On trivial zeroes of Euler systems for \mathbb{G}_m
aa r X i v : . [ m a t h . N T ] F e b On trivial zeroes of Euler systems for G m Dominik Bullach Martin Hofer
Iwasawa-theoretic descent calculations are particularly delicate in the pres-ence of additional totally split primes, a situation commonly referred to as thecase of trivial zeroes . We generalise a conjecture of Mazur-Rubin and Sano thatcontains the essential information needed in such a scenario to arbitrary higher-rank Euler systems for G m , and provide unconditional evidence for its validity.This includes a proof of the conjecture for the Euler system of elliptic units,which allows us to derive new cases of the p -part of the equivariant Tamagawanumber conjecture for abelian extensions of an imaginary quadratic field. Contents
Mathematics Subject Classification.
Primary: 11R42; Secondary: 11R23, 11R29. Introduction
The equivariant Tamagawa Number Conjecture (eTNC for short) as formulated by Burnsand Flach [BF01] (building on earlier work of Kato [Kat93a], [Kat93b] and, independently,Fontaine and Perrin-Riou [FPR94]) is an equivariant refinement of the seminal TamagawaNumber Conjecture of Bloch and Kato [BK90]. It both unifies and refines a great varietyof conjectures related to special values of motivic L -functions such as Stark’s conjectures,the Birch and Swinnerton-Dyer conjecture, and the central conjectures of classical Galoismodule theory (see [Bur07], [Kin11], [Bur01] for more details).The idea of deducing cases of the eTNC from a variant of the Iwasawa Main Conjecturealready appears in the original article of Bloch and Kato [BK90]. However, the neces-sary descent calculations are particularly involved in cases where the associated p -adic L -function posseses so-called trivial zeroes . To handle such cases, Burns and Greither[BG03] developed a descent machinery in their proof of the eTNC for the Tate motive( h (Spec K ) , Z [ ][Gal( K | Q )]), where K denotes an absolutely abelian field (the 2-partwas later resolved by Flach [Fla11]). This formalism uses the vanishing of certain Iwasawa µ -invariants, the known validity of the Gross-Kuz’min conjecture in this setting, and aresult of Solomon [Sol92] as crucial ingredients. Bley [Ble06] later proved partial resultsfor K an abelian extension of an imaginary quadratic field using the same strategy andan analogue [Ble04] for elliptic units of Solomon’s result.In [BKS17] Burns, Kurihara and Sano showed that an Iwasawa-theoretic version of a con-jecture proposed by Mazur-Rubin [MR16] and, independently, Sano [San14] constitutesan appropriate conjectural generalisation of the aforementioned result of Solomon’s andtherefore allows one to extend the Burns-Greither descent formalism to provide a gen-eral strategy for proving eTNC( h (Spec K ) , Z [Gal( K | k )]), where K | k is a finite abelianextension of number fields. Overview of results
In the present article we first provide a natural generalisation of theIwasawa-theoretic Mazur-Rubin-Sano conjecture to arbitrary Euler systems for G m andgive an equivalent formulation of the conjecture that is both more explicit and closer inspirit to Solomon’s original result on the valuation of cyclotomic p -units. We then provethe following (see Theorems (3.12) and (5.1) for precise versions). Theorem A.
The Iwasawa-theoretic Mazur-Rubin-Sano conjecture (3.10) holds for(a) any Euler system that is basic in the sense of Burns, Daoud, Sano and Seo [Bur+19],(b) the Euler system of elliptic units.We would like to point out that Theorem A (a) is essentially due to Burns, Kurihara andSano since the key arguments in its proof already appear in [BKS17]. We moreover remarkthat an equivalent formulation of a conjecture due to Coleman asserts that essentially allEuler systems are basic (see [Bur+19], [Dao20]). Theorem A therefore already providesstrong, and unconditional, evidence in favour of our conjecture.It is worth noting that part (b) of Theorem A also includes the technically difficult cases p = 2 and p being non-split in the imaginary quadratic base field. This was not possibleonly using techniques previously employed for the absolute abelian situation but is achievedby building on new ideas from [BH20]. 2 Introduction
We will then use part (b) of Theorem A and the formalism of Burns, Kurihara and Sano[BKS17] to prove new cases of the p -part of the eTNC (see Theorem (5.9)). Theorem B.
Let p be a prime number, k an imaginary quadratic field, and K | k a finiteabelian extension.(a) If p is split in k , then eTNC( h (Spec K ) , Z p [Gal( K | k )]) holds.(b) If p is not split in k , then eTNC( h (Spec K ) , Z p [Gal( K | k )]) is implied by the higher-rank equivariant Iwasawa Main Conjecture formulated by Burns, Kurihara and Sano(see Conjecture (5.6) and Remark (5.7) for a precise statement).In particular, Theorem B combines with a result of Bley [Ble98] on the Strong StarkConjecture to imply that eTNC( h (Spec( K )) , Z [Gal( K | k )]) holds if every prime factor of[ K : k ] is split in k (see Corollary (5.10) for more details).The first part of Theorem B generalises work of Bley [Ble06] which only covers primenumbers p > k . The second part of TheoremB grew out of the second presently named author’s thesis [Hof18] and settles the descentproblem in this previously widely open case. In addition, we expect that the programme onan equivariant theory of higher-rank Euler systems recently initiated by Burns, Sakamotoand Sano ([BS19a], [BSS19a], [BSS19b] and [BSS19c]) will soon lead to general results onthe equivariant Iwasawa Main Conjecture. We have therefore decided to not address thispoint here but instead leave it for later treatment.The proof of Theorem B (b) also requires the validity of an appropriate analogue of theGross-Kuz’min conjecture which is labelled ‘Condition (F)’ in [BKS17]. To this end,we prove the following general result that seems to not have previously appeared in theliterature (see Theorems (4.11) and (4.13) for the full statements). Theorem C.
Let K | k be an abelian extension of number fields and let p be a primenumber. If p = 2, assume that k is totally imaginary.(a) Let χ be a non-trivial character on Gal( K | k ) and let k ∞ | k be a Z p -extension. If p splits completely in k | Q and there is at most one finite place v of k that ramifiesin k ∞ | k and is such that χ ( v ) = 1, then the χ -part of condition (F) holds for the Z p -extension k ∞ · K of K .(b) If k is imaginary quadratic and p does not split in k | Q , then there are infinitely many Z p -extensions of k such that condition (F) holds for the Z p -extension k ∞ · K of K . Acknowledgements
The authors would like to extend their gratitude to Werner Bley,David Burns, Alexandre Daoud, S¨oren Kleine, Takamichi Sano and Pascal Stucky for manyilluminating conversations and helpful comments on earlier versions of this manuscript.
Notation
Arithmetic.
For any number field E we write S ∞ ( E ) for the set of archimedeanplaces of E , and S p ( E ) for the set of places of E lying above a rational prime p . Given anextension F | E we write S ram ( F | E ) for the places of E that ramify in F and S split ( F | E )for the places of E that split completely in F . If S is a set of places of E , we denote by S F the set of places of F that lie above those contained in S . We will however omit the3 Introduction explicit reference to the field in case it is clear from the context. For example, O F,S shalldenote the ring of S F -integers of F , and U F,S = Z p ⊗ Z O × F,S the p -completion of its units.We also define Y F,S to be the free abelian group on S F and set X F,S = n X w ∈ S F a w w ∈ Y F,S | X w ∈ S F a w = 0 o . Furthermore, if T is a finite set of finite places disjoint from S , then we let A S,T ( F ) bethe p -part of the S F -ray class group mod T F , i.e. the p -Sylow subgroup of the quotient ofthe group of fractional ideals of O F,S coprime to T F by the subgroup of principal idealswith a generator congruent to 1 modulo all w ∈ T F . If S = S ∞ ( E ) or T = ∅ , then we willsuppress the respective set in the notation.For any place w of F we write ord w : F × → Z for the normalised valuation at w . In caseof a finite extension H of Q p we also write ord H for the normalised valuation on H . If F | E is Galois and v unramified in F | E , then we let Frob v ∈ Gal( F | E ) be the arithmeticFrobenius at v . Algebra.
