An equivariant $p$-adic Artin conjecture
Ben Forrás
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Universität der Bundeswehr

	INF 1 Institut für Theoretische Informatik, Mathematik und Operations Research 

	Werner-Heisenberg-Weg 39

	85579 Neubiberg

	Germany
ben.forras@unibw.de
https://bforras.eu

[2020]11R23, 11R42, 11R80 





We formulate an equivariant version of Greenberg's p-adic Artin conjecture for smoothed equivariant p-adic Artin L-functions in the context of an arbitrary one-dimensional admissible p-adic Lie extension of a totally real number field.
Using results of the author on the Wedderburn decomposition of the total ring of quotients of the Iwasawa algebra Λ(), we deduce validity of the conjecture in several interesting cases.


An equivariant p-adic Artin conjecture
    Ben Forrás
    Version of April 23, 2024
======================================





§ INTRODUCTION

Integrality questions have accompanied the study of p-adic Artin L-functions since its inception. Indeed, in the seminal works of Pi. Cassou-Noguès <cit.> resp. Deligne and Ribet  <cit.>, it was shown that in the case of a linear totally even p-adic Artin character χ with open kernel, the associated p-adic L-function can be expressed as a power series G_χ(X) divided by an explicit polynomial. These results were generalised to arbitrary p-adic Artin characters with open kernel by Greenberg <cit.>, employing the method of Brauer induction. This led to a weakening of the integrality property above, with G_χ(X) now being a fraction of two power series. Greenberg posited two conjectures, namely the p-adic Artin conjecture and a refinement thereof, which assert that G_χ(X) is contained in the ring of power series over a certain ring of integers, possibly up to a constant factor. 
These conjectures are now theorems: the p-adic Artin conjecture follows from the main conjecture proven by Wiles <cit.>, and its refinement was proven by Ritter and Weiss <cit.>. In this work, we propose a p-adic Artin conjecture for equivariant p-adic Artin L-functions, and prove it in certain important cases.

o0.25
	

	

		[every label/.append style = font = , every arrow/.append style=no head, row sep=1cm]
			    L_∞^+ [d] [ldd, ""] 

			K_∞[d, "_p≃Γ"'] [ru, "H"]     L^+ [ld, no head] 

			|[alias=K]|K [d]     

			
	

	



Let  be the Galois group of an admissible one-dimensional p-adic Lie extension L^+_∞/K, where L^+/K is the maximal totally real subextension of a finite CM extension L/K such that L contains a primitive pth root of unity, and L^+_∞ is the cyclotomic _p-extension of L^+. Let S be a finite set of places of K containing all places ramifying in L_∞^+/K and all infinite places. The group  is isomorphic to the semidirect product H⋊Γ  where H(L^+_∞/K_∞) is a finite group and Γ≃_p. The results cited above are valid for any p-adic Artin character χ:→_p with open kernel, and the associated p-adic L-function L_p,S(χ,s) can be expressed as a fraction of a power series G_χ,S(X) and an explicit polynomial.

In an equivariant setting, the denominators caused by the trivial character cannot be avoided. A possible way to remedy this is the introduction of a nonempty smoothing set T disjoint from S, and adjusting the p-adic L-function by a modified Euler factor at primes in T. We obtain a smoothed equivariant p-adic Artin L-function Φ^T_S∈(())^×, where () denotes the total ring of quotients of the Iwasawa algebra Λ()=_p and (()) is the centre thereof. Then in analogy with Greenberg's p-adic Artin conjecture, we posit the following integrality statement:
[equivariant p-adic Artin conjecture]
	Let Γ_0Γ^p^n_0 where n_0 is large enough such that Γ_0 is central in .
	Let 𝔐 be a Λ(Γ_0)-order in () containing Λ(). Then the smoothed equivariant p-adic Artin L-function Φ_S^T is in the image of the composite map
	
    𝔐∩()^×→ K_1(()) (())^×,

	where the first arrow is the natural map sending an invertible element to the class of the 1× 1 matrix consisting of said element, and the second arrow is the reduced norm map.




 §.§ Main result

In this paper, we prove the following maximal order variant of the equivariant p-adic Artin conjecture:

	Assume the equivariant Iwasawa main conjecture (<ref>). Then the equivariant p-adic Artin conjecture holds for all maximal Λ(Γ_0)-orders in () containing Λ().


The proof of this statement rests upon three pillars: (i) a dimension reduction result, (ii) a description of the Wedderburn decomposition of (), and (iii) a smoothed main conjecture. We briefly discuss these ingredients here.

The dimension reduction result goes back to Nichifor and Palvannan <cit.>. They studied integrality of non-smoothed equivariant p-adic Artin L-functions attached to extensions cut out by mod p characters, with the assumption that  is a direct product of H and Γ. A key step in the proof is a dimension reduction statement involving Dieudonné determinants of matrices and employing Venjakob's noncommutative Weierstraß theory <cit.>. Our dimension reduction result is an adaptation of their method to the semidirect product case.

Addressing the equivariant p-adic Artin conjecture, in particular the reduced norm map, necessitates studying the Wedderburn decomposition of the semisimple artinian ring (). If one assumes  to be a direct product, the representation theory of () is rather straightforward: in this case, the Wedderburn decomposition of () is directly determined by that of _p[H], and the maximal order in Nichifor–Palvannan's theorem is induced by the unique maximal orders in each of the skew fields in the Wedderburn decomposition.
However, when  is a semidirect product, the difference between Wedderburn decompositions of _p[H] and () becomes substantial: for instance, when  is pro-p, all Schur indices for _p[H] are 1 due to a theorem of Schilling <cit.>, whereas () can have nontrivial Schur indices. A general description of the Wedderburn decomposition of () has been carried out in <cit.>.

In recent work of Johnston and Nickel <cit.>, an (S,T)-modified Iwasawa module Y_S^T was introduced. The module Y_S^T admits a free resolution of length one, and one may formulate a smoothed main conjecture relating the class of Y_S^T in K_0(Λ(),()) to a T-smoothed version Φ_S^T of Ritter–Weiss's equivariant p-adic Artin L-function, postulating the existence of a unique element ζ_S^T∈ K_1(()) such that (ζ_S^T)=Φ_S^T and ζ_S^T is mapped to the negative of the class of Y_S^T under the connecting homomorphism of relative K-theory; see <ref>. Under the assumption of the main conjecture, the statement of the equivariant p-adic Artin conjecture can also be reformulated in terms of ζ_S^T (<ref>).
The algebraic object in the main conjecture poses an important difference between the present work and that of Nichifor–Palvannan: indeed, the Iwasawa module they consider is the Pontryagin dual of a certain non-smoothed Selmer group, and a cohomological criterion due to Greenberg shows that this module only admits a free resolution of length 1 away from the trivial character. No such restriction applies to the Johnston–Nickel module Y_S^T.

The equivariant Iwasawa main conjecture has been proven in the μ=0 case by Ritter–Weiss <cit.> and Kakde <cit.>, as well as in the case when  has an abelian p-Sylow subgroup by Johnston–Nickel <cit.>. In these cases, we obtain the following result:

	Suppose that either L^+_∞/K satisfies the μ=0 hypothesis, or that  has an abelian p-Sylow subgroup. Then the equivariant p-adic Artin conjecture holds for every maximal Λ(Γ_0)-order in ().




 §.§ Special case: p-abelian extensions

The equivariant p-adic Artin conjecture is stated for all Λ(Γ_0)-orders 𝔐 in () containing Λ(). The smaller 𝔐 is, the stronger this statement becomes, the extreme being 𝔐=Λ(). While proving this in general seems to be out of reach at the present juncture, it can be proven in the so-called p-abelian case, in which it is an easy corollary of a theorem of Johnston and Nickel <cit.>.


	If p does not divide the order of the commutator subgroup of , then Φ_S^T∈(Λ( )). Moreover, <ref> holds with 𝔐=Λ( ).




 §.§ Outline

This paper is structured as follows. <Ref> recalls the algebraic background used in the sequel. <Ref> is of an analytic nature: we recall the definitions of characterwise and equivariant p-adic Artin L-functions, and define a smoothed version. We formulate the equivariant p-adic Artin conjecture in <ref>. In <ref>, we recall the relevant algebraic object, and state a smoothed equivariant main conjecture. We use this in <ref> to study the behaviour of the equivariant p-adic Artin conjecture under change of the input data. <Ref> is a generalisation of Nichifor–Palvannan's dimension reduction result. Finally in <ref>, we prove the equivariant p-adic Artin conjecture in the cases mentioned above.

The paper contains results obtained in Chapters 1 and 4 of the author's doctoral thesis <cit.>.



 §.§ Acknowledgements
 The results in the aforementioned thesis were obtained while the author was a member of the Research Training Group 2553 at the Essen Seminar for Algebraic Geometry and Arithmetic (ESAGA), funded by the Deutsche Forschungsgemeinschaft (DFG, project no. 412744520). The author wishes to extend his gratitude to Bharathwaj Palvannan for hosting him at IISc Bengaluru in November 2022, and for many helpful discussions. Furthermore, the author thanks Andreas Nickel for his guidance and support throughout his doctoral studies, as well as for his comments on a draft version of this article.



 §.§ Notation and conventions
 The letter p always stands for an odd prime number. We fix an algebraic closure _p^ of _p.

The word `ring' is short for `not necessarily commutative ring with unity'. A domain is a ring with no zero divisors; the term `integral domain' is reserved for commutative domains. A principal ideal domain (PID) is a domain such that all left ideals and all right ideals are principal; in particular PIDs are not necessarily commutative.  If R is a ring, (R) stands for the total ring of quotients of R, which is obtained from R by inverting all central regular elements; when R is an integral domain, this is the field of fractions, and to emphasise this, we use the notation (R) instead.

For a ring R, the ring of n× n matrices is M_n(R). The n× n identity matrix is 1_n.
The centre of a group G resp. a ring R is denoted by (G) resp. (R).
If S is a finite set, then # S denotes its cardinality.
For n≥1, μ_n stands for the group of nth roots of unity. For a field , we write μ() for the group of roots of unity in .

In our notation, we make a clear distinction between rings of power series and completed group algebras: we use double square brackets for the former and blackboard square brackets for the latter. E.g. _p[[X]] is a ring of power series, and _pΓ is a completed group algebra. If  is a profinite group, then we write () for the set of _p^-valued irreducible characters of  with open kernel.

We will abuse notation by writing ⊕ for a direct product of rings, even though this is not a coproduct in the category of rings.



§ ALGEBRAIC PRELIMINARIES
 
This section consists of a recollection of algebraic concepts used in subsequent sections.


 §.§ Iwasawa algebras, characters and idempotents
 
Let  be a profinite group isomorphic to H⋊Γ, where H is a finite group and Γ≃_p. The Iwasawa algebra of  over _p is
Λ()_p.
Let ()(_p) denote its total ring of quotients. We will also consider ^()_p^⊗__p(). These algebras are defined analogously for  an arbitrary profinite group. The ring () is semisimple artinian, as shown in the proof of <cit.>. On the other hand, the Iwasawa algebra Λ() is never semisimple.

