of Taking the cap product then gives We now define ϑ p . As before we view rec p H F as an element of H 0 (F * , C(R ′ , Z[G]) p ) and denote by its image under the map Here the cap product is induced by the map (23). We then define Proposition 5.8. Let V be a finite index subgroup of E + . Write Then Note that, since F * p ⊗ Z[G] only has an operation as a group, we have (∑ σ∈G a σ ⊗ σ) n = ∑ σ∈G a n σ ⊗ σ.
The cap-products in the top and bottom rows above include applying the isomorphisms (22) and (35), respectively. By Proposition 9.5 in [1, §3], we have following identities, Applying these identities with diagram (36) gives The proposition follows.

Explicit expression for u 2
Let V ⊆ E + be a finite index subgroup such that if V = ⟨ε 1 , . . . , ε n−1 ⟩ the ε i and π satisfy Lemma 4.5 and Lemma 4.6. For ease of notation we write ε n = π. In this section we calculate the value of u 2 (V ) = Eis 0 F ∩ ∆ * (c id ∩ ρ H F,V ) as an explicit multiplicative integral. Following §4.6 and Corollary 4.7, we choose the following generator for H n (V ⊕ ⟨π⟩, Z), We then calculate Here rec p H F ⋅ ½ F V is as defined in (23). It then follows that c id ∩ ρ H F,V is equal to Note that we have the action (x ⋅ f )(y) = f (yx −1 ) for a continuous map f and unit x. We also recall the definition of z id = z ½ O p ,id from §4.5. Then z id (ε τ (n) ) ∈ C ◇ c (F p , F * p ). We now apply the map ∆ * to this quantity. In §4.2.1 ∆ * is defined via the composition of three maps namely (30) * , (31) and (32) * . By Proposition 5.5 we have that the image of c id ∩ ρ H F,V under (30) * is given by Here, for ease of notation we have written It is now convenient for us to make a choice for F V . Let G V denote the group of fractional ideals of O F,p modulo the group of fractional principal ideals generated by elements of V , where O F,p denotes the ring of p integers of F . Let {b 1 , . . . , b h } be a set of integral ideals prime to R ′ ∪ λ representing G V . We may then choose . . , h we can choose the b i to be totally positive and prime to R ′ ∪ λ. This description of F V is similar to a construction given in [9,Page 14]. From this description of F V we have that Y τ (n) is trivial for all n. Thus (38) is equal to We now apply (31). By Proposition 5.6 we have that the image of (39) under (31) is equal to We can now finish calculating the effect of ∆ * on c id ∩ ρ H F,V by applying (32) to (41) and using Proposition 5.7 to calculate that Here . We now apply the measure Eis 0 Recall that the measure is defined in (25). We write µ F * p for the measure with values in F * p ⊗ Z[G] induced from the Eisenstein series Eis 0 F . We now consider the function (ε τ (1) . . . ε τ (n−1) ) ⋅ z id (ε τ (n) ). For ease of notation we define, for τ ∈ S n and z ∈ Z, We are then able to calculate, recalling that z id = z ½ O p ,id , To calculate the measure we first note that F * p ≅ ⟨π⟩ ⊕ O and that O ≅ lim m→∞ O 1 + p m O p . By (43) we are able to calculate the value at π and O separately. Let m ≥ 0 and α ∈ O 1 + p m O p . We write U α = α(1 + p m O p ). For σ ∈ G, we define the following maps For τ ∈ S n with τ (n) ≠ n we also define The construction of the measure given in §4.2 now allows us to calculate Here we have written Eis 0 Then, by Proposition 4.4 we have that where Φ = φ ⊗ φ λ and φ λ is as defined in (15). Let Recall from (16) that for s ∈ C with Re(s) > 1 we have We also write B = B n . Since we have chosen the ε i as in Lemma 4.5 we have Applying (46) and the definition in (45) to (44) we have, after noting (−1) n+1 (−1) (n−1)(n−1) = 1, We now calculate each term in the above expression, beginning with the limit term. Fix m ≥ 0 and σ ∈ G. Let α ∈ O 1 + p m O p . We also let b be a fractional ideal of F , coprime to S ∪ λ, and such that σ b = σ. We need to find the elements z ∈ Z such that φ Uα⊗σ −1 n,z is not trivial. For this we require that for some By the definition of F V and the reciprocity