For an abelian group A we denote by A tor its torsion-subgroup and by A tf = A/A tor its torsion-free part. If there is no confusion possible, we often shorten the functor( − ) ⊗ Z A to just ( − ) · A (or even ( − ) A ) and, if A is also a Z p -module, similarly for thefunctor ( − ) ⊗ Z p A . If A is finite, we denote by b A = Hom Z ( A, C × ) its character group,and for any χ ∈ b A we let e χ = 1 | A | X σ ∈ A χ ( σ ) σ − ∈ C [ A ]be the usual primitive orthogonal idempotent associated to χ . Furthermore, N A = P σ ∈ A σ ∈ Z [ A ] denotes the norm element of A .If R is a commutative Noetherian ring, then for any R -module M we write M ∗ =Hom R ( M, R ) for its dual. Let r ≥ r -th exterior bidual of M is defined as \ rR M = (cid:16)^ rR M ∗ (cid:17) ∗ . If R = Z [ A ] for a finite abelian group A , then the exterior bidual coincides with the latticefirst introduced by Rubin in [Rub96, § §
2] for anoverview. At this point we only remark that for r ≥ f ∈ M ∗ induces a map \ rR M → \ r − R M which, by abuse of notation, will also be denoted by f , and is defined as the dual of ^ r − R M ∗ → ^ rR M ∗ , g f ∧ g. Iterating this construction gives, for any s ≤ r , a homomorphism ^ sR M ∗ → Hom R (cid:0) \ rR M, \ r − sR M (cid:1) , f ∧ · · · ∧ f s f s ◦ · · · ◦ f . (1)Finally, we write Q ( R ) for the total ring of fractions of R .4 Euler systems
Fix a number field k and a rational prime p . We suppose to be given an abelian extension K of k that contains a Z p -extension k ∞ of k in which only finitely many finite places splitcompletely. Let Ω( K| k ) be the set of non-trivial finite extensions of k that are containedin K and write G E = Gal( E | k ) for every E ∈ Ω( K| k ).Fix a finite set S of places of k containing S ∞ ( k ) and let S ( E ) = S ∪ S ram ( E | k ) for every E ∈ Ω( K| k ). (2.1) Definition. Let r ≥ p -adic Euler system of rank r for ( K| k, S ) is a collection η = ( η E ) E ∈ Y E ∈ Ω( K| k ) Q p ^ r Z [ G E ] O × E,S ( E ) with the property that for any E, F ∈ Ω( K| k ) with E ⊆ F we haveN rF | E ( η F ) = (cid:16) Y v ∈ S ( F ) \ S ( E ) (1 − Frob − v ) (cid:17) η E in Q p ^ r Z [ G E ] O × E,S ( E ) , where N rF | E denotes the map Q p V r Z [ G F ] O × F,S ( F ) → Q p V r Z [ G E ] O × E,S ( E ) induced by the normmap N F | E : F × → E × . The set of all p -adic Euler systems of rank r for the data ( K| k, S )will be denoted as ES r ( K| k, S ). (2.2) Remark. We caution the reader that the notion of p -adic Euler system introducedabove differs from that in Rubin’s book [Rub00] or the articles [BS19a], [BSS19a] and[Dao20] since there the set S is assumed to always contain S p . This prevents Euler factorsat p from appearing in the norm relation which cause the “trivial zeroes phenomenon” weshall be concerned with in this article.One would hope that in applications the values u E of a rational Euler system u are in factcontained in a canonical sublattice of Q p V r Z [ G E ] O × E,S ( E ) . To address such questions thenotion of exterior bidual first introduced by Rubin [Rub96] has proven to be an adequatechoice. Furthermore, it is customary in this context to introduce an auxiliary set of places T that is disjoint from S ( E ) and such that the group of ( S ( E ) E , T ) E -units O × E,S ( E ) ,T = ker n O × E,S ( E ) → M w ∈ T E (cid:0) O E,S ( E ) (cid:30) w (cid:1) × o . is Z -torsion free. In fact, it is sufficient for this purpose to let T = { v } for a place v S ( E )that does not divide | µ E | , where µ E ⊆ E × is the group of roots of unity contained in E .We denote the p -completion Z p O × E,S ( E ) ,T of O × E,S ( E ) ,T by U E,S ( E ) ,T . Iwasawa theory
For any E ∈ Ω( K| k ) we define a Z p -extension of E by E ∞ = E · k ∞ , andwrite E n for the n -th layer of E ∞ | E . We further set Γ E = Gal( E ∞ | E ), Γ E,n = Gal( E n | E ),Γ nE = Gal( E ∞ | E n ), and V E = lim ←− n Z p [ G E n ].Fix a field K ∈ Ω( K| k ) and assume that no finite place contained in S ( K ) splits completelyin k ∞ | k . 5 Euler systems: Set-up
Moreover, we introduce the following notation: • Σ = S ∪ S ram ( K ∞ | k ), • V = S split ( K ∞ | k ), which is contained in S ∞ ( k ) ( Σ by our assumptions, and r itscardinality, • V ′ ⊆ S split ( K | k ) a subset of Σ of size r ′ containing V , • e = r ′ − r the cardinality of W = V ′ \ V .Finally, we assume that K is the maximal abelian extension of k in which all placescontained in V split completely.Observe that we have a (non-canonical) splitting Gal( K ∞ | k ) ∼ = G L × Γ L for some subfield L of K ∞ | k . (2.3) Lemma. The following hold:(a) The fields K n and L n agree for n big enough (up to an off-set in the numbering).(b) Let γ K ∈ Γ K be a topological generator, then the element γ K − V K . Proof.
By definition we have K ∞ = S n ≥ L n , hence there is n such that K ⊆ L n . Thatis, L n is an intermediate field of the Z p -extension K ∞ | K and therefore must agree with K m for some m . To prove (b) we fix a topological generator γ L ∈ Γ L . Then there is aunit a ∈ Z × p such that γ ap n L = γ p m K . The element γ ap n L − V K = Z p J Γ L K [ G L ]. It then follows from γ ap n L − γ p m K − γ K − · (1 + γ K + · · · + γ p m − K )that γ K − K ∞ | k . More precisely, our objects of interest are the following. (2.4) Definition. We define ES r ( K| k, K ∞ , S, T ) to be the subset of ES r ( K| k, S ) compris-ing all elements ( η E ) E such that η E ∈ \ r Z p [ G E ] U E,S ( E ) ,T whenever E ⊆ K ∞ .Assume that T is chosen in a way such that U E, Σ ,T is Z p -torsion free for every E ∈ Ω( K| k )contained in K ∞ . We remark that any non-empty set disjoint from Σ will in fact meetthis condition (see, for example, [Dao20, Lem. B.6]).We will give examples of arithmetically significant p -adic Euler systems that belong toES r ( K| k, K ∞ , S, T ) in § § Euler systems: The Euler system of Rubin-Stark elements
Let E ∈ Ω( K| k ) be a field and M ⊇ S ( E ) a finite set of places of k that is disjoint froma second finite set of places Z . Assume that V is a proper subset of M and choose anordering M = { v , . . . , v m } such that V = { v , . . . , v r } . For every i ∈ { , . . . , m } fix aplace v i of the algebraic closure Q of Q that extends v i and write w E,i for the place of E induced by v i .The Dirichlet regulator map λ E,M : O × E,M → R X E,M , a
7→ − X w ∈ M E log | a | w · w (2)then induces, by fixing an isomorphism C ∼ = C p , an isomorphism C p ^ r Z [ G E ] O × E,M ≃ −→ C p ^ r Z [ G E ] X E,M (3)that will also be denoted as λ E,M . If χ ∈ c G E , we moreover define the M -truncated Z -modified Artin L -function as L M,Z ( χ, s ) = Y v ∈ Z (1 − χ (Frob v )N v − s ) · Y v M (1 − χ (Frob v )N v − s ) − , where s is a complex number of real part strictly greater than 1. It is well-known thatthis defines a function on the complex plane by meromorphic continuation. By [Tat84,Ch. I, Prop. 3.4], the existence of the set V ( M then implies that the order of vanishingof L M,Z ( χ, s ) at s = 0 is at least r . This allows us to define the r -th order Stickelbergerelement as θ ( r ) E | k,M,Z (0) = X χ ∈ c G E e χ · lim s → s − r L M,Z ( χ, s ) ∈ R [ G E ]which we view as an element of C p [ G E ]. (2.5) Definition. The r -th order Rubin-Stark element ε VE | k,M,Z ∈ C p V r Z [ G E ] O × E,M isdefined to be the preimage of the element θ ( r ) E | k,M,Z (0) · V ≤ i ≤ r ( w E,i − w E, ) under theisomorphism (3) induced by the Dirichlet regulator map λ E,M .We remark that this definition does not depend on the choice of element v ∈ M \ V (see[San15, Prop. 3.13]).The p -part of Stark’s conjecture, in the formulation [Rub96, Conj. A ′ ], then predicts that ε VE | k,M,Z ∈ Q p V r Z [ G E ] O × E,M . Taking Z = ∅ and assuming r < | S ( E ) | for all E ∈ Ω( K| k ),[Rub96, Proposition 6.1] (see also [San14, Prop. 3.5]) implies that in this situation thecollection of elements { ε VE | k,S ( E ) , ∅ } E ∈ Ω( K| k ) constitutes a p -adic Euler system in the senseof Definition (2.1).Next we will explain how a slight modification of this Euler system yields (at least con-jecturally) an element of ES r ( K| k, K ∞ , S, T ).7 Euler systems: The Euler system of Rubin-Stark elements T -modification For every E ∈ Ω( K| k ) define an ideal A E | k ⊆ Z p [ G E ] by A E | k = Ann Z p [ G E ] ( µ E )where µ E ⊆ E × denotes the group of roots of unity contained in E . (2.6) Lemma. The following claims are valid:(a) If M is a finite set of places of k such that S ( E ) ∪ S p ( k ) ⊆ M , then A E | k is generatedas a Z p -module by { − N v · Frob − v | v M } .(b) The ideals A k := lim ←− E ∈ Ω( K| k ) A E | k ⊆ Z p J Gal( K| k ) K and A K ∞ := lim ←− n A K n | k ⊆ V are well-defined and the natural projection map A k → A K ∞ is surjective. Proof.
For a proof of (a) see [Tat84, Ch. IV, Lem. 1.1]. Now, (a) implies that, for E ⊆ F ,the natural restricition map Z p [ G F ] → Z p [ G E ] sends A E | k onto A F | k . In particular, boththe ideals A k and A K ∞ are well-defined.To prove the second claim of (b), we first note that each ideal A E | k is a finitely generated Z p -module, hence is endowed with the structure of a compact Hausdorff space. It followsthat A k is compact Hausdorff as well. Taking the limit over n of the surjective projectivemaps A k → A K n | k therefore gives a surjective map A k → A K ∞ , as claimed.For the next result we resume the notation and hypotheses introduced at the end of § ′ ] predicts that ε VE | k,S ( E ) ,T ∈ \ r Z p [ G E ] U E,S ( E ) ,T for any E ∈ Ω( K| k ) contained in K ∞ . (2.7) Corollary. Assume that the p -part of Stark’s Conjecture [Rub96, Conj. A ′ ] for thedata ( E | k, S ( E ) , V ) holds for all fields E ∈ Ω( K| k ) and, in addition, that the p -part of theRubin-Stark Conjecture [Rub96, Conj. B ′ ] holds for the data ( E | k, S ( E ) , V, T ) whenever E ⊆ K ∞ . Then there is an Euler system ( η E ) E ∈ ES r ( K| k, K ∞ , S, T ) such that η E = ε VE | k,S ( E ) ,T for all E ⊆ K ∞ . Proof.