Let Γ_0Γ^p^n_0 where n_0 is chosen such that Γ_0⊆ is central. This is the case when n_0 is large enough: indeed, since  is a semidirect product, failure of Γ to be central comes from the homomorphism ϕ:Γ→(H) defining conjugation by elements of Γ not being trivial. However, since H is finite, so is (H), wherefore ϕ has open kernel in Γ. All such open subgroups are of the form Γ^p^n for some n≥0. Therefore if (ϕ)=Γ^p^n then Γ^p^n_0 is central in  whenever n_0≥ n. Fix a topological generator γ of Γ, and let γ_0γ^p^n_0.

As () is semisimple, it can be studied further by investigating its Wedderburn decomposition, which leads to considering characters of  and H. Let () denote the set of absolutely irreducible _p^-valued characters of  with open kernel, and let χ∈(). Let η|^_Hχ be an irreducible constituent of the restriction of χ to H.

Following <cit.>, let us define the fields

    _p,χ   _p(χ(h): h∈ H) ;
    _p(η)    _p(η(h): h∈ H) .

Write w_χ(:_η) for the index of the stabiliser _η of η in . This is a power of p since H stabilises η. Moreover, neither _p(η) nor w_χ depend on the choice of the irreducible constituent η.
The field _p(η) is an extension of _p,χ.
Furthermore, the field _p(η), and thus also the subfield _p,χ, is abelian over _p, as it is contained in some cyclotomic extension of _p. For g∈ and σ∈(_p(η)/_p), the characters ^gη resp. ^ση of H are given by ^gη(h)=η(ghg^-1) resp. ^ση(h)=σ(η(h)).



 §.§ Skew power series rings
 
Let  be a noetherian pseudocompact ring, that is, it is a complete Hausdorff noetherian topological ring with a fundamental system (ℒ_i)_i∈ I of open neighbourhoods of zero such that ℒ_i⊆ is a left ideal and each /ℒ_i is a finite length -module. 
Let σ∈ be a ring endomorphism, and let δ:→ be a σ-derivation, that is, a homomorphism of additive groups such that for all r,s∈, δ(rs)=δ(r)s+σ(r)δ(s).
Suppose that δ is σ-nilpotent. We refrain from giving the definition of σ-nilpotency in general. In the case when σ and δ commute, which is the only case we shall encounter in this work, this is equivalent to δ being topologically nilpotent, that is, for all n≥1 there is an m≥1 such that for all k≥ m, there is an inclusion δ^k()⊆ ()^n, where  is the Jacobson radical of .

For such a ring , endomorphism σ, and derivation δ, one can define the formal skew power series ring [[X;σ,δ]] as the ring with underlying additive group [[X]] and multiplication defined by Xr=σ(r)X+δ(r) for r∈. Nilpotence of δ is needed to ensure that multiplication is well-defined, that is, the coefficients of any product converge in . We refer to <cit.> for details. In the sequel, we drop the adjective `formal'.

Let  be a local ring with maximal ideal = such that  is separated and complete with respect to the -adic topology. Then skew power series over  possess a Weierstraß theory, as exhibited in <cit.>. For a skew power series f(X)=∑_i=0^∞ a_i X^i∈[[X;σ,δ]], let 

    (f)inf{i:a_i∈^×}∈_≥0∪{∞}

be the lowest degree term with an invertible coefficient, called the reduced order of f. Then the following hold:

 
	If f∈[[X;σ,δ]] has finite reduced order, then
	
    [[X;σ,δ]]=[[X;σ,δ]] f⊕⊕_i=0^ f-1[[X;σ,δ]] X^i.



 
	If f∈[[X;σ,δ]] has finite reduced order, then there is a unique unit ϵ∈[[X;σ,δ]]^× and a unique distinguished skew polynomial F∈[X;σ,δ] such that f=ϵ F.

Here a monic skew polynomial being distinguished means that all coefficients but the leading one are contained in the maximal ideal .



 §.§ Algebraic K-theory

We recall the relevant K-groups used in equivariant Iwasawa theory, mostly following the exposition in <cit.>.
Let  be a noetherian integral domain,  a finite dimensional algebra over (), and  an -order in . Later we will set these to be Λ(Γ_0), () and Λ().

The groups K_0() resp. K_0() are the Grothendieck groups of the abelian categories () resp. () of finitely generated projective left - resp. -modules; see <cit.> for details. We let K_1() resp. K_1() denote the Whitehead groups of the rings  resp. , as defined e.g. in <cit.>.

Let (())  be the category of bounded complexes of finitely generated projective -modules, and let (()) be the full subcategory of such complexes with -torsion cohomology groups. We define the relative K_0-group K_0(,) to be the abelian group generated by the objects of (()), with the relations being [C^∙]=0 for C^∙ acyclic, and [C^∙]=[C'^∙]+[C”^∙] for short exact sequences 0→ C'^∙→ C^∙→ C”^∙→0 in (()). In fact, (()) is a Waldhausen category, and K_0(,)=K_0((())); see <cit.>.

An equivalent definition of the relative K_0-group is to let K_0(,) be the abelian group generated by triples (P,ϕ,Q) where P and Q are finitely generated projective -modules and ϕ:⊗_ P⊗_ Q is an isomorphism of -modules. The relations are

    (P,ϕ,Q)+(Q,ψ,R)   =(P,ψϕ,R), 
    
    	(P',ϕ',Q')+(P”,ϕ”,Q”)   =(P,ϕ,Q),

where P'↪ P↠ P” and Q'↪ Q↠ Q” are exact sequences of -modules, and the diagram

    ⊗_ P' [d, "∼"] [r]    ⊗_ P [d, "∼"] [r]    ⊗_ P”[d, "∼"] 
    ⊗_ Q' [r]                      ⊗_ Q [r]                      ⊗_ Q”

is commutative. See <cit.> for more on this. An isomorphism between this group and the one defined above is given by

    [(P,ϕ,Q)]    ↦P-2ϕQ-1

on triples (P,ϕ,Q) for which ϕ maps P to Q.
Here the complex on the right is concentrated in degrees -2 and -1.
For further details and other definitions, we refer the reader to <cit.>.

The aforementioned K-groups are connected by the localisation exact sequence of K-theory:

    K_1()→ K_1()K_0(,)→ K_0()→ K_0().

The first resp. last arrows are induced by the inclusion → using functoriality of Whitehead resp. Grothendieck groups. The connecting homomorphism is given as follows: if M∈_n(), and M∈(^n) such that 1_⊗M=M, then

    ∂([M])^n-2M^n-1,

where the complex is in degrees -2 and -1. Matrices M for which such an M exists generate K_1(A).
Finally, the third arrow sends a complex [P Q] concentrated in degrees -2 and -1 to [P]-[Q].

In the case Λ(Γ_0), () and Λ(), the connecting homomorphism ∂ is surjective by <cit.>; see <cit.> for further comments on this. Therefore the localisation sequence takes the following form:

    K_1(Λ())→ K_1(())∂ K_0(Λ(),())↪ K_0(Λ())→ K_0(()).


Finally, we recall some derived categorical language, which will make the formulation of some statements easier. We let () denote the derived category of -modules. A complex of -modules is perfect if, considered as an element of (), it is isomorphic to a complex in (()). The full subcategory of perfect complexes with -torsion cohomology modules will be denoted by (). With each object of (), one can naturally associate an element in K_0(,). In the notation (), the domain  is suppressed; in this work, we will always have Λ(Γ_0) and Λ(), so this should not lead to confusion.



 §.§ Maximal orders

We will use the fact that maximal orders behave well with respect to taking matrix rings or direct sums:
 
	Let  be a noetherian integrally closed integral domain with field of fractions . Then if Δ is a maximal -order in a separable -algebra , then the matrix ring M_n(Δ) is a maximal -order in M_n() for all n≥1.



	Suppose that a separable -algebra  has decomposition
	=⊕_i=1^n _i
	into simple components. Then if for all i=1,…,n, the ring Δ_i is a maximal -order in _i, then ⊕_i=1^n Δ_i is a maximal -order in .




 §.§ Skew fields over local fields
 
We begin with some generalities on cyclic algebras; see <cit.> for details. Let E/ be a finite cyclic Galois extension of fields of degree , with ς a generator of (E/). Let ∈^× be a unit. Then the cyclic algebra (E/,ς,) is defined to be the -algebra

    (E/,ς,)⊕_i=0^-1 E α^i,

where α is a formal th root of , and α obeys the multiplication rule α x α^-1 =ς(x) for all x∈ E. 

The theory of skew fields over local fields was originally laid out by Hasse in <cit.>, a presentation of which is available in <cit.>. We provide a brief review of the results.

Let  be a complete commutative discrete valuation ring, = its field of fractions. Let  be a skew field with ()= and Schur index .
By extending the valuation from  to , one can show that  contains a unique maximal -order, which is the integral closure of  in .

Under the additional assumption that the residue field of  is finite, the skew field  can be described explicitly: see <cit.>. In this case,  is a (1-dimensional) local field. Write  for the order of the residue field of . Let (ω) be the cyclotomic field extension of  obtained by adjoining a primitive (^-1)st root of unity ω. Then (ω) is a maximal subfield of , called the inertia field (unique up to conjugacy in ).

Fix a uniformiser π∈. Then there exists a uniformiser π_ in the maximal order such that π_^=π, and there exists 1≤≤ coprime to  such that π_ωπ_^-1=ω^^. The fraction /∈/ is called the Hasse invariant of . Let σ∈((ω)/) be the automorphism defined by σ(ω)=ω^^; this is a power of the Frobenius and a generator of the Galois group ((ω)/). The maximal _p-order in  is generated by π_ and ω as an -algebra. In other words,  is the cyclic algebra

    =( (ω) / , σ, π_)=⊕_i=0^-1(ω) π_^i.


Since (ω) is a maximal subfield of , it is also a splitting field. A splitting isomorphism can be described explicitly as follows.

    (ω)⊗_   M_((ω)) 
    
    	x⊗ 1   ↦ x1_
    
    	1⊗ x   ↦(x, σ(x),…,σ^-1(x)) 
    
    	1⊗π_   ↦[ 0 1;   1; ⋮ ⋱;   1; π … 0;   ]

Here x∈(ω). We let ϕ denote the composite map

    ϕ: →(ω)⊗_   M_((ω)),

where the first arrow is x↦ 1⊗ x and the second one is (<ref>). Then ϕ maps  isomorphically onto its image (as rings).



 §.§ Reduced norms and the Dieudonné determinant
 
We recall definitions of the reduced norm map, the Dieudonné determinant, and some of their key properties. Standard references on reduced norms include <cit.> and <cit.>. Caveat: while our definition of the reduced norm on a central simple algebra agrees with that of <cit.>, its extension to semisimple algebras differs from that in <cit.> and <cit.>.
For the Dieudonné determinant, see <cit.> and <cit.>. A survey with a view towards Iwasawa algebras is provided in <cit.>; we shall follow the exposition there.