map, the above equation is non-trivial only if z ∈ F V and ∏ v finite q −ordv(zv) v = b −1 (α)p ordp(α) for some (α) ∈ P f,1 . Recall that f is the conductor of H F and we have written P f,1 = {(α) α ∈ F * + , α ≡ 1 (mod f)}. By the description of F V we note that for each σ ∈ G there is a unique z ∈ Z that satisfies the above equation. Since v finite for (α) ∈ P f,1 we have that (α)p −ordp(α) must be coprime to p ∪ R ∪ λ since z and b −1 are. Thus, for all r ∈ (α)p −ordp(α) we have r −1 ∈ ∏ r∉S ′ ∪λ O r . We therefore have We now define, for a Shintani set A, U ⊆ F p compact open, fractional ideal b and s ∈ C with Re(s) > 1 Here the multiplicative integral is as in Definition 3.1. We can apply similar calculations for the other terms in (47) and thus deduce following the explicit expression for Eis 0 F ∩ (∆ * (c id ∩ ρ H F,V )): 6 Cohomological formula II (u 3 ) In [9] the first author and Spieß give two equivalent constructions for their formula. Since we require each of them in the later sections we give both here. We denote them by u 3 and u ′ 3 . In this work we adopt slightly different conventions from [9], namely we have u 3 = u 3 (DS) # . Here # denotes the involution on Z[G] given by g ↦ g −1 for g ∈ G and u 3 (DS) is the construction in [9]. This is done by adjusting the definitions of κ λ and ω p f,λ in §4.3. The key adjustment we have is to apply rec H F ((η)) −1 rather than rec H F ((η)) in (14).
We begin with u 3 . Recall that in §4.5 and §4.3 we have defined the following objects: Let r = #S p and η Sp ∈ H n+r−1 (E Sp,+ , Z) be as defined in (20). Label the elements of S p by p, p 2 . . . , p r . We now define a class Here the cup product is induced by the canonical map Definition 6.1. We define Adapted from [9, Conjecture 3.1] we have the following conjecture.
We now give the definition for u ′ 3 . Recall that in §4.3 we have defined ).
be the generator defined in (21). Then, we define The following Proposition follows from [9, Proposition 3.6].
As before, it follows from [3, Theorem 2.6] that ν p b,λ,V is a homogeneous (n − 1)-cocycle on V ⊕ ⟨π⟩ with values in the space of Z-distribution on F p . Hence we obtain a class We then define . The next proposition shows the relation between u ′ 3 and u ′ 3 (V ).
As before, it follows from [3, Theorem 2.6] that µ χ,b,λ,V is a homogeneous (n − 1)-cocycle on V ′ Sp with values in the space of Z-distribution on F Sp . Hence, we obtain a class We then define Then Proof. This proposition follows from the proof of [9, Proposition 3.6].
Following from this proposition we have a simple corollary which gives the result of Proposition 6.5, but for u ′ 3 .
Corollary 6.7. Let V ′ be a finite index subgroup of E + . Then, Proof. We calculate Here the first and last equality follow from Proposition 6.6 and the middle equality follows from Proposition 6.5.
We recall the definition of κ b,λ,V from §6.1. Using that we have chosen V and π through Lemma 4.5 we note that for τ ∈ S n and a compact open U ⊆ O p , we have by definition: Here L is as defined in (14). Thus, Applying (51) and (52), and piecing together the appropriate Shintani sets we further deduce Considering the first two terms on the right hand side of (53) it is clear that We now consider the sum on the right hand side of (53). It is straightforward to see that

3
For later calculations we also require an explicit expression for u ′ 3 . Let V be a finite index subgroup of E + (f) such that V = ⟨ε 1 , . . . , ε n−1 ⟩ where ε 1 , . . . , ε n−1 and π = ε n are chosen to satisfy Lemma 4.5 and Lemma 4.6. For i = 1, . . . , n write Let B = B n . Recall we write V p = V ⊕ ⟨π⟩. Following §4.6 and Corollary 4.7, we choose the following generator for H n (V p , Z), We can now calculate We recall the definition of ω p f,b,λ,V from §4.3. For τ ∈ S n and a compact open U ⊆ O p , we have Returning to our main calculation, using (54) we have 1) . . . ε τ (n−1) ]), x, 0)).