Lemma (2.6) implies that there is ( δ E ) E ∈ A k such that δ E = Q v ∈ T (1 − N v · Frob − v )whenever E ⊆ K ∞ . Thus, the Euler system ( η E ) E given by η E = δ E · ε VE | k,S ( E ) for all E ∈ Ω( K| k )satisfies η E = ε VE | k,S ( E ) ,T if E ⊆ K ∞ , and hence has the required properties. (2.8) Examples. (a) ( cyclotomic units ) Take k = Q , S = S ∞ ( Q ), K the maximalreal abelian extension of Q , and V = S ∞ ( Q ) = { v } . Then the assumptions ofCorollary (2.7) are satisfied, and for any E ∈ Ω( K| k ) we have ε VE | Q ,S ( E ) , ∅ = 12 ⊗ N Q ( ξ m ) | E (1 − ξ m ) ∈ Q p ⊗ Z O × E,S ( E ) , Euler systems: Modifed ´etale cohomology complexes where m = m E is the conductor of E and ξ m = ι − ( e πi/m ) for the embedding ι : Q ֒ → C corresponding to the choice of place v fixed at the beginning of thesection. (See [Tat84, Ch. IV, §
5] for a proof.)(b) (
Stickelberger elements ) Let k be a totally real field, S = S ∞ ( k ), K the maximalabelian CM extension of k , and V = ∅ . In this setting the assumptions of Corol-lary (2.7) hold true due to the work of Deligne-Ribet [DR80] and the Rubin-Starkelement is given by ε VE | k,S ( E ) , ∅ = θ E | k,S ( E ) , ∅ (0) . (c) ( elliptic units ) Let k be an imaginary quadratic field and f ⊆ O k a non-zero idealsuch that O × k → ( O k / f ) × is injective. Take S = S ∞ ( k ) ∪ { q | f } , K the maximalabelian extension of k , and V = S ∞ ( k ) = { v } . Then the Rubin-Stark Conjectureholds for all fields E ∈ Ω( K| k ) satisfying | S ( E ) | > f ), see, for example, [Tat84, Ch. IV, Prop. 3.9].To describe the Rubin-Stark element in this setting, we fix such a field E and write m = m E for its conductor. Let k ( fm ) be the ray class field of k modulo fm andchoose an auxiliary prime ideal a ( O k coprime to 6 fm . Using the elliptic function ψ introduced by Robert [Rob92] we set ψ fm , a = ι − ( ψ (1; fm , a )) ∈ O × k ( fm ) ,S ( k ( fm )) for the embedding ι : Q ֒ → C corresponding to v , where ψ (1; fm , a ) is a commonshort hand for what would be ψ (1; fm , a − fm ) in Robert’s original notation. It thenfollows from Kronecker’s second limit formula, e.g. [Fla09, Lem. 2.2 e)], that ε VE | k,S ( E ) , ∅ = (Frob a − N a ) − · N k ( fm ) | E ( ψ (1; fm , a )) ∈ Q p ⊗ Z O × E,S ( E ) , where we have used that (Frob a − N a ) is invertible in Q p [ G E ]. This definition doesnot depend on the choice of a . For any field E ∈ Ω( K| k ) and finite set of places M ⊇ S ( E ) that is disjoint from a secondfinite set of places Z , Burns-Kurihara-Sano have constructed [BKS16, Prop. 2.4] a canon-ical Z -modified, compactly supported Weil-´etale cohomology complex RΓ c,Z (( O E,M ) W , Z )of the constant sheaf Z on the ´etale site of Spec O E,M . In the sequel we shall need thecomplex C • E,M,Z = R Hom Z (RΓ c,Z (( O E,M ) W , Z ) , Z )[ − p -completion D • E,M,Z = Z p ⊗ LZ C • E,M,Z . This complex has the following properties. (2.9) Proposition.
Let E ∈ Ω( K| k ) be a field and M ⊇ S ( E ) a finite set of places of k which is disjoint from Z .(a) The complex D • E,M,Z is perfect as an element of the derived category D ( Z p [ G E ]) andacyclic outside degrees zero and one. 9 Euler systems: Modifed ´etale cohomology complexes (b) There is a canonical isomorphism H ( D • E,M,Z ) ∼ = U E,M,Z and an exact sequence0 A M,Z ( E ) H ( D • E,M,Z ) X M,U . π E (4)(c) If Z ′ is a finite set of places of k that contains Z and is disjoint from M , then thereis a canonical exact triangle D • E,M,Z ′ D • E,M,Z (cid:16) M w ∈ ( Z ′ \ Z ) E (cid:0) O E,M (cid:30) w (cid:1) × (cid:17) [0] . (5)(d) Given a finite set M ′ of places of k that contains M and is disjoint from Z , thereexists a canonical exact triangle of the form D • E,M,Z D • E,M ′ ,Z M v ∈ M ′ \ M (cid:20) Z p [ G E ] (1 − Frob − v ) −→ Z p [ G E ] (cid:21) , (6)where in the complex on the right hand side the first term is placed in degree zero.(e) For any pair of fields E, F ∈ Ω( K| k ) such that E ⊆ F and S ( F ) ⊆ M , there existsa natural isomorphism D • F,M,Z ⊗ LZ p [ G F ] Z p [ G E ] ∼ = D • E,M,Z . Proof.
This follows directly from the corresponding properties of C • E,M,Z as listed, forexample, in [Bur+19, Prop. 3.1].Recall that we have previously fixed a set T disjoint from Σ = Σ L such that U E, Σ ,T istorsion free for any E ∈ Ω( K| k ) contained in K ∞ . For Iwasawa-theoretic purposes it willprove useful to also record the corresponding properties of the complex D • K ∞ , Σ ,T = R lim ←− n D • K n , Σ ,T , (7)where the limit is taken with respect to the transition maps induced by the property inProposition (2.9) (e). Each complex D • K n , Σ ,T is a complex of compact Hausdorff spaces,hence the inverse limit functor commutes with taking cohomology and so we have a ca-nonical isomorphism H ( D • K ∞ , Σ ,T ) ∼ = U K ∞ , Σ ,T and an exact sequence0 lim ←− n A Σ ,T ( K n ) H ( D • K ∞ , Σ ,T ) lim ←− n X K n , Σ π K, ∞ (8)obtained by taking the limit of (4).Resume the notation and hypotheses introduced at the end of § § { v , . . . , v s } such that V = { v , . . . , v r } and W = { v r +1 , . . . , v r + e } . Moreover, we fix a place v i of Q extending v i and write w E,i forthe place of E ∈ Ω( K| k ) lying above v that is induced by v i .Choose a surjection pr ∞ : Π ∞ ։ H ( D • K ∞ , Σ ,T ) , Euler systems: Modifed ´etale cohomology complexes where Π ∞ is a finitely generated free V K -module of large enough rank d with basis { b , . . . , b d } , such that the compositionΠ ∞ ։ H ( D • K ∞ , Σ ,T ) π K, ∞ ։ X K ∞ , Σ := lim ←− n X K n , Σ satisfies π K, ∞ ( b i ) = ( w K n ,i − w K n , ) n ≥ for all 1 ≤ i ≤ s. (9)Write Π n = Π ∞ ⊗ V Z p [ G K n ] for each n ≥
0, then we obtain an induced surjectionpr n : Π n → H ( D • K n , Σ ,T ) and the method of [BKS16, § D • K n , Σ ,T that is of the form [ Q n → Π n ] with Q n a Z p [ G n ]-projective (hence free) module. By a standard argument in representationtheory, the Dirichlet regulator (3) induces a rational isomorphism Q p H ( D • K n , Σ ,T ) ∼ = Q p H ( D K n , Σ ,T ), therefore we may identify Q n ∼ = Π n by Swan’s theorem [CR81, Thm.(32.1)].We shall now give an explicit description of the transition maps used in (7) in terms ofthese fixed representatives [Π n φ n −→ Π n ]. This will give rise to a representative of D • K ∞ , Σ ,T that is key to our study. For this purpose, we set D • n = D • K n , Σ ,T for simplicity and write γ n for the isomorphism D • n +1 ⊗ LZ p [ G Kn +1 ] Z p [ G K n ] ∼ = D • n in the derived category D ( Z p [ G K n ]).As a morphism between perfect complexes, this map can be represented by a commutativediagram of the form0 H ( D • n +1 ⊗ L Z p [ G K n ]) Π n Π n H ( D • n +1 ⊗ L Z p [ G K n ]) 00 H ( D • n ) Π n Π n H ( D • n ) 0 . H ( γ n ) ≃ φ n +1 γ n γ n pr n +1 H ( γ n ) ≃ φ n pr n (10)Here φ n +1 and pr n +1 denote the maps induced by φ n +1 and pr n +1 , respectively. Byconstruction, pr n = H ( γ n ) ◦ pr n +1 and so exactness of the bottom line in (10) yields thatthe image of γ n − id Π n is contained in the image of φ n . Choose a map h : Π n → Π n suchthat φ n ◦ h = γ n − id Π n and set f = γ n − h ◦ φ n +1 , then h defines a chain homotopybetween ( γ n , γ n ) and ( f, id). We may therefore assume that γ n is the identity map. Giventhis, we can appeal to the Five Lemma to deduce from (10) that γ n is an isomorphism aswell. Finally, we may now pass to the limit to obtain a representative [Π ∞ φ −→ Π ∞ ] of thecomplex D • K ∞ , Σ ,T . In particular, the latter complex is perfect as an element of the derivedcategory D ( V K ) and, for each E ⊆ K ∞ , the complex D • E, Σ ,T = D • K ∞ , Σ ,T ⊗ LV Z p [ G E ] isrepresented by [Π E φ E −→ Π E ], where Π E = Π ∞ ⊗ V Z p [ G E ]. We write { b E, , . . . , b E,d } forthe basis of Π E induced by { b , . . . , b d } .In summary, the complexes D • K ∞ , Σ ,T and D • E, Σ ,T are represented by exact sequences0 U K ∞ , Σ ,T Π ∞ Π ∞ H ( D • K ∞ , Σ ,T ) 0 φ (11)and 0 U E, Σ ,T Π E Π E H ( D • E, Σ ,T ) 0 , φ E (12)respectively. We remark that these representatives satisfy the requirements of a quadraticstandard representative as defined in [BS19a, Def. A.6].11 Euler systems: Basic Euler systems