Let  be a field. Let  be a central simple -algebra. Let / be a splitting field for , meaning that there is an isomorphism

    ϕ:⊗_M_n()

of -algebras. A splitting field always exists by <cit.>.
The reduced characteristic polynomial of a is defined to be

    _/(a)(ϕ(1⊗ a)),

where (ϕ(1⊗ a)) is the characteristic polynomial of the matrix ϕ(1⊗ a) over the field . The reduced characteristic polynomial does not depend on the choice of the splitting field , and its coefficients lie in .

The reduced norm, which can be seen as a noncommutative generalisation of the determinant, is defined as follows:

    _/(a)(-1)^_/(a)_/(a)(0).

For every n≥1, the matrix ring M_n() is also a central simple -algebra, and the above definition provides maps _M_n()/. On invertible matrices, these take values in ^×, and they induce a reduced norm map on the colimit, which factors through the abelianisation, thus defining a map : K_1()→^×. The kernel of this map is denoted by _1()().

The following result shows that elements in orders have integral reduced norms.
 
	Let  be a noetherian integrally closed integral domain, =() its field of fractions,  a central simple -algebra, and Δ an -order in . Then the reduced characteristic polynomial of an element in Δ has coefficients in .


The notion of a reduced norm extends to arbitrary semisimple -algebras as follows: if =⊕_i=1^n _i, where each _i is a central simple algebra with centre _i, then there is a reduced norm map __i/_i in each component, and their product defines a homomorphism

    _/()⊕_i=1^n __i/_i: →()=⊕_i=1^n _i.


There is another generalisation of the usual determinant: the Dieudonné determinant. Let  be a semilocal ring (that is, / is semisimple artinian).
Define the subgroup W()≤^× as

    W(){(1+rs)(1+sr)^-1: (1+rs)∈^×}.

Then the Dieudonné determinant :_n()→^×/W() is the homomorphism characterised by the following two properties: 

	
  * Elementary matrices have Dieudonné determinant 1. An elementary matrix is a matrix which differs from the identity matrix by a single off-diagonal entry.
	
  * If r∈ and M∈_n() then ((r,1,…,1) M)=r (M).

In particular, the Dieudonné determinant of a(n upper or lower) triangular matrix is the product of the elements in its diagonal. This makes sense since the Dieudonné determinant is only defined up to W(), which contains the commutator subgroup of ^×. (This is not obvious; consult either <cit.> or <cit.> for a proof.)

We indicate how to compute A for A∈ M_n(). 
We say that an n× n matrix is a permutation matrix if it is obtained from the identity matrix 1_n by swapping two of its columns. This is the terminology used in <cit.>; some authors also call a product of such matrices a permutation matrix.
Suppose that we can write A=UBV where B is diagonal, and U and V are products of elementary, permutation and scalar matrices. In this case it is said that A admits a diagonal reduction via elementary operations.
As explained above, we can compute the Dieudonné determinant of any diagonal matrix, and the same goes for elementary and scalar matrices. Permutation matrices have order at most 2, hence their Dieudonné determinants are second roots of unity—in fact, the Dieudonné determinant of a permutation matrix is always -1, see <cit.>.

Jacobson's elementary reduction theorem provides an important class of rings for which there is such  a decomposition A=UBV. We say that  is a PID if all left ideals and all right ideals are principal, see <cit.>. Then the elementary reduction theorem states the following:
 
	If  is a PID, then every square matrix over  admits a diagonal reduction via elementary operations.


In the colimit, the Dieudonné determinant naturally extends to a homomorphism K_1()→^×/W().
Due to results of Vaserstein, there is an isomorphism ^×/W()≃ (^×)^ whenever  is a local ring or 2∈^×, see <cit.>. Moreover, <cit.> states that in this case, the Dieudonné determinant provides an isomorphism

    K_1()(^×)^.

The ring () is semisimple and therefore semilocal. The Iwasawa algebra Λ() is finitely generated as a module over the commutative local ring Λ(Γ_0), hence it is semilocal, see <cit.>. Therefore the Dieudonné determinant is defined for both () and Λ().

If = is a skew field, then the Dieudonné determinant is a homomorphism : _n()→ (^×)^, and the reduced norm factors over it, see <cit.>

    _n() [rr, "_M_n()/()"] [rd, ""']       ()^×
        (^×)^[ru, "_/()"']


We will sometimes talk about reduced norms of elements in Λ(). In this case, the map is understood to be the reduced norm map _()/(()), as the ring Λ() is not necessarily semisimple but it is contained in the semisimple ring (). In particular, the reduced norm of such an element need not lie in Λ(), but see <ref> for a special case in which this is true.



§ SMOOTHED EQUIVARIANT P-ADIC ARTIN L-FUNCTIONS
 
In this section, we work on the analytic side. We first recall the definitions of characterwise resp. equivariant p-adic Artin L-functions, then introduce a smoothed version of the latter, and investigate its basic properties.



 §.§ Characterwise p-adic Artin L-functions

Let p an odd rational prime, and let K be a number field.
Let K_∞ be the cyclotomic _p-extension of K. Let /K be a Galois extension such that  is totally real, contains K_∞, and such that the degree (:K_∞) is finite. In other words, /K is an admissible one-dimensional p-adic Lie extension. The Galois group (/K) is isomorphic to the semidirect product of the profinite group Γ(K_∞/K)≃_p and the finite group H(/K_∞), with H being a normal subgroup: ≃ H⋊Γ. Finally, let S be a finite set of places of K containing all infinite places.

For all places v∉ S of K, fix a place w_∞ of  above v. Let I_w_∞ be the associated inertia group, and let ϕ_w_∞∈/I_w_∞ denote the Frobenius at w_∞. Let 𝔑(v) denote the absolute norm of v.


	Let χ be an irreducible complex character of  with open kernel (i.e. /χ is finite). Define the S-truncated Artin L-function as
	
    L_S(χ,s)∏_v∉ S( 1-ϕ_w_∞𝔑(v)^-s| V_χ^I_w_∞)^-1,

	where V_χ is the representation affording χ, that is, V_χ is a simple [/χ]-module with character χ; in other words, V_χ is a -vector space, ρ_χ: ↠/χ→ V_χ an Artin representation, and χ=ρ_χ. Then V_χ^I_w_∞ denotes the subspace fixed by I_w_∞. 

The definition is independent of the choices for the places w_∞: changing w_∞ leads to replacing ϕ_w_∞ by a conjugate, and the determinant is conjugation invariant. The function L_S(χ,s) is absolutely convergent for s>1, and admits a meromorphic conti­nuation to the complex plane.

Let χ_:(K(μ_p^∞)/K)→_p^× be the cyclotomic character. The decomposition _p^×≃μ_p-1×(1+p_p) gives rise to characters

    ω: (K(μ_p^∞)/K)    →(K(ζ_p)/K)→μ_p-1 ;
    κ: (K(μ_p^∞)/K)    →(K_∞/K)→ 1+p_p .

Here ω is the Teichmüller character.

The construction of p-adic Artin L-functions for linear characters was given by Pi. Cassou-Noguès resp. Deligne–Ribet in <cit.> resp. <cit.>. From now on, we assume that S contains all places ramifying in /K (so in particular, all p-adic places) as well as all infinite places. Moreover, we fix an isomorphism ι: _p of fields.
 
	For each linear character χ: →_p^× with open kernel, there exists a unique continuous function (called the S-truncated p-adic Artin L-function associated with χ)
	
    L_p,S(χ,-)   :_p→_p  →_p→for χ;
    
    		L_p,S(,-)   :_p-{1}→_p.

	satisfying the interpolation property
	
    L_p,S(χ,1-n)= ι( L_S(ι^-1∘χω^-n,1-n))   ∀ n≥1.

	Moreover, there exist power series G_χ,S(X)∈__p(χ)[[X]] such that
	
    L_p,S(χ,1-s)=G_χ,S(u^s-1)/H_χ(u^s-1),

	where
	γ is the fixed topological generator of Γ, and it  corresponds to 1+X under the (non-canonical) isomorphism __p(χ)Γ≃__p(χ)[[X]],
	uκ(γ), and
	
    H_χ(X)χ(γ)(1+X)-1    if H⊆χ;
     1    otherwise.


Note that due to work of Siegel <cit.>, the interpolation property does not depend on the choice of the isomorphism ι.

The construction of p-adic Artin L-functions was extended to not necessarily linear Artin characters via Brauer induction by Greenberg in <cit.>. This, however, weakens the above statement, with G_χ,S(X) now being a quotient of two power series. The interpolation property is also weakened from n≥1 to n≥2. Both of these weakenings are due to possibly negative coefficients in the Brauer induction theorem. Ellerbrock and Nickel recently verified the interpolation property at n=1, see <cit.>.

	Let χ:→_p be a nontrivial character with open kernel. There exists a unique continuous function
	
    L_p,S(-,χ):_p→_p

	satisfying the interpolation property
	
    L_p,S(1-n,χ)=ι(L_S(1-n,ι^-1∘χω^-n))   ∀ n≥1.

	Moreover, there exists a fraction G_χ,S(X)∈(__p(χ)[[X]]) of power series such that
	
    L_p,S(1-s,χ)=G_χ,S(u^s-1)/H_χ(u^s-1),

	where u, T and H_χ(X) are as in <ref>.




 §.§ Equivariant p-adic Artin L-functions

Ritter and Weiss defined equivariant p-adic Artin L-functions as follows; see <cit.> for the original version and <cit.> for the formulation used here.

	Let R_p() be the additive group generated by _p^-valued characters of  with open kernel. Let ^*(R_p(), ^(Γ)^×) denote the group of homomorphisms f:R_p()→^(Γ)^× satisfying the following transformation properties: 
	
		
  * f(χ⊗ρ)=ρ^♯ (f(χ)) for all characters ρ such that ^_Hρ= (so-called  type W characters), where ρ^♯∈(^(Γ)) is the automorphism given by ρ^♯(γ)=ρ(γ)γ for all γ∈Γ;
		
  * f(^σχ)=σ(f(χ)) for all σ∈(_p^/_p).
	


 
	There is an isomorphism
	
    (())^×^*(R_p(), ^(Γ)^×).


We give a brief description of the isomorphism in <ref>. Ritter and Weiss proved that there exists a unique element 

    γ_χ∈(^()e_χ)

which acts trivially on the space V_χ and it is of the form γ_χ=g_χ c_χ, where g_χ∈ is mapped to γ^w_χ∈Γ under the projection ↠Γ, and c_χ∈ (_p^[H]e_χ)^×; see <cit.>. We will refer to γ_χ as the Ritter–Weiss element. In fact, this construction also works over a sufficiently large finite extension of _p instead of _p^, see <cit.>. Let Γ_χ denote the procyclic group generated by γ_χ. Then (^()e_χ)= ^(Γ_χ), which gives rise to a group homomorphism

    j_χ: (^())↠(^()e_χ)≃^(Γ_χ)^(Γ)

Then the isomorphism of <ref> is given by 

    z↦ (χ↦ j_χ(z)).