We note that taking the cap product gives another factor of (−1) which cancels the factor from before. Applying (51) and (52) and piecing together the appropriate Shintani sets we further deduce It is clear that we can then write 7 Equality of u 2 and u 3 In this section we prove the following theorem.
Proof. Let V be finite index subgroup of E + which satisfies Lemma 4.5. We will show Then by Proposition 5.8 and Proposition 6.5 we have that We note that since F * p ⊗Z[G] only has an operation as a group we have (∑ σ∈G a σ ⊗ σ) n = ∑ σ∈G a n σ ⊗σ. It then follows that u 2 (σ) [E+∶V ] = u 3 (σ) [E+∶V ] . Considering Lemma 4.9, one can make two choices for V in Lemma 4.5 and Lemma 4.6, say . This then gives the result of the theorem.
We now recall the explicit calculations for u 2 (V ) and u 3 (V ) as given in §5.3 and §6.2 respectively.
Note that for U ⊆ O p , i = 1, . . . , n, k = 1, . . . , h and s ∈ C, Re(s) > 1 that We write Recall that in §4.2 we defined for L = ∑ σ∈G a σ ⊗σ ∈ Z[G] and α ∈ F * p , It then follows from the definitions of L R,λ and L R,λ that for α ∈ F * p , This completes the proof that u 2 (V ) = u 3 (V ).
8 Equality of u 1 and u 2 In this section we prove the following theorem.
We prove the theorem by showing that for each σ ∈ G we have u 1 (σ) = u 2 (σ). We show a strong enough functorial property for each of u 1 , u 2 that forces them to be equal.
Let H ⊂ H ′ be two finite abelian extensions of F in which p splits completely. Let ff ′ be the conductor of H ′ F where, as before, f is the conductor of H F . Let σ ∈ G. Write u 1 (σ, H) and u 2 (σ, H) for σ components of the formulas u 1 and u 2 , for the extension H F and Galois group element σ. We show that, for i = 1, 2, We refer to (56) as norm compatibility. For now we assume that the above equality holds for i = 1, 2.
In the following sections of this section we prove this. We now prove the following proposition.
Proposition 8.2. We have Proof. Let V be a free finite index subgroup of E + (f) of rank n − 1 satisfying the conditions given in the statement of Proposition 3.8. Furthermore we choose V such that if V = ⟨ε 1 , . . . , ε n−1 ⟩ then the ε i with π satisfy Lemma 4.5. We then let V ′ be a free finite index subgroup of E + of rank n − 1, By Theorem 7.1 and Proposition 6.6 we have Recall that we have defined 1) . . . ε τ (n−1) ]).
Thus u 1 (V, B, b) ≡ u ′ 3 (V, σ) (mod E + (f)) and hence u 1 (V, B, b) ≡ u 2 (V ′ , σ) (mod E + (f)). Working with coprime choices for V using Lemma 4.9, as in the proof of Theorem 7.1, allows us to complete the proof of the proposition. Note that here we are also using Proposition 3.7.
Assuming that (56) holds we can give the proof of the main theorem of this section.
Proof of Theorem 8.1. From Proposition 8.2 we have that for each τ ∈ G ′ , Our assumption that (56) holds then gives that for each σ ∈ G, we have u 1 (σ, H) = u 2 (σ, H).
This completes the proof.

Norm compatibility for u 1
The reciprocity map identifies Gal(H ′ H) with We let D f be a Shintani domain for E + (f) and define where the union is over a set of representatives {γ} for E + (ff ′ ) in E + (f). Let e ′ be the order of p in G ff ′ , and suppose that p e ′ = (π ′ ) with π ′ totally positive and π ′ ≡ 1 (mod ff ′ ). We can choose π ′ such that π ′ = π α for some α ≥ 1. We then define Let B denote a set of totally positive elements of O F which are relatively prime to S and λ and whose images in (O F ff ′ ) * are a set of distinct representatives for (57).