In this section we recall the construction of basic Euler systems that was carried outin [Bur+19]. These constitute a large, and arithmetically significant, submodule of themodule ES r ( K| k, K ∞ , S, T ). For this purpose we define, as the evident p -adic analogue of[Bur+19, Def. 3.6], the module of vertical determinantal systems asVS( K| k ) = lim ←− E ∈ Ω( K| k ) Det Z p [ G E ] ( D • E,S ( E ) ) , where, for any pair E, F ∈ Ω( K| k ) such that E ⊆ F , the transition map i F | E : Det Z p [ G F ] ( D • F,S ( F ) ) → Det Z p [ G E ] ( D • E,S ( E ) )is defined as the composite homomorphism of Z p [ G F ]-modulesDet Z p [ G F ] ( D • F,S ( F ) ) −→ Det Z p [ G F ] ( D • F,S ( F ) ) ⊗ Z p [ G F ] Z p [ G E ] ≃ −→ Det Z p [ G E ] ( D • F,S ( F ) ⊗ LZ p [ G F ] Z p [ G E ]) ≃ −→ Det Z p [ G E ] ( D • E,S ( F ) ) ≃ −→ Det Z p [ G E ] ( D • E,S ( E ) ) . Here the first map is the canonical projection, the second is induced by the base changeproperty of the determinant functor, the third is the isomorphism from Proposition (2.9) (d),and the last isomorphism results from the exact triangle in Proposition (2.9) (c) combinedwith the trivialisation isomorphismDet Z p [ G E ] (cid:0)h Z p [ G E ] (1 − Frob − v ) −→ Z p [ G E ] i(cid:1) ∼ = Z p [ G E ] (13)for all v ∈ S ( F ) \ S ( E ).We now assume that S contains a proper subset V ⊆ S split ( K| k ) of size r (as introducedat the end of § E ∈ Ω( K| k ) and set M ⊇ S ( E ) we define anidempotent of e ( E,M ) of Q p [ G E ] as e ( E,M ) = X χ e χ , (14)where the sum ranges over all characters χ ∈ G E such that e χ X E,M \ V vanishes. Giventhis, we have an isomorphism e ( E,M ) Q p X E,M = e ( E ) Q p Y E,V ∼ = e ( E,M ) Q p [ G E ] r . (15)If Z is a finite set of places of k disjoint from M , then we thus have a composite morphismΘ rE | k,M,Z defined asDet Q p [ G E ] ( Q p D • E,M,Z ) ≃ −→ Det Q p [ G E ] ( Q p H ( D • E,M,Z )) ⊗ Q p [ G E ] Det Q p [ G E ] ( Q p H ( D • E,M,Z )) − · e ( E,M ) −→ e ( E,M ) Q p ^ r Z p [ G E ] U E,M,Z ⊗ Q p [ G E ] e ( E,M ) Q p ^ r Z p [ G E ] X ∗ E,M ≃ −→ e ( E,M ) Q p ^ r Z p [ G E ] U E,M,Z , (16)12 Euler systems: Basic Euler systems where the first map is the passage-to-cohomology map, the second arrow is multiplicationby the idempotent e ( E,M ) , and the last map is induced by the isomorphism (15).The collection of morphisms (Θ rE | k,S ( E ) , ∅ ) E then induces a homomorphismΘ r K| k : VS( K| k ) → Y E ∈ Ω( K| k ) Q p ^ r Z p [ G E ] U E,S ( E ) . Define an element δ T,K ∞ = ( δ T,n ) of the ideal A K ∞ defined in Lemma (2.6) by δ T,n = Q v ∈ T (1 − N v · Frob − v ) ∈ Z p [ G n ] for all n ∈ N . We let B T,k be the kernel of the compositemap A k → A K ∞ → A K ∞ /δ T,K ∞ V . (2.10) Proposition. The module of T -modified basic Euler systems ES b ( K| k, S, V, T ) := B T,k · im Θ K| k , is contained in ES r ( K| k, K ∞ , S, T ). Proof.
It is proved in [Bur+19, Thm. 3.8 (i)] that Θ K| k maps into ES r ( K| k, S ), and theproof of [Bur+19, Thm. 3.8 (ii)] shows that B T,k · im Θ K| k is contained in ES r ( K| k, K ∞ , S, T ).For the convenience of the reader, we give a sketch of the arguments involved in the latterproof.Fix a field E ∈ Ω( K| k ) contained in K ∞ . The triangle (5) then implies thatDet Z p [ G E ] ( D • E,S ( E ) ) = A E | k · Det Z p [ G E ] ( D • E,S ( E ) ,T ) = δ T,E · Det Z p [ G E ] ( D • E,S ( E ) ,T ) . We have already noted that, due to our choice of T , the complex D • E,S ( E ) ,T is a quadraticstandard representative in the sense of [BS19a, Def. A.6] and so the general result [BS19a,Prop. A.11 (ii)] combines with the discussion above to imply that δ T,E · im Θ rE | k,S ( E ) , ∅ = im Θ rE | k,S ( E ) ,T ⊆ \ r Z p [ G E ] U E,S ( E ) ,T , which proves the Proposition. (2.11) Remark. (a) It follows from [Bur+19, Thm. 3.8 (iv)] that ES b ( K| k, S, V, T ) = 0if there is a field E ∈ Ω( K| k ) and a character χ ∈ G E such that L ( r ) S ( E ) ,T ( χ, = 0.Since r was chosen to be the size of V = S split ( K| k ) this condition is automaticallysatisfied.(b) We remark that the T -modification used in [Bur+19] differs from the one used hereaway from the extension K ∞ | k , and can be viewed as the combination of all possiblechoices for the set T along K ∞ | k . This is achieved by using the annihilators ofroots of unity and is done in a way that the basic Euler systems defined in loc. cit. take integral values not only on all intermediate fields of K ∞ | k but even all fields E ∈ Ω( K| k ). 13 Iwasawa-theoretic congruences
Recall that we have fixed K ∈ Ω( K| k ) such that no finite place contained in S ( K ) splitscompletely in k ∞ | k . In this section we will shorten the notation G K , Γ K , G K n , Γ K,n , V K etc. introduced in § G , Γ , G n , Γ n , V , respectively. We also use the followingnotation: I (Γ n ) = ker (cid:8) Z p [Γ n ] → Z p (cid:9) and I Γ n = ker (cid:8) Z p [ G n ] → Z p [ G ] (cid:9) . Note that I Γ n = I (Γ n ) · Z p [ G n ] and hence for any Z p [ G n ]-module M we have an isomorphism M ⊗ Z p [ G n ] I e Γ n (cid:30) I e +1Γ n ∼ = M ⊗ Z p I (Γ n ) e (cid:30) I (Γ n ) e +1 . (17)Moreover, we have an isomorphism I (Γ) := ker { Z p J Γ K → Z p } ∼ = lim ←− n I (Γ n ) . In particular, the latter ideal is generated by γ − γ ∈ Γ,and we have an isomorphism I (Γ) e (cid:30) I (Γ) e +1 ∼ = Γ , ( γ − e γ. We recall that in [BD20] the module of universal norms of rank r and level n was definedas UN rn = \ m ≥ n N rK m | K n (cid:16) \ r Z p [ G m ] U K m , Σ ,T (cid:17) ⊆ \ r Z p [ G n ] U K n , Σ ,T . Let v be a finite place of k and fix a place w of K lying above v . Consider the mapOrd v : U K, Σ ,T → Z p [ G ] , a X σ ∈G ord w ( σa ) σ − . (18) (3.1) Lemma. (a) Let q be a prime of k that is unramified but not completely split in K ∞ | K , and let r ≥ r is contained in the kernel ofOrd q : \ r Z p [ G ] U K, Σ ,T → \ r − Z p [ G ] U K, Σ ,T . (b) Let χ ∈ b G be a character and Q p ( χ ) = Q p (im χ ). Thendim Q p ( χ ) e χ Q p ( χ )UN = r S ∞ ( χ ) , where for any finite set M of places of k we set r U ( χ ) = ( |{ v ∈ M | χ ( G v ) = 1 }| if χ = 1 , | M | − χ = 1 . Here G v ⊆ G denotes the decomposition group at v .14 Iwasawa-theoretic congruences: Preliminaries on universal norms (c) For every r, n ≥
0, the natural map \ r V U K ∞ , Σ ,T → \ r Z p [ G n ] U K n , Σ ,T induces an isomorphism (cid:0) \ r V U K ∞ , Σ ,T (cid:1) ⊗ V Z p [ G n ] ∼ = UN rn . Proof.