Characterwise p-adic Artin L-functions satisfy the transformation properties listed in <ref>, see <cit.>. Therefore they can be seen as elements on the right hand side of the isomorphism of <ref>, so the following definition makes sense:


	Let _/K,S∈^*(R_p(), ^(Γ)^×) be defined by the assignment
	
    _/K,S: χ↦G_χ,S(γ-1)/H_χ(γ-1).

	The S-truncated equivariant p-adic Artin L-function Φ_S(/K)∈(())^× is the corresponding element under the isomorphism of <ref>.




 §.§ Smoothing

Recall that S is a finite set of places of K containing all places ramifying in /K as well as all infinite places.
Let T be another finite set of places of K such that S∩ T=∅; we shall refer to T as the smoothing set.
We define a smoothed version of the equivariant p-adic Artin L-function, following <cit.>. The element Ψ_S,T there differs from our Φ_S^T by a twist by the cyclotomic character, which is in line with the fact that Ψ_S,T is associated with a CM extension, whereas Φ_S^T corresponds to the totally real extension /K.


	For each v∈ T fix a place w_∞ of . Define
	
    Φ_S^T(/K) Φ_S(/K) ·∏_v∈ T(1-ϕ_w_∞) .

	Here the reduced norm map is applied to the class of the map
	
    Λ() Λ()

	in K_1(()). If there is no risk of confusion, we will omit the field extension from the notation and simply write Φ_S^T.


It would have been possible to introduce smoothing already in the complex L-function, as is done in <cit.>. Instead, we make the following:


	Let _/K,S^T denote the element of ^*(R_p(), ^(Γ)^×) corresponding to Φ_S^T under the isomorphism of <ref>.


In the rest of this section, we describe how _/K,S^T and Φ_S^T(/K) transform under replacing  by a quotient. Such properties are available for non-smoothed equivariant p-adic Artin L-functions, see <cit.>. These go back to corresponding statements for non-equivariant p-adic Artin L-functions, see <cit.>, and ultimately to characterwise complex Artin L-functions, see <cit.>.


Let /'/K be a tower of fields such that '/K satisfies the same conditions as imposed upon /K. Then all notions defined for /K are also defined for '/K. We shall distinguish the objects associated with '/K from those defined for /K by a prime symbol, e.g. '('/K). Note that Γ'=Γ. We choose the places relevant to the construction of smoothed p-adic L-functions such that they remain compatible with the non-primed versions, that is, we require w_∞| w_∞'. In particular, there is a natural map π:↠' sending ϕ_w_∞ to ϕ_w_∞'.
The natural projection π induces surjective maps Λ()↠Λ(') and ()↠('), which we will, by abuse of notation, also denote by π. Functoriality of K-groups induces a morphism K_1(π):K_1(())→ K_1((')).

 
	The elements _/K,S^T and Φ_S^T(/K) satisfy the following transformation properties:
	in the setup above,
	
    _/K,S^T ∘inf_'^   = _'/K,S^T,
    π(Φ_S^T(/K))   =Φ'^T_S('/K),

	where inf_'^ denotes inflation from ' to .



	For T=∅, this is <cit.>. It remains to verify compatibility of the smoothing factors (1-ϕ_w_∞) on both sides.
	
	There is a commutative diagram:
	
    K_1(()) [dd, "K_1(π)"] [rr] [rd,""]       ^*(R_p(), ^(Γ)^×) [dd, "-∘inf^_'"] 
       (())^×[ru,"∼"]  
    
    			K_1((')) [rr] [rd,""]       ^*(R_p('), ^(Γ)^×) 
       (('))^×[ru,"∼"] [from=2-2, crossing over, "π" near start]

	The square containing K_1-groups and ^*-groups is <cit.> for deflation maps, and the two triangles are <cit.>, originally formulated in <cit.>. The left front square is obviously commutative. 
	
	Commutativity of the right front square can be seen from the definition of the northeast-pointing arrows: these are given by (<ref>). In this proof, we write γ_χ' for the Ritter–Weiss element of the quotient extension. 
	For commutativity of the square, one only needs to check that π maps γ_χ to γ_χ', which follows from the uniqueness property of γ_χ, as shown in <cit.>. 
	
	Finally, the left vertical arrow sends (1-ϕ_w_∞) to (1-ϕ_w'_∞) because the places w_∞ and w_∞' were chosen compatibly. The claims follow.


In <ref>, these properties will be used to establish functoriality for integrality of Φ_S^T under the assumption of the main conjecture.


	One may also consider replacing  by an open subgroup '. Then in the statement of <ref>, one can replace inflation by induction, as done in <cit.>. See also <cit.>. This, however, does not directly lead to a functoriality statement as for quotients, the crux being the definition of a map ()^×→(')^× making the following square commutative.
	
    ()^×[d,""] [r, dashed]    (')^×[d,""] 
    (())^×[r]    (('))^×





§ THE EQUIVARIANT P-ADIC ARTIN CONJECTURE
 
We begin by recalling the complex Artin conjecture. As before, let S be a finite set of places of K containing all places ramifying in /K as well as all infinite places.

Let χ be an irreducible character of  with open kernel. Then the function L_S(χ,s) is entire whenever χ is nontrivial.

The assertion has been proven for abelian extensions <cit.>, but remains open in general. 
The corresponding statement for p-adic Artin L-functions was conjectured by Greenberg in <cit.> for the depletion set S=S_p∪ S_∞.
 
	For a character χ:→_p^ with open kernel,
	
    G_χ,S(X)∈_p[[X]]⊗__p_p^,

	where G_χ,S∈(__p(χ)[[X]]) is the object appearing in the numerator of the expression for the p-adic Artin L-function in (<ref>).

In <cit.>, Greenberg showed that the appropriate characterwise main conjecture implies the p-adic Artin conjecture. Since the main conjecture is known due to the work of Wiles, so is the p-adic Artin conjecture, see <cit.>.

Greenberg also considered the following stronger version, and proved that it follows from the main conjecture under the additional assumption of the vanishing of Iwasawa's μ-invariant, see <cit.>. The statement was proven unconditionally by Ritter and Weiss in <cit.>.
 
	For a character χ:→_p^ with open kernel,
	
    G_χ,S(X)∈__p(χ)[[X]].



In analogy with these conjectures, we posit the following equivariant version of the p-adic Artin conjecture. Let T be a finite set of places of K such that S∩ T=∅ and T∅.
 
	Let 𝔐 be a Λ(Γ_0)-order in () containing Λ(). Then the smoothed equivariant p-adic Artin L-function Φ_S^T is in the image of the composite map
	
    𝔐∩()^×→ K_1(()) (())^×,

	where the first arrow is the natural map sending an invertible element to the class of the 1× 1 matrix consisting of said element.



	<Ref> is a generalisation of a question of Nichifor–Palvannan <cit.>, who considered extensions of totally real fields cut out by a non-trivial mod p character, with the Galois group being a direct product of Γ and a finite group. They showed that the answer to their question is positive for a certain maximal order, see <cit.>. 
	
	We shall provide further comments on how the equivariant p-adic Artin conjecture is analogous to the non-equivariant version in <ref>.


	The smaller 𝔐 is, the stronger the statement of <ref> becomes. In particular, it is at its strongest resp. weakest when 𝔐 is Λ() resp. a maximal order.




§ MAIN CONJECTURES
 
In order to state a smoothed version of the equivariant Iwasawa main conjecture—in particular, to define the object on the algebraic side—we impose further restrictions on our setup. Let L/K be a finite Galois extension of number fields such that L is a CM number field containing ζ_p (we will address these assumptions in <ref>). Let L^+ denote its maximal totally real subfield. Let L_∞ K_∞ L and L_∞^+ K_∞ L^+. In the notation above, set L^+_∞. We keep the assumption that S and T are finite non-empty disjoint sets of places of K, S contains all places ramifying in L^+_∞/K and all infinite places.
Then the objects Γ, H,  and Φ_S^T are defined as in <ref>. Finally, let (L_∞/K).


    [remember picture, every label/.append style = font = , every arrow/.append style=no head]
    	       |[alias=Li]| L_∞[dl] [d] 
        L_∞^+ [d] [ldd, ""]     |[alias=L]| L [dl] 
    
    	K_∞[d, "_p≃Γ"'] [ru, "H"]     L^+ [ld, no head] 
    
    	|[alias=K]|K                                                                
    [overlay,remember picture]
    	 (K) .. controls +(1,0) and +(4,-1) .. (Li) node [midway,shift=(.2,-.2)] ;
    	[right=-1mm of L] ∋ζ_p;




 §.§ The Johnston–Nickel module Y_ST
 
The object on the algebraic side of the smoothed equivariant main conjecture is given rise to by the module Y_S^T, which was defined by Johnston and Nickel <cit.>. We now recall the properties of this module.

Let j∈ denote complex conjugation. Every Λ()-module M decomposes as M^+⊕ M^-, where j acts as +1 resp. -1 on M^+ resp. M^-. 
Let 
    Λ()_-Λ()/(1+j).
 The Λ()-module structure on M induces Λ()- resp. Λ()_--module structures on the direct summand M^+ resp. M^-.

For a Λ()-module M and r∈, let M(r) denote the rth Tate twist, which has the same underlying abelian group as M with group action twisted by the rth power of the cyclotomic character: g⋆ mχ_(g)^r· g* m where g∈ and m∈ M, “⋆” is the new action and “*” is the old one. We note two properties of Λ()-modules which shall be used in the sequel.

 
	Let M be a Λ()-module. Then (M(-1))^-(1)= M^+.


	Since χ_(j)=-1, we have (M(-1))^-(1)=(M(-1)(1))^+=M^+.


 
	There is an isomorphism Λ()≃Λ()_-(1) of Λ()-modules.


	The ring Λ()_- has a natural Λ()-module structure via the natural projection Λ()↠Λ()_-. It follows that Λ()_- as a Λ()-module is generated by a single element m∈Λ()_-:
	
    ⟨ m⟩_Λ()   =Λ()_-. 
    Taking Tate twists, we get⟨ m(1)⟩_Λ()   =Λ()_-(1).

	
	The element j∈ acts as -1 on Λ()_-=Λ()/(1+j). Since χ_(j)=-1, this means that after a Tate twist, j acts trivially on Λ()_-(1). Since =/⟨ j⟩, it follows that the action of Λ() on Λ()_- factors through Λ().
	
	Consequently, (<ref>) can be sharpened to ⟨ m(1)⟩_Λ()=Λ()_-(1), which means that there is a surjection of Λ()-modules
	
    Λ()↠Λ()_-(1).

	The group Γ_0 is a central open subgroup both in  and in , and the ranks of Λ() and Λ()_-(1) as Λ(Γ_0)-modules agree. Therefore (<ref>) is an isomorphism.


Let X_S denote the S-ramified Iwasawa module associated with L_∞. More explicitly, X_S is the Galois group of the maximal abelian pro-p extension of L_∞ unramified outside S over L_∞.
The maximal j-invariant submodule of X_S is the S-ramified Iwasawa module X_S^+ associated with L_∞^+. This is a finitely generated torsion Λ(Γ_0)-module by <cit.>. For each v∈ T, we fix places w_∞ resp. w_∞ of L^+_∞ resp. L_∞ such that w_∞|w_∞.