The following theorem is stated without proof by the first author in [5,Theorem 7.1]. For completeness we include a proof of this result here. Theorem 8.3 (Theorem 7.1, [5]). We have The key to the proof of Theorem 8.3 is to use translation properties of Shintani sets. For a subset A of equivalence classes of (57), let This definition extends to ζ A R,λ as in (6). Throughout this section we will use the following simple equality: This follows from Lemma 2.11. Recall the following definition. For β ∈ B, It is clear from the definition of B that Theorem 8.3 follows from the following proposition.
The proof of Theorem 8.4 is largely an exercise in explicit calculation. We begin by considering the multiplicative integral in u T (b(β), β −1 D ff ′ ).
By changing variable and then factoring out π i we have We now note that we can write, for i = 1, . . . , α − 1, Then, .
Thus we have, noting that B = AE = {ae a ∈ A, e ∈ E}, We now consider the powers of π given in the definition of u T (b(β), β −1 D ff ′ ) and arising in the statement of Lemma 8.5. Recall that π ′ = π α . Lemma 8.6.
Recalling the definition of D ff ′ we also note that for i = 0, . . . , α − 1 we have Thus we can calculate, using the fact that ζ R, By Lemma 2.11 we have for i = 1, . . . , α, We can then write

Remarking that {π
Noting that B = AE completes the proof.
We now consider the error term in the definition of u T (b(β), β −1 D ff ′ ) and the products of elements of E + (f) that arise in Lemma 8.5. Considering Lemma 8.5 and Lemma 8. 6 we can see that to prove Proposition 8.4 it is enough to prove the following. Then For clarity, we shall perform the calculations required for this proposition in a few lemmas.
Lemma 8.8. We have Proof. Considering the definition of D ff ′ we calculate Similarly we have We also calculate for i = 1, . . . , α − 1 We now note the following equalities, both of which hold via telescoping sum arguments. 1.
If α = 2 it is straightforward to see that Lemma 8.9 is equivalent to Proposition 8.7 and thus we are also finished in the case α = 2. From this point on we assume that α > 2. From Lemma 8.9 one can see that to prove Proposition 8.7 it is enough for us to show To do this we first show the following lemma.
Lemma 8. 10. We have that for j = 1, . . . , α − 1 the right hand side of (64) is equal to Note that for j = α − 1 the last product is empty. We also remark that it is implicit in the statement of this lemma that e(1) = ⋅ ⋅ ⋅ = e(α − 1).
Proof. We prove this by induction. The case j = 1 holds trivially. We now assume it holds for j and prove the result for j + 1, i.e., we show e(j) = e(j + 1). To do this we note that for i = j + 2, . . . , α, we have Thus, e(j) is equal to the product of the following elements: We remark that the first bracketed term in (65), and (68) are already products in e(j + 1). We now consider (67) and calculate that it is equal to We now consider the way the terms in (69) interact with (66). Multiplying (69) by (66) gives The first term in (70) is the term we were missing from e(j + 1). Thus it only remains to show that the second bracketed term in (65) multiplied by the second bracketed term in (70) is equal to 1. This is shown by the following calculation, We therefore deduce that e(j) = e(j + 1) as claimed. This completes the proof of the lemma.
We are now ready to prove Proposition 8.7.
This completes the proof of Proposition 8.7 and thus proves Theorem 8.3.

Norm compatibility for u 2
We recall the definition Theorem 8.11. We have for any σ ∈ G, Remark 8.12. This theorem has been stated without proof by the first author with Spieß in Proposition 5.1. We include the proof for completeness. We note also that the proof of the norm compatibility for u 2 is much simpler than that for u 1 . This is a result of the additional structure we have due to the cohomological nature of the construction.
Proof of Theorem 8.11. We consider the natural map Then, on the one hand, On the other hand ψ(u 2 (H ′ )) = ψ(Eis 0 The only equality of note here is the final one. This follows since we can commute ψ * with ∆ * , which follows from the calculations of §5.1 and §5.3. Then since ψ * ρ H ′ F = ρ H F , the desired result follows.