By [BD20, Theorem 3.8 (c)] we have an identification UN r = T r Z p [ G ] UN , thereforeit suffices to check that UN is contained in the kernel of Ord q .Let x ∈ UN and take ( x n ) n ∈ lim ←− n U K n , Σ ,T to be a norm-coherent sequence with x asits bottom value. Since q is unramified and not completely split in K ∞ | K , we can take N to be an integer big enough such that σ − q is inert in K ∞ | K N , where σ ∈ G is fixed. Forevery n ≥
1, let Q n be a place of K n above σ − q and denote by ord Q n : U K n , Σ ,T → Z p themap obtained from valuation at Q n by Z p -linear extension. Then for any m ≥ N we haveord Q N ( x N ) = ord Q m (N K m | K N ( x m )) = p m − N · ord Q m ( x m ) . This shows that the valuation of x N at Q N is infinitely divisible by p and so it followsthat ord Q N ( x N ) = 0. Hence ord q ( σx ) = 0 as well.For part (b) we define a regular height-one prime (see page 38 for this terminology) by p = ker { V χ −→ Q p ( χ ) } , and note that there is an isomorphism V p (cid:30) p V p ∼ = Q p ( χ )of Z p [ G ]-modules. This implies (as can be seen, for example, via the arguments used toprove [BD20, Thm. 3.8 (b)]) that we have an isomorphism e χ ( Q p ( χ ) ⊗ Z p UN ) = Q p ( χ ) ⊗ Z p [ G ] UN ∼ = Q p ( χ ) ⊗ V U K ∞ , Σ ,T ∼ = Q p ( χ ) ⊗ V p ( U K ∞ , Σ ,T ) p . It is therefore sufficient to compute the V p -rank of the free module ( U K ∞ , Σ ,T ) p . Let V χ = { v ∈ S ∞ ( K ) | χ ( G v ) = 1 } . Since we assume that no finite place contained in Σ splitscompletely in k ∞ | k , the module X K ∞ , Σ \ V χ is V p -torsion. Further, it is well-known thatlim ←− n A Σ ,T ( K n ) is V -torsion. Let Q ( V p ) be the total ring of fractions of V p , then the exactsequence (8) therefore gives that Q ( V p ) ⊗ V H ( D • K ∞ , Σ ,T ) = Q ( V p ) ⊗ V X K ∞ ,V χ . This combines with the vanishing of the Euler characteristic of the complex D • K ∞ , Σ ,T toimply that rk Q ( V p ) ( Q ( V p ) ⊗ V U K ∞ , Σ ,T ) = rk Q ( V p ) ( Q ( V p ) ⊗ V H ( D • K ∞ , Σ ,T ))= rk Q ( V p ) ( Q ( V p ) ⊗ V X K ∞ ,V χ )= r S ∞ ( χ ) . Finally, assertion (c) is proved in the same way as [BD20, Thm. 3.8 (b)] by using Lemma(2.3) (b) for the analysis of the spectral sequence (15) in loc. cit. Iwasawa-theoretic congruences: Darmon derivatives
We suppose to be given an Euler system η = ( η E ) E ∈ ES r ( K| k, K ∞ , S, T ) and observethat this defines a norm-coherent sequence η K ∞ = ( η K ∞ ,n ) n ∈ lim ←− n ∈ N \ r Z p [ G n ] U K n , Σ ,T = \ r V U K ∞ , Σ ,T satisfying η K ∞ ,n = η K n for big enough n . (3.2) Conjecture. For every η ∈ ES r ( K| k, K ∞ , S, T ) we have η K ∞ ,n ∈ I e Γ · \ r V U K ∞ , Σ ,T . (3.3) Remark. (a) Variants of Conjecture (3.2) have previously appeared in the lit-erature in many places, with its archetypical relative being the ‘guess’ formulatedby Gross for the Euler system of Stickelberger elements [Gro88, top of p. 195]. Aversion for arbitrary rank was then formulated in [Bur07], see also [San14, Conj. 4].In the form stated above the conjecture has for example been studied in [BS19b,Conj. 2.7].(b) We will see below that Conjecture (3.2) for the Euler system of Rubin-Stark ele-ments is implied by a (relevant variant of a) Iwasawa Main Conjecture and this isindeed already well-known. A direct proof by analytic means for the Euler systemof Stickelberger elements is given in [DS18].(c) A containment as in the statement of Conjecture (3.2) should be thought of asan order of vanishing statement. In fact, it can be directly linked to the order ofvanishing of a p -adic L -function in many cases (see, for example, [BS19b, Lem. 4.9]).The following observation will be useful later on. (3.4) Lemma. Let u ∈ T r V U K ∞ , Σ ,T be a norm-coherent sequence. Then u ∈ I e Γ · \ r V U K ∞ , Σ ,T ⇔ u ∈ I e Γ · ^ r V Π ∞ , where Π ∞ is the module appearing in the choice of representative for D • K ∞ , Σ ,T fixed in § Proof.
By [Sak20, Lem. B.12], the exact sequence (11) induces an exact sequence0 \ r V U K ∞ , Σ ,T ^ r V Π ∞ Π ∞ ⊗ V ^ r − V Π ∞ , φ (19)whence the implication ‘ ⇒ ’ is clear. If we now fix a topological generator γ ∈ Γ, then u = ( γ − e κ for some κ ∈ V r V Π ∞ implies that0 = φ ( u ) = φ (( γ − e κ ) = ( γ − e · φ ( κ ) , hence φ ( κ ) = 0 since ( γ − e ∈ V is not a zero divisor by Lemma (2.3) (b) and Π ∞ ⊗ V V r − V Π ∞ is torsion-free. The exact sequence (19) thus reveals that κ ∈ T r V U K ∞ , Σ ,T .16 Iwasawa-theoretic congruences: Darmon derivatives (3.5) Proposition.
Conjecture (3.2) holds in all of the following cases:(a) W ∩ S = ∅ and e ≤ η is basic,(c) η is the Rubin-Stark Euler system and the higher rank Iwasawa Main conjecture[BKS17, Conj. 3.1] is valid. Proof. If W = ∅ , there is nothing to show. Let us therefore assume that W = { p } for aplace p S . Since p is assumed to split completely in K | k , we must have p ∈ Σ \ S ( K ).Choose n big enough such that S ( K n ) = Σ, hence also η K ∞ ,n = η K n , then the Eulersystem norm relations imply that η K ∞ , = N rK n | K ( η K ∞ ,n ) = N rK n | K ( η K n ) = (cid:16) Y v ∈ Σ \ S ( K ) (1 − Frob − v ) (cid:17) η K = 0since p is assumed to split completely in K | k . This means that η ∞ belongs to the kernelof the natural codescent maplim ←− n \ r Z p [ G n ] U K n , Σ ,T → \ r Z p [ G ] U K, Σ ,T and by Lemma (3.1) (c) this kernel is exactly I Γ K · T r V U K ∞ , Σ ,T .Next we note that, for n big enough, the image of the projection map Θ rK n ,S ( K n ) ,T iscontained in I e Γ n · V r Z p [ G ] Π n , see the proof of [BD20, Lem. 3.20] for details of this deduction.By Lemma (3.4) this shows (b) and (c). (Alternatively, one can use Theorem (3.12) andProposition (3.13).) (3.6) Definition. Let η be an Euler system that verifies Conjecture (3.2). The (Iwasawa-theoretic) Darmon derivative of η with respect to a topological generator γ ∈ Γ is thebottom value κ of the unique norm-coherent sequence κ = ( κ n ) n with the property( γ − e · κ = η K ∞ . (3.7) Remark. (a) If η is the Euler system of cyclotomic units, the notion of Darmonderivative recovers the element considered by Solomon in [Sol92]. Lemma (3.1) (a)moreover provides an easy proof for the analogue of [Sol92, Prop. 2.2 (i)], namelythat the valuation of κ at a prime coprime to p is almost always trivial. In contrast,we will see later that, as first observed by Solomon, the valuation of κ at a primeabove p can encode important arithmetic information. Similarly, Theorem (4.7)implies that the question if κ vanishes is related to information about class groups.(b) Our terminology follows [BKS19] where an element defined via a closely relatedconstruction is referred to as the Iwasawa-Darmon derivative (see [BKS19, Def. 4.6]).This points to Darmon [Dar95] who first interpreted this construction as a derivativeprocess (see also the discussion in [San14, Rk. 4.8]).17
Iwasawa-theoretic congruences: The conjecture of Mazur-Rubin and Sano
Recall that by [BS19a, Prop. A.4] (see also [San14, Lem. 2.11]) for every n ∈ N there isan isomorphism \ r Z p [ G ] U K, Σ ,T ≃ −→ (cid:16) \ r Z p [ G n ] U K n , Σ ,T (cid:17) Γ n ⊆ \ r Z p [ G n ] U K n , Σ ,T which gives rise to an injection ν n : (cid:16) \ r Z p [ G ] U K, Σ ,T (cid:17) ⊗ Z p I (Γ n ) e (cid:30) I (Γ n ) e +1 ֒ → (cid:16) \ r Z p [ G n ] U K n , Σ ,T (cid:17) ⊗ Z p Z p [Γ n ] (cid:30) I (Γ n ) e +1 . We note that this injection satisifes ν n (N rK n | K a ⊗ x ) = N K n | K a ⊗ x (20)for any a ∈ T r Z p [ G n ] U K n , Σ ,T and x ∈ I (Γ n ) e /I (Γ n ) e +1 , see [San14, Rk. 2.12]. Finally, wedefine Darmon’s twisted norm operator N n : \ r Z p [ G ] U K n , Σ ,T → (cid:16) \ r Z p [ G ] U K, Σ ,T (cid:17) ⊗ Z p Z p [Γ n ] (cid:30) I (Γ n ) e +1 , a X σ ∈ Γ n σa ⊗ σ − . (3.8) Lemma. Fix a topological generator γ ∈ Γ and let u, κ ∈ T r V U K ∞ , Σ ,T be norm-coherent sequences satisfying u = ( γ − e κ . Then we have N n ( u n ) = ν n ( κ ⊗ ( γ − e ) for all n ∈ N . Proof.