	There exists a Λ()_--module Y_S^T(-1) sitting in an exact sequence of Λ()_--modules
	
    0→ X_S^+(-1) → Y_S^T(-1) →(⊕_v∈ T__w_∞^_p(-1) )^-→_p(-1)→0.

	Moreover, Y_S^T(-1) has projective dimension ≤1 over Λ()_-, and it is torsion over Λ(Γ_0).


	Note that our assumptions on S and T are sufficient for this result. Indeed, the proof relies only upon Assumptions (i–iv) on <cit.>; our setup covers (i–iii), and (iv) follows from the fact that T is non-empty and it contains no p-adic places.


Since the module Y_S^T(-1) has a projective resolution of length one over Λ()_-, there exist a projective Λ()_--module P and an integer n≥0 such that there is an exact sequence

    0→ P (Λ()_-)^n→ Y_S^T(-1)→0

Since Y_S^T(-1) is torsion over Λ(Γ_0), the difference of the K_0-classes over Λ()_- of the first two terms in this short exact sequence becomes zero over the ring of quotients ()_-:

    K_0(Λ()_-)    ↪ K_0(()_-) 
    [Λ()_-^n]-[P]    ↦ 0

Here the map is injective by a result of Witte, see (<ref>).
Hence the first two terms of the projective resolution (<ref>) are stably isomorphic over Λ()_-. That is, there is a projective Λ()_--module Q such that Λ()_-^n⊕ Q≃ P⊕ Q. 

Since projective modules are direct summands of free modules, by possibly enlarging Q, there are isomorphisms
Λ()_-^m≃Λ()_-^n⊕ Q≃ P⊕ Q for some m≥0. Consequently, there is a projective resolution

    0→(Λ()_-)^m (Λ()_-)^m→ Y_S^T(-1)→0.

Twist by (+1), using <ref>:

    0→Λ()^m Λ()^m→ Y_S^T→0,

where α is some m× m matrix. This allows us to associate with Y_S^T a class in the relative K-group K_0(Λ(),()): considering [Y_S^T] as a complex concentrated in degree 0, the resolution (<ref>) shows that in the derived category (Λ()), this is isomorphic to 

    [Y_S^T]=Λ()^m-1αΛ()^m0,

where the complex on the right is concentrated in degrees -1 and 0. In particular, [Y_S^T] is in (Λ()), and thus it defines a class in K_0(Λ(),()).



 §.§ The smoothed equivariant main conjecture

Let us first recall the equivariant Iwasawa main conjecture (EIMC) with uniqueness and without smoothing, as formulated in <cit.> and <cit.>. The original formulations are due to Ritter–Weiss <cit.> and Kakde <cit.>; we refer to <cit.> for a comparison of these approaches.

 Consider the following complex:
	
    C_S^∙(L_∞^+/K)R( RΓ_(_L_∞^+,S, _p/_p), _p/_p),

	where _L_∞^+,S is the ring of S-integers of L_∞^+, consisting of elements of L_∞^+ with non-negative valuation away from S, and _p/_p is a constant sheaf.


The nonzero cohomology groups of the complex C_S^∙(L_∞^+/K) are H^-1(C_S^∙(L_∞^+/K))≃ X_S^+ and H^0(C_S^∙(L_∞^+/K))≃_p.
Thus C_S^∙(L_∞^+/K) is a complex in (Λ()). In particular, it defines a class in K_0(Λ(),()).

 
	Let L/K be any finite Galois CM extension with S a finite set of places as above.
	There exists a unique element ζ_S(L_∞^+/K)∈ K_1(()) such that (ζ_S(L_∞^+/K))=Φ_S(L^+_∞/K). Moreover,  ∂(ζ_S(L_∞^+/K))=-[C_S^∙(L_∞^+/K)], where [C_S^∙(L_∞^+/K)] is the class of C_S^∙(L_∞^+/K) in K_0(Λ(),()).
	
    [every label/.append style = font = ]
    		    K_1(()) [rr, "∂"]        K_0(Λ(),())
    ∃! ζ_S(L_∞^+/K) [d, maps to] [rrr, maps to]           -[C_S^∙(L_∞^+/K)] 
    Φ_S(L^+_∞/K)    (())^×[from=1-2, "" near start, crossing over]




	It is known that the boundary homomorphism is surjective, see (<ref>).
	Vanishing of _1(())=() is equivalent to the uniqueness part of the conjecture.


According to <cit.>, the middle two terms of (<ref>), considered as 0th resp. (-1)st terms of a complex, represent C_S^∙(L_∞^+/K)(-1):

    Y_S^T(-1)-1(⊕_v∈ T__w_∞^_p(-1) )^-0=[C_S^∙(L_∞^+/K)(-1)]∈ K_0(Λ()_-,()_-).

Tate twisting this by +1 yields the following, using <ref>:

    Y_S^T-1(⊕_v∈ T__w_∞^_p )^+0=[C_S^∙(L_∞^+/K)]∈ K_0(Λ(),()).

Note that for all v∈ T, we have (__w_∞^_p )^+≃__w_∞^_p. Moreover, there is a short exact sequence

    0→Λ(_w_∞) Λ(_w_∞) →_p→0,

where the labelled arrow is right multiplication by (1-ϕ_w_∞).
These factors correspond exactly to those by which we perturbed the analytic p-adic L-function in <ref>.
In light of this, we posit the following smoothed version of <ref>:

  
	Let L/K be as above, and let S and T be finite non-empty disjoint sets of places of K, with S containing all places ramifying in L^+_∞/K and all infinite places.
	Then there is a unique element ζ_S^T(L_∞^+/K)∈ K_1(()) such that (ζ_S^T(L_∞^+/K))=Φ_S^T. Moreover, ∂(ζ_S^T(L_∞^+/K))=[Y_S^T].
	
    [every label/.append style = font = ]
    		    K_1(()) [rr, "∂"]        K_0(Λ(),())
    ∃! ζ_S^T [d, maps to] [rrr, maps to]          [Y_S^T] 
    Φ_S^T(L_∞^+/K)    (())^×[from=1-2, "" near start, crossing over]



 
	Let L/K, S and T be as in <ref>. Then <ref> is equivalent to <ref>.


	The ingredients of the proof are all contained in the paragraphs preceding <ref>.	
	We first deal with the existence and `moreover' parts of the two main conjectures. Assume that <ref> holds. Then let 
	
    ζ_S^Tζ_S·∏_v∈ T[(1-ϕ_w_∞)]∈ K_1(()).

	Its reduced norm is
	
    (ζ_S^T)=(ζ_S)·∏_v∈ T(1-ϕ_w_∞)=Φ_S·∏_v∈ T(1-ϕ_w_∞)=Φ_S^T,

	as desired. We compute the image under the boundary homomorphism ∂.
	
    ∂(ζ_S^T)
    		   =∂(ζ_S)+∑_v∈ T∂(1-ϕ_w_∞) 
       =-[C^∙(L_∞^+/K)]+∑_v∈ T∂(1-ϕ_w_∞) 
    The equality (<ref>) in K_0(Λ(),()) lets us rewrite the first term as the complex represented by Y_S^T→(⊕_v∈ T__w_∞^_p )^+. Then (<ref>) allows us to write:   =[Y_S^T]-⊕_v∈ T__w_∞^_p 0+∑_v∈ T∂(1-ϕ_w_∞) 
    We rewrite the sum using the definition of ∂ and the isomorphism (<ref>).   =[Y_S^T]-∑_v∈ T__w_∞^_p 0+∑_v∈ TΛ()-21-ϕ_w_∞Λ()-1
Using the exact sequence (<ref>), we find that the two summations cancel each other out, and we get ∂(ζ_S^T)=[Y_S^T], as desired. In conclusion, <ref> implies <ref> without uniqueness. By the same computation, <ref> implies <ref> without uniqueness by setting 
	
    ζ_Sζ_S^T·∏_v∈ T[(1-ϕ_w_∞)]^-1∈ K_1(()).

	
	Lastly, the two uniqueness statements are equivalent to each other, because they are both equivalent to _1(())=1.




 §.§ The equivariant p-adic Artin conjecture for zeta_ST̂


We also formulate an integrality conjecture regarding the element ζ_S^T, in the setup of <ref>.

	Assume <ref> without assuming uniqueness; let ζ_S^T∈ K_1(()) such that ζ_S^T=Φ_S^T and ∂(ζ_S^T)=[Y_S^T]. Let 𝔐 be a Λ(Γ_0)-order in () containing Λ(). Then ζ_S^T is in the image of the natural map
	
    𝔐∩()^×→ K_1(()).



	<Ref> is formulated without assuming uniqueness. However, if it holds for one ζ^T_S satisfying the conditions in the main conjecture, then it holds for all such elements. Indeed, let ζ^T_S,ζ'^T_S ∈ K_1(()) such that ∂(ζ_S^T)=∂(ζ'^T_S). Using the localisation sequence (<ref>), we find a class [β]∈ K_1(Λ() such that ζ'^T_S=ζ^T_S·[β]. Then Vaserstein's theorem (<ref>) allows us to find a representative b∈Λ() of β in Λ(). So if x∈𝔐∩()^× be a preimage of ζ^T_S under the natural map in the conjecture, then ζ'^T_S is the image of xb. We will encounter the same argument again in the proof of <ref>.


 
	Assume <ref> without uniqueness. Then the following hold:
	
		
  *  <Ref> implies <ref>.
		
  *  If in addition, we assume uniqueness, then <ref> implies <ref>.
	


	(<ref>) Let ζ_S^T be a preimage of Φ_S^T. If this is in the image of the natural map, then (ζ_S^T)=Φ_S^T is in the image of the composite map (<ref>), so <ref> implies <ref>. 
	
	(<ref>) For the converse, let ζ_S^T be the unique preimage of Φ_S^T under the reduced norm map, and let x∈𝔐∩()^× be a preimage of Φ_S^T under the map (<ref>). Then the class [(x)]∈ K_1(()) is also a preimage of Φ_S^T under the reduced norm, therefore [(x)]=ζ_S^T by uniqueness of ζ_S^T.



	One can also consider the weakening of the above conjectures without involving an order 𝔐.
	That is, one may ask whether Φ_S^T comes from a 1× 1 matrix over (), or more explicitly, whether it is in the image of the map
	
    ()^×→ K_1(())(())^×.

	This is not known in general. However, it is true that Φ_S^T is in the image of (<ref>) if and only if it is in the image of : K_1(())→(())^×. Indeed, by (<ref>) there is a commutative diagram:
	
    ()^×[rrrdd, "_()/(())", bend left] [rd, two heads] [rdd, two heads, bend right]          
        K_1(()) [d, ""', "∼"] [rrd, ""]       
       (()^×)^[rr, "_()/(())"']       (())^×

	The vertical arrow is an isomorphism by (<ref>). Since the map ()^×↠(()^×)^ is surjective, so is the natural map ()^×→ K_1(()) by commutativity of the left triangle. The claimed equivalence follows. In particular, Φ_S^T is in the image of (<ref>) if <ref> holds.