We calculate: N n ( u n ) = N n (( γ − e κ n )= X σ ∈ Γ n σ ( γ − e κ n ⊗ σ − = X σ ∈ Γ n σκ n ⊗ σ − ( γ − e = X σ ∈ Γ n σκ n ⊗ ( γ − e = N K n | K ( κ n ) ⊗ ( γ − e . Here the third equality from the bottom is obtained by reparametrising the sum and thepenultimate equality follows from σ − ( γ − e − ( γ − e = ( σ − − γ − e ≡ I (Γ n ) e +1 . The property (20) then yields ν n ( κ ⊗ ( γ − e ) = ν n (N rK n | K ( κ n ) ⊗ ( γ − e ) = N K n | K ( κ n ) ⊗ ( γ − e . and this finishes the proof of the Lemma. (3.9) Remark. Lemma (3.8) implies that, in particular, the element κ ⊗ ( γ − e ∈ T r Z p [ G ] U K, Σ ,T ⊗ Z p I (Γ) e /I (Γ) e +1 does not depend on the choice of topological generator γ . 18 Iwasawa-theoretic congruences: The conjecture of Mazur-Rubin and Sano
Let v ∈ W and recall that we have fixed a place w of K above v . Denote by Γ w thedecomposition group of w inside Γ and writerec w : K × ֒ → K × w → Γ w ⊆ Γfor the local reciprocity map at w , where K w denotes the completion of K at w . Considerthe map Rec v : K × → I Γ (cid:30) I , a X σ ∈G (rec w ( σa ) − σ − , which by [San14, Prop. 2.7] and (17) induces the mapRec W = ^ v ∈ W Rec v : \ r ′ Z p [ G ] U K, Σ ,T → (cid:18)\ r Z p [ G ] U K, Σ ,T (cid:19) ⊗ Z p I (Γ) e (cid:30) I (Γ) e +1 . (21)In addition, we define an idempotent e r ∈ Q p [ G ] as the sum of all primitive orthogonalidempotents e χ for characters χ ∈ b G such that e χ Q p ( χ ) U K, Σ \ W,T has Q p ( χ )-dimension r .Note that e r is equal to the idempotent = e ( K, Σ \ W ) defined in (14).With this notation in place, there is an isomorphismOrd W = ^ v ∈ W Ord v : e r Q p \ r ′ Z p [ G ] U K, Σ ,T ≃ −→ e r Q p \ r Z p [ G ] U K, Σ \ W,T , where Ord v is the map defined in (18), see [Rub96, Lem. 5.1].To state our adaptation of [BKS17, Conj. 4.2] we assume that V ′ is maximal in thefollowing sense: • V ′ = S split ( K | k ) ∩ (Σ \ S fin ). (3.10) Conjecture (Iwasawa-theoretic Mazur-Rubin-Sano) . Let η ∈ ES r ( K| k, K ∞ , S, T ).Then, there exists an element k = ( k n ) n ∈ N ∈ (cid:18)\ r Z p [ G ] U K, Σ ,T (cid:19) ⊗ Z p I (Γ) e (cid:30) I (Γ) e +1 = (cid:18)\ r Z p [ G ] U K, Σ ,T (cid:19) ⊗ Z p lim ←− n I (Γ n ) e (cid:30) I (Γ n ) e +1 such that ν n ( k n ) = N n ( η K n ) for all n big enough, and e r · k = (Rec W ◦ Ord − W ) (cid:16) Y v ∈ Σ \ ( W ∪ S ( K )) (1 − Frob − v ) · e r · η K (cid:17) , where the equality takes place in Q p (cid:16)T r Z p [ G ] U K, Σ ,T (cid:17) ⊗ Z p I (Γ) e /I (Γ) e +1 . (3.11) Remark. (a) The above conjecture was originally formulated for the Euler sys-tem of Rubin-Stark units in [BKS17, Conj. 4.2] (Conjecture (3.10) recovers the con-jecture in loc. cit. after taking into account [San14, Prop. 3.6]) and is an Iwasawa-theoretic version of a conjecture that was independently proposed by Mazur-Rubin[MR16] and Sano [San14]. The latter of which, in turn, unify the central conjecturesin [Bur07] and [Dar95].It is conjectured that all Euler systems essentially arise from the Euler system ofRubin-Stark units (see [Bur+19, Conj. 2.5]), this explains our motivation to believein a more general version of the conjecture stated in [BKS17].19 Iwasawa-theoretic congruences: The conjecture of Mazur-Rubin and Sano (b) Conjecture (3.10) is known for the Euler system of Rubin-Stark elements (which isthe system provided by Corollary (2.7)) in the following cases: • k = Q and K is totally real, in this case the conjecture follows from a classicalresult of Solomon [Sol92] (see [BKS17, Thm. 4.10]), • k is totally real and K is CM, in this case the conjecture follows from thevalidity of the Gross-Stark conjecture that has been settled in [DKV18] (see[BKS17, Thm. 4.9]), • every place in W is finite and unramified in K ∞ | K (this follows from the proofof [San14, Prop. 3.13]).The following is essentially an observation of Burns, Kurihara and Sano [BKS17]. (3.12) Theorem. Conjecture (3.10) holds if η is basic. Proof.
Choose n big enough such that Σ = S ( K n ). To prove the theorem we will showthe commutativity of the following diagram:Det Z p [ G n ] ( D • K n , Σ ,T ) e r I (Γ n ) · ^ r Z p [ G n ] Π n ^ r Z p [ G ] Π ⊗ Z p I (Γ n ) e (cid:30) I (Γ n ) e +1 Det Z p [ G ] ( D • K, Σ ,T ) \ r ′ Z p [ G ] U K, Σ ,T (cid:18)\ r Z p [ G ] U K, Σ ,T (cid:19) ⊗ Z p I (Γ n ) e (cid:30) I (Γ n ) e +1 Det Z p [ G ] ( D • K,S ( K ) ,T ) \ r Z p [ G ] U K,S ( K ) ,T Θ rKn | k, Σ ,T ν − n ◦N n Θ r ′ K | k, Σ ,T ( − re Rec W ( − re Ord W Q (1 − Frob − v ) · Θ rK | k,S ( K ) ,T Here the composition of the downward arrows on the left is the transition map i K n | K appearing in the definition of VS( K| k ). The upper rectangle of the above diagram iscommutative by [BKS17, Lem. 5.17]. As for the lower square, we first note that thediagram Det Z p [ G ] ( D • K, Σ ,T ) \ r ′ Z p [ G ] U K, Σ ,T Det Z p [ G ] ( D • K, Σ \ M,T ) \ r ′ Z p [ G ] U K,S ( K ) ∪ W,T Θ rK | k, Σ ,T Θ rK | k,S ( K ) ∪ W,T Q (1 − Frob − v ) was already obtained in the proof of [Bur+19, Thm. 3.8] (see the first diagram in § loc. cit. ).It remains to check that the diagramDet Z p [ G ] ( D • K,S ( K ) ∪ W,T ) \ r ′ Z p [ G ] U K,M,T
Det Z p [ G ] ( D • K,S ( K ) ,T ) \ r Z p [ G ] U K, Σ \ W,T Θ rK | k,S ( K ) ∪ W,T ( − er Ord W Θ rK | k,S ( K ) ,T Iwasawa-theoretic congruences: The conjecture of Mazur-Rubin and Sano commutes. To do this, we may first base change to Q p [ G ]. By the very definition of the pro-jection maps Θ r ′ K | k,S ( K ) ∪ W,T and Θ rK | k,S ( K ) ,T (which involves the passage-to-cohomologymap and trivialising the top degree cohomology, see (16)) it suffices to determine the map e r Q p ^ r ′ Z p [ G ] U K,S ( K ) ∪ W,T = Det Q p [ G ] ( Q p H ( D • K,S ( K ) ∪ W,T )) −→ Det Q p [ G ] ( Q p H ( D • K,S ( K ) ,T )) = e r Q p ^ r Z p [ G ] U K, Σ \ W,T , induced by the exact triangle (6) and the trivialisation (13). By the argument of [BF98,Prop. 3.2] (note that Artin-Verdier duality identifies the complex denoted Ψ S in loc. cit. with C • E,S , see [Bur08, Prop. 3.5 (e)]) the long exact sequence in cohomology induced bythe triangle (6) splits into the two short exact sequences0 Q p U K,S ( K ) ,T Q p U K,S ( K ) ∪ W,T Q p Y K,W Q p X K,S ( K ) Q p X K,S ( K ) ∪ W Q p Y K,W , f where the map f is given by f : U K,S ( K ) ∪ W,T → Y K,W , a X w ∈ W K ord w ( a ) w. Given this, the claim follows by an explicit calculation.In the remainder of this section we will give a more explicit version of Conjecture (3.10)in cases where Conjecture (3.2) holds true. (3.13) Proposition.
Let η ∈ ES r ( K| k, K ∞ , S, T ). The following are equivalent:(i) Conjecture (3.10) is valid for η ,(ii) Conjecture (3.2) holds for η and we have an equality e r · κ ⊗ ( γ − e = (Rec W ◦ Ord − W ) (cid:16) Y v ∈ Σ \ ( W ∪ S ( K )) (1 − Frob − v ) · e r · η K (cid:17) , (22)where κ denotes the Darmon derivative of η with respect to a fixed topologicalgenerator γ ∈ Γ. Proof.
Let us first assume that statement (ii) holds. In light of Lemma (3.8), the element k = ( k n ) n given by k n = ν n ( κ ⊗ ( γ − e ) ∈ (cid:16)T r Z p [ G n ] U K n , Σ ,T (cid:17) ⊗ Z p I (Γ n ) e /I (Γ n ) e +1 satis-fies the requirements of Conjecture (3.10).Conversely, suppose that Conjecture (3.10) holds true for η . By assumption ν n ( k n ) = N n ( η K n ) for n big enough, so it follows from [BKS16, Prop. 4.17] that η K ∞ ,n ∈ I e Γ n · V r Z p [ G n ] Π n . Thus, we can write η K ∞ ,n = ( γ − e x n for some x n ∈ V r Z p [ G n ] Π n . The ele-ment x n defines a unique class modulo (cid:16)V r Z p [ G n ] Π n (cid:17) Γ n , hence ( x n ) n is a norm-compatiblesequence modulo these modules. Observe that in the commutative diagram of transitionmaps 21 Iwasawa-theoretic congruences: The conjecture of Mazur-Rubin and Sano ^ r Z p [ G ] Π ^ r Z p [ G n +1 ] Π n +1 ^ r Z p [ G n +1 ] Π n +1 (cid:30) (cid:18)^ r Z p [ G n +1 ] Π n +1 (cid:19) Γ n +1 ^ r Z p [ G ] Π ^ r Z p [ G n ] Π n ^ r Z p [ G n ] Π n (cid:30) (cid:18)^ r Z p [ G n ] Π n (cid:19) Γ n N Kn +1 | Kn the vertical map on the left is multiplication by p . Taking inverse limits (these are allfinitely generated Z p -modules, so compact and therefore taking limits is exact), we get anisomorphism lim ←− n ≥ ^ r Z p [ G n ] Π n ∼ = lim ←− n ≥ V r Z p [ G n ] Π n (cid:30) (cid:16)V r Z p [ G n ] Π n (cid:17) Γ n . Consequently, the family ( x n ) n ≥ can be regarded as an element of lim ←− n ≥ V r Z p [ G n ] Π n = V r V Π. By construction, we have( γ − e · ( x n ) n = ( η K ∞ ,n ) n ∈ \ r V U K ∞ , Σ ,T . and Lemma (3.4) gives ( x n ) n ∈ T r V U K ∞ , Σ ,T .For the second part of (ii) it suffices to note that Lemma (3.8) now implies that for n bigenough we have ν n ( x ⊗ ( γ − e ) = N n ( η K ∞ ,n ) = N n ( η K n ) = ν n ( k n ) , so x ⊗ ( γ − e = k n by the injectivity of ν n .The appearance of Ord − W in (22) suggests the application of the map Ord W to said equa-tion. This is possible in cases where r ≥ e and we summarise this observation in thefollowing Lemma. (3.14) Lemma. Assume r ≥ e . The equality (22) holds if and only if the equalityOrd W ( e r · κ ) ⊗ ( γ − e = ( − e · Rec W (cid:16) Y v ∈ Σ \ ( W ∪ S ( K ) (1 − Frob − v ) · e r · η K (cid:17) , (23)holds in Q p (cid:16)T r − e Z p [ G ] U K, Σ \ W,T (cid:17) ⊗ Z p I (Γ) e /I (Γ) e +1 . Proof.