 §.§ A maximal order main conjecture

The `maximal order main conjecture' (a theorem of Ritter and Weiss) admits a smoothed version. We first recall the statement in the non-smoothed case.
 
	Let 𝔐 be a maximal Λ(Γ_0)-order in () containing Λ(). If x_S∈ K_1(()) is such that ∂(x_S)=-[C_S^∙(L_∞^+/K)], then (x_S)Φ_S^-1∈(𝔐)^×.

 
	Let 𝔐 be a maximal Λ(Γ_0)-order in () containing Λ(). If x_S^T∈ K_1(()) is such that ∂(x_S^T)=[Y_S^T], then (x_S^T)(Φ_S^T)^-1∈(𝔐)^×.


	Let x_S^T be as in the statement, and let
	
    x_S x_S^T·∏_v[(1-ϕ_w_∞)]^-1∈ K_1(()).

	Then the same argument as in the proof of <ref> shows that
	
    ∂(x_S)=∂(x^T_S)+∑_v∈ T∂([1-ϕ_w_∞]^-1)=[Y^T_S]-∑_v∈ T∂(1-ϕ_w_∞)=-[C_S^∙(L_∞^+/K)].

	Consequently, (x^T_S)(Φ_S^T)^-1 = (x_S ) (Φ_S)^-1∈(𝔐)^× by <ref>.


 
	Let 𝔐 be a maximal Λ(Γ_0)-order in () containing Λ(). Then Φ_S^T∈(𝔐).


	Let α∈ M_m(Λ()) be as in (<ref>). On the one hand, since M_m(Λ())⊆ M_m(𝔐), and the latter is a maximal Λ()-order in M_m(()) by <ref>, the reduced norm (α) is contained in 𝔐 by <ref>.
	On the other hand, ∂([(α)])=[Y_S^T], and so  <ref> is applicable: (α)· (Φ_S^T)^-1∈(𝔐)^×. The assertion follows.



	The integrality statement of <ref> is unconditional. It can be seen as a special case of the equivariant p-adic Artin conjecture.
	Indeed, let 𝔐 be a Λ(Γ_0)-order in () containing Λ(). Assuming <ref> for this 𝔐, we obtain that Φ_S^T∈(𝔐) because of <ref>.


 
	We demonstrate how the equivariant p-adic Artin conjecture (<ref>) is analogous to its non-equivariant counterparts. Let 𝔐 be a maximal Λ(Γ_0)-order in () containing Λ(), and assume the validity of <ref> for this 𝔐 (see <ref> for when this holds). Then <ref> shows that Φ^T_S∈(𝔐). Let _p^ denote the integral closure of _p in _p^. The map in <ref> induces an isomorphism (𝔐)≃^*(R_p(), _p^⊗__pΛ(Γ)), as shown by Ritter–Weiss in <cit.>. Consulting <ref>, one sees that this map sends Φ^T_S to the map given by
	
    χ↦G_χ,S(γ-1)/H_χ(γ-1)·δ_T(γ-1),

	where δ_T(γ-1)=∏_v∈ T(1-ϕ_w_∞ | _^_p[H](V_χ,())), see <cit.> or <cit.>. It follows that G_χ,S(γ-1) is integral up to the factor δ_T(γ-1)/H_χ(γ-1), which can be made explicit by choosing the smoothing set T appropriately.
	
	Indeed, consider the case of a type W character, that is, when χ⊇ H, and assume that χ is non-trivial. In this case, χ factors through a finite abelian group, so in particular, it is linear. Using Tchebotaryov's density theorem, one finds a place v of K such that ϕ_w_∞ is conjugate to γ in . Set T{v}. Then 
    δ_T(γ-1)=(1-γ | _^_p[H](V_χ,()))=1-γχ(γ^-1).

	Indeed, if f∈_^_p[H](V_χ,()), then 
    ((1-γ)f)(v)=f(v)-(γ f)(v)=f(v)-γ· f(γ^-1 v)=f(v)-γ· f(χ(γ^-1 v)) = (1-γχ(γ)^-1)· f(v).

	Since χ is of type W, we have H_χ(γ-1)=χ(γ)γ-1. Passing to power series by replacing γ↦ 1+X, we find that δ_T(X)/H_χ(X) has constant term
	
    1-χ(γ)^-1/χ(γ)-1 = χ(γ)·(1-χ(γ)^-1)/χ(γ)·(χ(γ)-1)=1/χ(γ).

	The denominator on the left hand side is nonzero because χ is non-trivial.
	Since χ(γ) is a p-power root of unity, the constant term is invertible, and thus δ_T(X)/H_χ(X) is a unit in _p^[[X]]. Hence G_χ,S(X)∈_p^[[X]], and since χ is invariant under (_p^/_p(χ)), this even shows G_χ,S(X)∈__p(χ)[[X]], as stated by the refined p-adic Artin conjecture (<ref>).




§ BEHAVIOUR OF THE EQUIVARIANT P-ADIC ARTIN CONJECTURE UNDER CHANGING S, T, AND G
  
We note some generalities regarding <ref>. 
We first describe how the conjecture behaves under adding more places to either S or T. We will work with a fixed extension /K as in <ref>. For the sake of brevity, we will suppress the extension in our notation, and simply write Φ_S^TΦ_S^T(/K).

 
	Let 𝔐⊂() be a Λ(Γ_0)-order in () containing Λ(). Let S and T satisfy the hypotheses above, and let v^* be a place of K not in S∪ T. If Φ_S^T is in the image of the map (<ref>), then so is Φ_S^T∪{v^*}.


	By <ref>, the relationship between the two smoothed equivariant p-adic L-functions is Φ_S^T∪{v^*}=Φ_S^T·(1- ϕ_w^*_∞) where w^*_∞ is the (previously) fixed place of L_∞^+ above v^*.
	The first term is in the image of the map by assumption, while the second one is the reduced norm of the class of the map given by left multiplication by the element (1-ϕ_w^*_∞)∈Λ(). This element is invertible in (), and Λ()⊆𝔐, hence the claim.


 
	Retain the assumptions of <ref>. Then if Φ_S^T is in the image of the map (<ref>), then so is Φ_S∪{v^*}^T.


	Noting that χ_(ϕ_w^*_∞)=𝔑(v^*), we have Φ^T_S∪{v^*}=Φ^T_S·(1-ϕ_w^*_∞χ_(ϕ_w^*_∞)^-1). Indeed, this can be seen e.g. by comparing Proposition 3.2.13 and Lemma 4.2.1 of <cit.>. Then the same argument as in <ref> works.


Consider the setup for changing the top field in <ref>.
In particular, if L/L'/K is a tower of fields such that L'/K and L/K both satisfy the assumptions at the beginning of <ref>, then  =L^+_∞ and '=L'^+_∞ fit into this setup.

There is a natural projection π:↠' which induces ()↠(') and K_1(π):K_1(())→ K_1((')).
Let 𝔐 be a Λ(Γ_0)-order in () containing Λ(), and let 𝔐'π(𝔐). Then by surjectivity of the natural maps, 𝔐' is a Λ(Γ_0)-order in (') containing Λ('). Write Φ_S^TΦ_S^T(/K) and Φ'^T_SΦ_S^T('/K) for brevity.

 
	If Φ_S^T is in the image of the composite map 
	
    𝔐∩()^×→ K_1(())(())^×,

	then Φ'^T_S is in the image of the composite map
	
    𝔐'∩(')^×→ K_1(('))(('))^×.



	The maps in the statement fit into a commutative diagram:
	
    𝔐∩()^×[r] [d,"π"]     K_1(()) [d, "K_1(π)"] [r,""]    (())^×[d,"π"] 
    𝔐'∩(')^×[r]     K_1((')) [r,""]    (('))^×

	By <ref>, we have π(Φ_S^T)=Φ'^T_S. 
	Hence if Φ_S^T is the image of some x∈𝔐∩()^× under the top row, then Φ'^T_S is the image of π(x) under the bottom row.



	Using the same methods as in the proof of <ref>, the results of <ref> can be reformulated for ζ_S^T under the assumption of <ref>.


 
	In order to formulate the smoothed equivariant Iwasawa main conjecture (<ref>), we need to assume that L is a CM field containing a primitive pth root of unity ζ_p. <Ref> shows that we may restrict our attention to such extensions when studying the equivariant p-adic Artin conjecture (<ref>). Indeed, if F/K is an arbitrary finite extension of totally real number fields, then let F_∞ FK_∞ be the cyclotomic _p-extension, and let '(F_∞/K). Then L F(ζ_p) is a CM field with maximal totally real subfield L^+=F(ζ_p+ζ_p^-1), and thus ' is a quotient of =(L^+_∞/K).




§ DIMENSION REDUCTION FOR INTEGRAL MATRICES OVER SKEW POWER SERIES RINGS

Let  be a local field with valuation ring _, and fix a uniformiser π_. Let / be a cyclic Galois p-extension of local fields, and let τ∈(/) be a generator. Let  be a finite dimensional skew field with centre , and let _ denote the unique maximal _-order in . Let / be the Hasse invariant of . Then  is obtained by adjoining an element π_ to (ω) where ω is a primitive root of unity of order ^-1 with q being the order of the residue field of , moreover π_^=π_ and π_ωπ_^-1=ω^^.

Suppose that the Schur index  of  divides #-1 where # is the order of the residue field of .
Then by <cit.>, τ admits a unique extension to an automorphism of  of order (:).


The skew power series ring _[[X;τ,τ-𝕀]] is well defined, and has the following properties:

	
  * The total ring of quotients (_[[X;τ,τ-𝕀]]) is a skew field.
	
  *  The ring  resp. its skew field of fractions  have centres 
	
    ()   = _[[(1+X)^(:)-1]] _(), 
    ()   =(_[[(1+X)^(:)-1]]),

	where _ denotes the ring of integers of .
	
  *  The ring  is the unique maximal _()-order in .
	
  *  There is an embedding 
	
    Φ:   → M_(:) ((_(ω)[[(1+X)^(:)-1]])) 
    of (_[[(1+X)^(:)-1]])-algebras, sending an element α∈ resp. 1+X toα   ↦[          ϕ(α);                     τ(ϕ(α));                           ⋱;               τ^(:)-1(ϕ(α)) ]
    
    	1+X    ↦[                      1_;                       ⋱;                      1_; (1+X)^(:)1_ ]

	where ϕ is the map from (<ref>), and 1_ denotes the × identity matrix.


We refer to <cit.> for the proof.

In this section, we study matrices over the skew field . The main result is <ref>, which allows one to do dimension reduction: by this we mean that an invertible matrix with entries in a maximal order can be replaced by a smaller matrix with the same Dieudonné determinant. This is a generalisation of a result of Nichifor and Palvannan, see <cit.>, and our proof indeed follows the outline of theirs.

 
	Let m≥1, n≥1, and
	
    A ∈ M_m( M_n( ) ) ∩_m ( M_n( ) ).

	Then there exists some C∈ M_n()∩_n() such that A= C where  is the Dieudonné determinant.



	An equivalent formulation of <ref> would be to state it for n=1 instead of for all n≥1. In fact, the proof goes by showing it for n=1, and then creating an n× n matrix by taking this as the (1,1)-entry, filling up the rest of the diagonal with ones, and setting all off-diagonal entries to be zero. The 2.13 is stated this way because this is the form in which it will be used in the proof of <ref>.



	We begin with the following observation:
	
		The reduced norm of A is integral: 
		
    _M_mn()/() A∈_().

	
	
		Actually, even more is true: the reduced characteristic polynomial of A has coefficients in _(). This follows directly by applying <ref> with _(), = (), M_mn(), and Δ M_mn().
		These satisfy the conditions of <ref>: indeed, <ref> states that  is a maximal _()-order in , and consequently M_mn() is a maximal _()-order in M_mn() by <ref>.
	
	
	Recall that we call a ring a PID if all left ideals and all right ideals are principal.
	 
		[π_^-1] is a (noncommutative) PID.
	
	
		We will show that all left ideals of [π_^-1] are principal; the right version can be shown in the same way, as every statement we will use has a right counterpart. The proof uses Weierstraß theory of skew power series rings, for which we refer back to <ref>.
		
		Let I⊆[π_^-1] be a nontrivial left ideal. Then I∩(0): indeed, if 0 x∈ I, then π_^a x∈ I∩ for a≥0 large enough. Therefore not all reduced orders are ∞; let f∈ I∩ be an element of minimal reduced order. Choosing f so that it is in  is necessary in order to be able to apply Weierstraß preparation (<ref>). This allows us to write f uniquely as 
    f=ϵ F,
 where ϵ∈^× is a unit and F∈ is a distinguished skew polynomial. It follows that F∈ I, and (F)=(f). No such theorem is available in [π_^-1].
		
		Clearly [π_^-1]F⊆ I. We will show that the converse inclusion also holds, whence I is the principal left ideal generated by F. For this, let g∈ I be arbitrary. Weierstraß division (<ref>) carries over from  to [π_^-1] as follows. The Weierstraß division theorem for  states that there is a decomposition
		
    =  F⊕⊕_i=0^(F)-1_X^i.    
    Inverting π_, or in other words, tensoring with _[π_^-1] over _ then yields[π_^-1]    = [π_^-1] F⊕⊕_i=0^(F)-1_[π_^-1]X^i .
    With respect to this decomposition, g can be written as
    			g   =hF+r.

		Since g and F are in I, we have r∈ I, but a priori not necessarily in .
		
		We claim that r=0.  If r0, then since r has only finitely many coefficients, there is a unique a∈ such that π_^a r ∈_[X]-π__[X]. We have the following inequalities:
		
    (π_^a r)≤(π_^a r)=(r)<(F).

		But since π_^a r∈ I, this contradicts the minimality condition in our choice of f. 
		
		Now we have r=0, so g=hF. That is, g∈[π_^-1] F, as claimed.
	
	
	<ref> allows us to utilise <ref>, which tells us that A admits a diagonal reduction via elementary operations. That is, A=UBV where B∈ M_mn([π_^-1]) is a diagonal matrix, and U,V∈_mn([π_^-1]) are products of elementary, permutation, and scalar matrices.
	
	We want to use Weierstraß preparation again: for this, we need to clear denominators. Let b∈ be the minimal integer such that π_^b B∈ M_mn(). Then 
	
    (B)=(π_^-b·π_^b B)=π_^-b(π_^bB)=π_^-bβ_B J_B,
 
	where β_B∈^× is a unit and J_B∈ is a monic polynomial.
	Similarly, let u,v∈ be the minimal integers such that π_^u U, π_^v V ∈ M_mn(). Then there exist units β_U,β_V∈^× and monic polynomials J_U,J_V∈ such that 
	
    (U)=π_^-uβ_U J_U,   (V)=π_^-vβ_V J_V.
 
	We claim that J_U=J_V=1. For this, we go back to the fact that U and V are products of elementary, permutation and scalar matrices. Elementary matrices have Dieudonné determinant 1 by definition. A permutation matrix M satisfies M^2=1 and hence (M)^2=1; in particular, (M) is invertible in [π_^-1]. So the elementary and permutation factors only contribute  β-factors in (<ref>). Moreover, since U and V are invertible over [π_^-1], their scalar matrix parts may only contribute units in [π_^-1], which can be written as a power of π_ times a unit in : that is, these are accounted for in the π_- and β-factors in (<ref>). We conclude that J_U=J_V=1, as claimed.
	
	Since the Dieudonné determinant factors through the abelian group K_1(), we have
	
    (A)   = (U)(B)(V) = (β_Uβ_Bβ_V)· J_B·π_^-u-b-v=(C),

	where C=((β_Uβ_Bβ_V)· J_B·π_^-u-b-v,1,…,1) is an n× n matrix. In order to prove C∈ M_n()∩_n(), it remains to show that u+b+v≤0.
	
	In (), we have the following equality:
	
    _M_mn()/()(A)_∈_()   = _/()((A)) 
       = _/()(β_Uβ_Bβ_V)_∈_()^×·_/()(J_B)_monic·_/()(π_^-(u+b+v)).

	The first equality comes from the relationship between reduced norms and the Dieudonné determinant: for invertible matrices, _M_mn()/()=_/()∘, see (<ref>). Here we use the assumption that A is invertible. The second equality is (<ref>).
	
	On the left hand side of (<ref>), _M_mn()/()(A) is integral, i.e. contained in _() by <ref>. On the right hand side of (<ref>), the first term is a unit because β_Uβ_Bβ_V∈^×.
	The second term is a monic polynomial by the upcoming <ref> because J_B is monic. 
	
	Let π_ be a uniformiser of _. Then neither the first nor the second term is divisible by π_, so in order for the right hand side to be integral, the third term must also be integral. The π_-adic valuation of _/()(π_^-(u+b+v)) is a multiple of -(u+b+v). We conclude that u+b+v≤0. This completes the proof.


 
	Let G(X)∈=(_[[X;τ,τ-𝕀]]) be an element which is a monic polynomial in X.
	Then its reduced norm _ / () (G)
	in ()=(_[[(1+X)^(:)-1]]) is also a monic polynomial in (1+X)^(:). In particular, it is also a monic polynomial in X.


	
	We use the embedding Φ from <ref>.
	By definition of the reduced norm, we have
	
    _ / () (G) = (Φ( G)) .

	We compute the leading term of this determinant. For this, write G=∑_i=0^ Gα_i · (1+X)^i where α_i∈, G is the degree of G, and α_ G=1 as G was assumed to be monic.
	
    Φ(G) 
    		   = ∑_i=0^ G[          ϕ(α_i);                       τ(ϕ(α_i));                               ⋱;                 τ^(:)-1(ϕ(α_i)) ][                      1_;                       ⋱;                      1_; (1+X)^(:)1_ ]^i 
       = ∑_i=0^ G[          ϕ(α_i);                       τ(ϕ(α_i));                               ⋱;                 τ^(:)-1(ϕ(α_i)) ][                                              1_                        ;                                                           ⋱            ;                                                                      1_; (1+X)^(:)1_                                    ;                       ⋱                        ;                         (1+X)^(:)1_            ;             ] (1+X)^(:)j
    Here i=(:) j+ℓ and 0≤ℓ<(:). In the second block matrix inside the summation, the first (:)-ℓ rows have entries 0_ and 1_, and the last ℓ rows have entries 0_ and (1+X)^(:)1_.   = ∑_i=0^ G[                                                                                                  ϕ(α_i)                                                    ;                                                                                                                                 ⋱                          ;                                                                                                                                               τ^i-1(ϕ(α_i));     τ^i(ϕ(α_i))·(1+X)^(:)                                                                              ;                                                   ⋱                                                    ;                                                     τ^(:)-1(ϕ(α_i))·(1+X)^(:)                          ;                           ] (1+X)^(:)j

	The determinant of Φ(G) may be computed using Laplace expansion: we find that it is a polynomial in (1+X)^(:). Since α_ G=1, we have ϕ(α_ G)=1_. Therefore there is a unique highest degree term with respect to the variable (1+X)^(:) in the expansion of (Φ(G)), namely 
	
    ((1+X)^(:))^· (:) j·((1+X)^(:))^· r=((1+X)^(:))^· G,

	where G=(:) j+r and 0≤ r<(:). Hence the determinant is monic with respect to (1+X)^(:). Since (1+X)^(:) is a monic polynomial in X, the determinant is also a monic polynomial in X.




§ KNOWN CASES OF THE EQUIVARIANT P-ADIC ARTIN CONJECTURE
 


 §.§ Maximal orders
 
The ring () is semisimple artinian, and it has the following Wedderburn decomposition:

    () ≃⊕_χ∈()/∼__p M_n_χ(D_χ).

Here D_χ is a skew field, and χ runs through equivalence classes of irreducible characters with open kernel, where two characters χ∼__pχ' are equivalent if ^_Hχ=σ(^_Hχ') for some σ∈(_p,χ/_p). The group ring _p[H] is semisimple by Maschke's theorem, and it has the following Wedderburn decomposition:

    _p[H]=⊕_η∈(H)/∼__p M_n_η(D_η).

Here the equivalence relation ∼__p is given as follows: η∼__pη' if η=^ση for some σ∈(_p(η)/_p). Each D_η is a finite dimensional skew field over its centre _p(η). In particular, the statements in <ref> apply to D_η. Let _D_η denote the unique maximal _p-order in D_η. The relationship between the skew fields occurring in the Wedderburn decompositions of () and _p[H] is described as follows. For details, we refer to <cit.>.


Let χ∈(), and let η|^_Hχ be an irreducible constituent of its restriction.

	
  * Let
	
    v_χmin{0<i≤ w_χ : ∃τ∈(_p(η):_p), ^γ^iη=^τη}.

	Then v_χ is independent of the choice of η, and v_χ| w_χ. Moreover, there is precisely one τ∈(_p(η)/_p) such that ^γ^v_χη=^τη. From now on, τ denotes this automorphism.
	
  *  The automorphism τ∈(_p(η)/_p) has fixed field _p,χ. Moreover, the Galois group (_p(η)/_p,χ) is cyclic of order w_χ/v_χ, and τ is a generator.
	
  * The automorphism τ∈(_p(η)/_p,χ) admits a unique extension as an automorphism of D_η of order w_χ/v_χ, which we will also denote by τ.


 
	Let D^*_η=D_η^⟨τ^e⟩ denote the skew field fixed by τ^e inside D_η. 
	Restricting τ from D_η to the subfield _p(η)(ω), let  denote the image of τ|__p(η)(ω) under the natural map (_p(η)(ω)/_p,χ)↠(_p(η)(ω)^⟨τ^e⟩/_p,χ), as well as its unique extension to D^*_η as an automorphism of order e. Then the skew field D_χ is isomorphic to
	
    D_χ   ≃(_D^*_η[[X; , -𝕀]]), 
    and its centre is(D_χ)   ≃(__p,χ[[(1+X)^e-1]]).