By scalar extension, the map Ord W induces a map (cid:18)\ r Z p [ G ] U K, Σ ,T (cid:19) ⊗ Z p I (Γ) e (cid:30) I (Γ) e +1 → (cid:18)\ r − e Z p [ G ] U K, Σ \ W,T (cid:19) ⊗ Z p I (Γ) e (cid:30) I (Γ) e +1 (24)which we also denote by Ord W . Since Ord W is injective on (1 − e r )-torsion by [Rub96,Lem. 5.1] and I (Γ) e /I (Γ) e +1 ∼ = Γ is isomorphic to Z p , the map (24) is injective on the e r -part as well. Thus, the equation (22) holds if and only ifOrd W ( e r κ ) ⊗ ( γ − e = (Ord W ◦ Rec W ◦ Ord − W ) (cid:16) Y v ∈ Σ \ ( W ∪ S ( K ) (1 − Frob − v ) · e r · η K (cid:17) holds. To prove the Lemma it then suffices to observe that, by virtue of (1) being ahomomorphism, we haveOrd W ◦ Rec W = ( − e · (Rec W ◦ Ord W ) = ( − e · (Rec W ◦ Ord W ) . The Gross-Kuz’min conjecture and condition (F)
In this section we will investigate connections to a conjecture due to Gross [Gro81] and,independently, Kuz’min [Kuz72].
We resume the notation of § k ∞ | k denotes a Z p extension in which onlyfinitely many finite places split completely, none of them contained in S ( K ), and we set E ∞ = E · k ∞ for every E ∈ Ω( K| k ). However, we will suppress any subscripts K (as inthe previous section) whenever there is no risk of confusion. In addition, we put A M,T ( E ∞ ) = lim ←− n ∈ N A M,T ( E n ) for any E ∈ Ω( K| k ) and M ⊇ S ∞ ( k ) . If M = S ∞ ( k ) or T = ∅ , we will suppress the respective set in the notation. (4.1) Conjecture (Gross-Kuz’min) . If K cyc ∞ | K is the cyclotomic Z p -extension, then themodule of Γ-coinvariants ( A Σ ( K cyc ∞ )) Γ is finite. (4.2) Remark. (a) It is necessary to work with Σ-class groups in this context becausein general it is not true that the Γ-coinvariants of A ( K cyc ∞ ) are finite (see [Kol91,Prop. 1.17] and [Gre73, Prop. 2] for examples).(b) We remind the reader that for any finitely generated Z p J Γ K -module M the module M Γ is finite if and only if M Γ is finite (see, for example, [CS06, App. A.2, Prop. 2]).Conjecture (4.1) can therefore also formulated as the statement that ( A Σ ( K cyc ∞ )) Γ isfinite.We follow Burns, Kurihara and Sano in considering the following condition which is mo-tivated by the above Conjecture of Gross-Kuz’min and plays a crucial role in their descentformalism (see [BKS17, Thm. 5.2]). (4.3) Condition (F) . The Z p -extension K ∞ | K is such that the module of Γ-coinvariants( A Σ ( K ∞ )) Γ is finite. (4.4) Remark. The validity of Condition (F) is known in the following important cases(see also [HK20, §
2] for an overview of further results):(a) If k = Q , then Conjecture (4.1) is (implicitly) proved by Greenberg [Gre73].(b) If there is exactly one prime of K that ramifies in K ∞ | K , then Condition (F) is aconsequence of Chevalley’s ambiguous class number formula (cf. [Kle19, Ex. 2.7]).(c) If | S p ( K ) | ≤
2, then the validity of Conjecture (4.1) follows from (a) and a result ofKleine [Kle19, Thm. B].Said result of Kleine hinges upon the following fact (see the proof of [Kle19, Lem. 3.5]):Let N | Q be a normal extension and suppose that x ∈ O N,S p ( N ) is such that we havelog p N N p | Q p ( x ) = 0 for all p ∈ S p ( N ). Then the valuation ord p ( x ) is the same for all p ∈ S p ( N ).However, the proof of this assertion given in loc. cit. contains an inaccuracy and we23 The Gross-Kuz’min conjecture and condition (F): Coinvariants of class groups therefore take the opportunity to provide a better argument. Let p ∈ S p ( N ) be suchthat n := ord p ( x ) is minimal among { ord p ( x ) | p ∈ S p ( N ) } . Write e p for the ramific-ation degree of N | Q at p , then x e p p − n is a unit at p and integral at any other finiteplace of N . By assumption log p N N p | Q p ( x ) = 0, hence also log p N N p | Q p ( x e p p − n ) = 0and we can find an integer m ≥ N p | Q p ( x e p p − n ) m = 1. Let G p ⊆ Gal( N | Q ) be the decomposition group at p and set M = N G p . Then we haveN N p | Q p ( x me p p − mn ) = N N | M ( x me p p − mn ) = 1 , so x me p p − mn is a unit in O N and it follows thatord p ( x me p p − mn ) = 0 ⇔ ord p ( x ) = n for all p ∈ S p ( N ). This finishes the proof of the claim.(d) Suppose that Condition (F) holds for a fixed Z p -extension K ∞ of K . Kleine hasproved [Kle17, Cor. 3.6] that there exists an integer n ≥ Z p -extensions K ′∞ of K with the following property: The n -thlayers K ′ n and K n agree, and S ram ( K ′∞ | K ) ⊆ S ram ( K ∞ | K ).In § (4.5) Lemma. The following hold:(a) ( A Σ ,T ( K ∞ )) Γ is finite if and only if ( A Σ ( K ∞ )) Γ is finite,(b) if Σ ′ is a finite set of places of k which contains S ∞ ( k ) ∪ S ram ( K ∞ | K ) and is suchthat Σ ′ ⊆ Σ, then ( A Σ ,T ( K ∞ )) Γ is finite as soon as ( A Σ ′ ,T ( K ∞ )) Γ is. Proof.
The exact sequence F × T Kn := M w ∈ T Kn (cid:0) O K n (cid:30) w (cid:1) × A Σ ,T ( K n ) A Σ ( K n ) 0implies that it is sufficient to show that the module lim ←− n F × T Kn has finite Γ K -coinvariantsin order to prove (a). Now taking the limit of the exact sequences (which are obtained asrepresentatives of the complexes L v ∈ T RΓ f ( K v , Ind G k G Kn ( Z p (1))), see [BF01, (19)])0 M v ∈ T Z p [ G K n ] M v ∈ T Z p [ G K n ] F × T Kn (1 − N v − · Frob v ) v (25)gives an exact sequence0 M v ∈ T V M v ∈ T V lim ←− n F × T Kn . (26)By taking Γ-coinvariants of (26) we obtain the exact sequence M v ∈ T Z p [ G ] M v ∈ T Z p [ G ] lim ←− n F × T Kn ! Γ . − N v − · Frob v The Gross-Kuz’min conjecture and condition (F): Coinvariants of class groups
Comparing with (25), we deduce that (cid:0) lim ←− n F × T Kn (cid:1) Γ = F × T K , hence, in particular, is finite. Finally, (b) is clear since A Σ ,T ( K ∞ ) is a quotient of A Σ ′ ,T ( K ∞ ).To state the main result of this section we shall make the following assumption. (4.6) Hypothesis. If p = 2, then we assume that all archimedean places split completelyin K ∞ | k . (4.7) Theorem. Let χ ∈ b G be a non-trivial character such that r Σ ( χ ) = r ′ for the number r Σ ( χ ) defined in Lemma (3.1) and assume Hypothesis (4.6). The following assertions areequivalent:(i) The module e χ Q p ( χ ) A Σ ,T ( K ∞ ) Γ vanishes.(ii) The map M v ∈ W Rec v,χ : e χ Q p ( χ ) U K, Σ ,T → e χ Q p ( χ ) Y K,W ⊗ Z p I (Γ) (cid:30) I (Γ) ,a e χ X v ∈ W X σ ∈G χ ( σ ) w ⊗ (cid:0) rec w ( σ − a ) − (cid:1) is surjective.(iii) The mapRec W : e χ Q p ( χ ) ^ r ′ Z p [ G ] U K, Σ ,T → e χ Q p ( χ ) ^ r Z p [ G ] U K, Σ ,T ⊗ Z p I (Γ) e (cid:30) I (Γ) e +1 defined in (21) is non-zero (equivalently, injective).(iv) If η is a basis of ES b ( K| k, K ∞ , S, V, T ) with associated norm-coherent sequence η K ∞ ∈ T r V U K ∞ , Σ ,T , then e = r ′ − r is maximal with respect to the property e χ η K ∞ ∈ I e Γ · (cid:0) ^ r V U K ∞ , Σ ,T (cid:1) p , where p = ker { V → Z p [im χ ] } .(v) If κ is the Darmon derivative of a basis η of ES r b ( K| k, K ∞ , S, T ), then e χ κ is a Q p ( χ )-basis of e χ Q p ( χ )UN r .The proof of this result will be given in § (4.8) Remark. If K ∞ | K is the cyclotomic Z p -extension, then the equivalence (i) ⇔ (ii)in Theorem (4.7) is already known due to [Kol91, Thm. 1.14]. In general, the implication(i) ⇒ (iii) is proved in [BKS17, § K is a CM extension of a totally real field k and χ is totally odd, then (iii) is equivalentto the non-vanishing of the Gross regulator . In this setting, Gross has proved in [Gro81,Prop. 1.16] that condition (iii) holds if there is at most one prime p of k above p such that χ ( p ) = 1. 25 The Gross-Kuz’min conjecture and condition (F): Computation of Bocksteinhomomorphisms
To prove Theorem (4.7) we will perform a computation of
Bockstein maps as in [BKS17, § § n is defined as themap β n : U K, Σ ,T → H ( D • K n , Σ ,T ) ⊗ Z p I (Γ n ) → H ( D • K, Σ ,T ) ⊗ Z p I (Γ n ) (cid:30) I (Γ n ) , where the first arrow is the connecting homomorphism arising from the exact triangle D • K n , Σ ,T ⊗ Z p [ G n ] I Γ n D • K n , Σ ,T D • K n , Σ ,T ⊗ Z p [ G n ] Z p [ G ] . By taking the limit over n , we obtain a map β ∞ : U K, Σ ,T → H ( D • K, Σ ,T ) ⊗ Z p lim ←− n I (Γ n ) (cid:30) I (Γ n ) ∼ = H ( D • K, Σ ,T ) ⊗ Z p I (Γ) (cid:30) I (Γ) (27)that is identified with the map U K, Σ ,T δ → H ( D • K ∞ , Σ ,T ) ⊗ Z p I (Γ) → H ( D • K, Σ ,T ) ⊗ Z p I (Γ) (cid:30) I (Γ) , where the map δ is the connecting homomorphism induced by the triangle D • K ∞ , Σ ,T ⊗ V I Γ D • K ∞ , Σ ,T D • K ∞ , Σ ,T ⊗ V Z p [ G ] . To make the definition of β ∞ more explicit we now fix a topological generator γ ∈ Γand note that this choice gives rise to an identification of the above triangle with thetriangle induced by multiplication by γ −
1. In particular, it allows to view δ as theboundary homomorphism arising from an application of the snake lemma to the followingcommutative diagram: 0 H ( D • K n , Σ ,T )0 Π ∞ Π ∞ Π
00 Π ∞ Π ∞ Π H ( D • K ∞ , Σ ,T )0 φ · ( γ − φ φ · ( γ − Using that ( U K ∞ , Σ ,T ) Γ ∼ = UN , the snake lemma also shows that this boundary map δ fitsinto the exact sequence0 UN U K, Σ ,T H ( D • K ∞ , Σ ,T ) Γ . δ (28)26 The Gross-Kuz’min conjecture and condition (F): Computation of Bocksteinhomomorphisms
Our fixed choice of topological generator γ ∈ Γ also induces an isomorphism I (Γ) /I (Γ) ∼ = Z p , hence β ∞ can be identified with the composite map U K, Σ ,T δ −→ H ( D • K ∞ , Σ ,T ) Γ ⊆ H ( D • K ∞ , Σ ,T ) → H ( D • K ∞ , Σ ,T ) Γ ∼ = H ( D • K, Σ ,T ) . (29) (4.9) Lemma. Let v ∈ Σ and recall that we have previously fixed a place w of K lyingabove v . The composite map β w : U K, Σ ,T β ∞ −→ H ( D • K, Σ ,T ) ⊗ Z p I (Γ) (cid:30) I (Γ) π K → X K, Σ ⊗ Z p I (Γ) (cid:30) I (Γ) w ∗ −→ Z p [ G ] ⊗ Z p I (Γ) (cid:30) I (Γ) is zero if v ∈ V , and coincides with Rec v if v ∈ W . Here we have used the map π K appearing in (4) and the notation w ∗ for the Z p [ G ]-linear dual of w considered as anelement of Y K, Σ . Proof.