To ease notation, we will write D^*_η for the skew field D_η^⟨τ^e⟩, and W^* resp. Z^* for its inertia subfield resp. centre. Then W^*=Z^*(ω^*), and the multiplication rule in D^*_η is given by π^* ω^* = σ^*(ω^*)π^*.
Let Σ_χ be the preimage of _D^*_η[[X; , -𝕀]] in D_χ under the isomorphism (<ref>), and let _(D_χ) denote the preimage of __p,χ[[]] under the isomorphism (<ref>), where we write (1+X)^e-1 for brevity.
 	
	One can also determine the size n_χ of the matrix ring, as well as the Schur index of D_χ in terms of the corresponding data for _p[H], but this is not needed to address the equivariant p-adic Artin conjecture.


The results of <ref> are applicable with Z^*, _p,χ, and D^*_η. Indeed, the extension is cyclic due to <ref>, and the condition on the Schur index follows from a theorem of Witt <cit.>: this shows that the Schur index of D^*_η divides p-1, which in turn divides #_p,χ-1.

 
	Let L/K be a finite Galois extension of number fields such that L is a CM number field containing ζ_p, with =(L^+_∞/K). Let S and T be finite non-empty disjoint sets of places of K, with S containing all places ramifying in L^+_∞/K and all infinite places.
	
		
  *  Let 𝔐() be the preimage of 
		
    ⊕_χ∈()/∼__p M_n_χ(Σ_χ)
 
		under the Wedderburn isomorphism (<ref>). This is a maximal Λ(Γ_0)-order in (). 
		
  *  Assume <ref> without uniqueness, and let ζ_S^T∈ K_1(()) such that ζ_S^T=Φ_S^T and ∂(ζ_S^T)=[Y_S^T]. Then ζ_S^T is in the image of the natural map 𝔐()∩()^×→ K_1(()).
		In particular, <ref> holds with 𝔐=𝔐().
	



	This can be seen as (a special case of) an equivariant version of Greenberg's result that the main conjecture implies the p-adic Artin conjecture (<ref>).


	Nichifor and Palvannan's proof in the direct product case, see <cit.>, can be modified to work in our setup, with their Proposition 2.13 replaced by our <ref>. We now explain this in more detail.
	
	First we prove <ref>. According to <ref>, for each χ we have that _D^*_η[[X; , -𝕀]] is a maximal __p,χ[[(1+X)^e-1]]-order in the skew field (_D^*_η[[X; , -𝕀]]). <Ref> then shows that Σ_χ is a maximal _(D_χ)-order in D_χ. Therefore for all χ, the matrix ring M_n_χ(Σ_χ) is a maximal _(D_χ)-order in M_n_χ(D_χ) by <ref>. 
	The module _(D_χ) is finitely generated over Λ(Γ_0). It follows that M_n_χ(Σ_χ) is also a maximal Λ(Γ_0)-order in M_n_χ(D_χ).
	Finally, 𝔐() is a maximal order in () by <ref>, which establishes <ref>.
	
	For the proof of <ref>, recall from (<ref>) that there is a free resolution Λ()^m αΛ()^m↠ Y_S^T where 
	
    α∈ M_m(Λ())∩_m(()).
 
	In particular, α defines a class [α]∈ K_1(()). <Ref>, the validity of which has been assumed, provides the following equality in K_1(()):
	
    ∂(ζ_S^T)=∂([α]).

	Recall the localisation exact sequence of K-theory from (<ref>):
	
    K_1(Λ())→ K_1(())K_0(Λ(),()).

	Using this exact sequence, there exist some n≥1 and β∈_n(Λ()) such that the class [β]∈ K_1(Λ()) satisfies
	
    ζ_S^T=[α]·[β].

	
	Recall from <ref> that the Dieudonné determinant is defined on the K_1-groups of Λ() and (). According to a result of Vaserstein, see (<ref>), it is actually an isomorphism onto the abelianisation of the units of Λ() resp. ():
	
    : K_1(Λ())   (Λ()^×)^
    : K_1(     ())   (     ()^×)^
 
	Since [β]∈ K_1(Λ()), it follows that β has a representative in Λ(). This, together with the two isomorphisms, reduces showing integrality of ζ_S^T to showing integrality of α. That is, it remains to show that (α) is in the image of the map
	
    𝔐()∩()^×→( ()^×)^.

	This can be done Wedderburn component-wise: that is, we need to show that for every mn_χ× mn_χ matrix
	
    α_χ∈ M_m(M_n_χ(Σ_χ))∩_m(M_n_χ(D_χ))

	there exists an n_χ× n_χ matrix
	
    A_χ∈ M_n_χ(Σ_χ)∩_n_χ(D_χ)

	such that their Dieudonné determinants agree:
	
    α_χ= A_χ.

	This is the statement of <ref>. The last assertion in <ref> follows from <ref>.<ref>.


 
	Retain the setting of <ref>.
	Then every maximal Λ(Γ_0)-order in () is of the form
	
    𝔐_𝐮()⊕_χ∈()/∼__p u_χ M_n_χ( Σ_χ) u_χ^-1,

	where 𝐮=(u_χ)_χ with u_χ∈_n_χ(D_χ). Moreover, assuming <ref>, ζ_S^T is in the image of the natural map 𝔐_𝐮()∩()^×→ K_1(()).
	In particular, <ref> holds for every maximal Λ(Γ_0)-order in (), assuming <ref>. 


	The first assertion is a consequence of <ref> and basic properties of maximal orders, see <cit.>.
	
	<Ref> states that there is a 𝐳=(z_χ)_χ∈𝔐()∩()^× such that [(𝐳)]=ζ^T_S∈ K_1(()). Clearing denominators, we write u_χ=a^-1_χ U_χ where a_χ∈Σ_χ and U_χ∈ M_n_χ(Σ_χ)∩_n_χ(D_χ). Applying dimension reduction (<ref>), we find a b_χ∈Σ_χ∩ D_χ^× such that (b_χ)=(U_χ). It follows that in K_1(()), we have
	
    ζ^T_S=[(𝐳)]=[((z_χ))_χ]=[((a_χ^-1 b_χ z_χ b^-1_χ a_χ))_χ].

	This shows the second assertion. The third one follows from <ref>.<ref>.



	Suppose that L^+_∞/K satisfies the μ=0 hypothesis (i.e. X_S is finitely generated as a _p-module). Then <ref> holds for every maximal Λ(Γ_0)-order in ().


	In the μ=0 case, the main conjecture is due to Ritter–Weiss <cit.> and Kakde <cit.>. Using <ref>, this implies that <ref> holds, and therefore the conditions of <ref> are met.




 §.§ p-abelian extensions
 
In this section, ' denotes the commutator subgroup of . Since /H is abelian, the commutator subgroup is always contained in H, hence it is finite.

	 is called p-abelian if its commutator subgroup ' has prime-to-p order.

In particular,  is p-abelian whenever H has order coprime to p.

 
	The following hold:
	
		
  *   is p-abelian if and only if Λ()≃⊕_j M_n_j(R_j), where the R_j are complete commutative local rings.
		
  *  If  has an abelian p-Sylow subgroup, then <ref> holds.
	

 
	If  is p-abelian, then  has an abelian p-Sylow subgroup. Indeed, a p-Sylow subgroup of a profinite group is—by definition—a closed pro-p subgroup whose index is coprime to p, and by Sylow's theorem, these are all conjugate. If  is p-abelian, then for every p-Sylow subgroup P⊆ we have '∩ P=1; since P'≤'∩ P, this means that P is abelian.  
	
	The converse does not hold.
	If some p-Sylow subgroup P⊆ is abelian, then all p-Sylow subgroups are abelian, because they are conjugate to P by Sylow's theorem. We still have P'≤'∩ P, but this need not be an equality.
	
	As an explicit counterexample, consider ≃𝔖_p ×Γ, where 𝔖_p is the symmetric group of degree p. The symmetric group has order p!, hence all p-Sylows of 𝔖_p are of order p, so in particular abelian. Every p-Sylow of  is a direct product of a p-Sylow of 𝔖_p with Γ, hence abelian. Since p2, the commutator subgroup is the alternating group '≃𝔖_p'≃ A_p (see <cit.>), whose order p!/2 is divisible by p.


In particular, <ref>.<ref> combined with <ref> shows that the smoothed main conjecture (<ref>) holds whenever  has an abelian p-Sylow subgroup, so in particular also for p-abelian extensions. As a direct consequence of <ref>, we obtain the following:


	Suppose that L^+_∞/K has an abelian p-Sylow subgroup. Then <ref> holds for every maximal Λ(Γ_0)-order in ().

In the p-abelian case, <ref>.<ref> allows for an even stronger integrality statement.
 
	If  is p-abelian, then Φ_S^T∈(Λ( )). Moreover, <ref> holds with 𝔐=Λ( ).


	As in the proof of <ref>, using <ref> (which holds for p-abelian extensions), we have
	
    ζ_S^T    = [α]·[β], 
    where α∈ M_m(Λ())∩_m(()) and β∈_n(Λ()). Replacing α by αβ, and potentially enlarging m in case n>m, applying the reduced norm map on K_1(()) to (<ref>) yieldsΦ_S^T=ζ_S^T    =  [α] = _M_n(())/(())α,

	where the first equality is again due to <ref>.
	
	Let us now study the reduced norm map on M_n(Λ()) in general. This is the restriction of the reduced norm map M_n(())→(()). Recall that () is semisimple while Λ() is not, therefore a priori, the reduced norm on M_n(Λ()) takes values in (()).
	However, using the description of Λ() from <ref>, we can see that in the p-abelian case, the reduced norm actually maps to (Λ()). Indeed, the following diagram commutes (without the dashed arrow), and thus the left vertical arrow induces the dashed arrow.
	
    M_m(Λ()) [rr, "∼"] [rd, "", dashed] [d,""]       ⊕_j M_mn_j(R_j) [d, "⊕_j"] 
    (())    (Λ()) [l,hook'] [r, "∼"]    ⊕_j R_j

	We conclude that the reduced norm of α on the right hand side of (<ref>) is in (Λ()), and hence the same is true for Φ_S^T. The first assertion is now proven.
	
	Let z∈(Λ()) be an arbitrary central element, and let m 1 in the diagram above. Let z_j∈ R_j be the image of z in R_j under the bottom horizontal map. Let Z̃ be the tuple in ⊕_j M_n_j(R_j) whose jth component is (z_j,1,…,1). (Note that this element is not central unless  n_j=1 for all indices j.) 
	Then the image of Z̃ under ⊕_j is (z_j)_j.
	Let z̃∈Λ() be the element corresponding to Z̃ under the top horizontal isomorphism. So in the diagram above, we have
	
    z̃[r,mapsto] [d,mapsto,dashed,""]    Z̃[d,mapsto,"⊕_j"] 
    
    		z [r,mapsto]     (z_j)_j

	In particular, we have (z̃)=z. Setting zΦ_S^T, it follows that Φ_S^T is in the image of
	
    Λ()∩()^×→ K_1(())(())^×.

	This finishes the proof.



	The `if' part of <ref> shows that when  is not p-abelian, the reduced norm won't just be a product of determinants, and the argument above does not generalise.



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