This follows immediately from the corresponding results on β n , see [BKS16, Lemma5.20 and Lemma 5.21]. Proof (of Theorem (4.7)):
The assumption r Σ ( χ ) = r ′ implies that χ ( v ) = 1 if and only if v ∈ V ′ . It follows that e χ Q p ( χ ) X K, Σ = e χ Q p ( χ ) Y K,V ′ because χ = 1. By (29) and Lemma(4.9), the map L v ∈ W Rec v,χ in (ii) coincides with the composite e χ β ∞ : e χ Q p ( χ ) U K, Σ ,T δ −→ e χ Q p ( χ ) H ( D • K ∞ , Σ ,T ) Γ α −→ e χ Q p ( χ ) H ( D • K, Σ ,T ) π K ∼ = e χ Q p ( χ ) Y K,V ′ , and actually has image inside e χ Q p ( χ ) Y K,W . The first map δ is already surjective in anycase by the exact sequence (28), so the above composite map surjects onto e χ Q p ( χ ) X K,W if and only if α does. By Lemma (3.1) (b) (in combination with Hypothesis (4.6)) we havedim Q p ( χ ) e χ Q p Y K,W = e = r ′ − r = dim Q p ( χ ) e χ Q p ( χ ) U K, Σ ,T − dim Q p ( χ ) e χ Q p ( χ )UN = dim Q p ( χ ) e χ Q p ( χ ) H ( D • K ∞ , Σ ,T ) Γ , where the last equality uses the exact sequence (28), hence the map α is surjective if andonly if it is injective (and if and only if it is an isomorphism).Now, we have a commutative diagram e χ Q p ( χ ) H ( D • K ∞ , Σ ,T ) Γ e χ Q p ( χ ) H ( D • K, Σ ,T ) e χ Q p ( χ ) Y Γ K ∞ ,V ′ e χ Q p ( χ ) Y K,V ′ απ K, ∞ π K ≃ f (30)where the map f is induced by ( w K n ,i ) n w K,i for all 1 ≤ i ≤ r ′ . Observe that Y K ∞ ,V isa direct summand of Y K ∞ ,V ′ which is Z p J Γ K -projective in light of Hypothesis (4.6). Giventhis, we have Y Γ K ∞ ,V = Y Γ K ∞ ,W . We now claim that the bottom map f in (30) maps Y Γ K ∞ ,W isomorphically onto Y K,W .To do this, we write l i for the index of the decomposition group of w i inside Γ. For27 The Gross-Kuz’min conjecture and condition (F): Proof of condition (F) in specialcases ≤ i ≤ r ′ , each w i splits completely in K | k and hence K l i coincides with the decompositionfield of w i inside the extension K ∞ | k . It follows that Y Γ K ∞ ,W = (cid:0) r ′ M i =1 Z p [ G l i ] w i (cid:1) Γ = r ′ M i =1 (cid:0) Z p [ G l i ] w i (cid:1) Γ = r ′ M i =1 ( Z p [ G l i ] w i ) Γ li As a consequence, the map f corresponds with the inverse of the isomorphism r ′ M i =1 Z p [ G ] ∼ = r ′ M i =1 Z p [ G l i ] Γ li −→ r ′ M i =1 Z p [ G l i ] Γ li which, in each component, is given by multiplication by N Γ li . This shows the claim and,because the image of π K ◦ α is contained in e χ Q p ( χ ) Y K,W , we are therefore reduced to thequestion of when the map labelled π K, ∞ in the diagram (30) is injective. From the exactsequence (8) we obtain the exact sequence0 ( A Σ ,T ( K ∞ )) Γ H ( D • K ∞ , Σ ,T ) Γ ( X K ∞ , Σ ) Γ , π K, ∞ (31)hence π K, ∞ is injective after extending scalars to e χ Q p ( χ ) if and only if e χ Q p ( χ )( A Σ ,T ( K ∞ )) Γ =0. This establishes (i) ⇔ (ii).By counting dimensions using Lemma (3.1) (b) and the exact sequence (28) as above wesee that e χ Q p ( χ ) ker β ∞ has dimension at least r . As a consequence, the exterior power e χ Q p ( χ ) V r Z p [ G ] ker β ∞ is non-zero and so the equivalence (ii) ⇔ (iii) follows by appealingto [BKS16, Lem. 4.2].Next we note that (v) implies (iii) by means of Theorem (3.12) and Proposition (3.13).For the converse recall that by the commutative diagram in the proof of Theorem (3.13)the element e χ · ω := e χ · Ord − W ( Q Σ \ ( W ∪ S ( K )) (1 − Frob − v ) · η K ) is the image of a basisunder the surjective mapΘ rK | k, Σ ,T : e χ Q p ( χ ) Det Z p [ G ] ( D • K, Σ ,T ) → e χ Q p ( χ ) ^ r ′ Z p [ G ] U K, Σ ,T . In particular, e χ ω is nonzero and hence is a basis of e χ Q p ( χ ) V r ′ Z p [ G ] U K, Σ ,T . As a con-sequence, Theorem (3.12) implies that e χ κ ⊗ ( γ − e is nonzero as soon as Rec W is notthe zero map. It follows that e χ κ = 0 and, since e χ Q p ( χ )UN r is one-dimensional, saidelement is therefore a basis of e χ Q p ( χ )UN r .Finally, (iv) ⇔ (v) is clear from the fact that the kernel of the natural codescent map V p ⊗ V \ r V U K ∞ , Σ ,T → V p (cid:30) p V p ⊗ V \ r V U K ∞ , Σ ,T ∼ = e χ Q p ( χ )UN r is exactly I e Γ · (cid:0) T r V U K ∞ , Σ ,T (cid:1) p by Lemma (3.1) (c). In this section we shall explain how one can prove the equivalent conditions of Theorem(4.7) in special cases. Crucial ingredient in these arguments is the following Lemma, whichis a direct consequence of Brumer’s p -adic analogue of Baker’s Theorem from transcendencetheory. 28 The Gross-Kuz’min conjecture and condition (F): Proof of condition (F) in specialcases (4.10) Lemma.
Let v be a place of k that splits completely in K . Then there is anelement a ∈ O × K, { v } such that X σ ∈G χ ( σ ) · log p ( ι w ( σ − a )) = 0for all non-trivial characters χ ∈ b G . Proof.
Let a ∈ O × K, Σ be an element that is only divisible by w and no other finite primeof K . For example, such an element is given by any generator of w h K , where h K is theclass number of K . Fix a non-trivial character χ ∈ b G and suppose that X σ ∈G χ ( σ ) · log p ( ι w ( σ − a )) = 0 . (32)The elements χ ( σ ) are algebraic over Q and not zero, hence Brumer’s p -adic analogue ofBaker’s theorem [Bru67] (see also [NSW08, Thm. 10.3.14]) asserts the existence of integers n σ ∈ Z , not all of them zero, such thatlog p (cid:0) ι w (cid:0) X σ ∈G n σ σa (cid:1)(cid:1) = 0 . After multiplying by a suitable integer if necessary we may therefore assume that P σ ∈G n σ σa is a power of p . In fact, this power of p needs to be p n ord w ( a ) /e w | p by choice of a . Here e w | p denotes the ramification degree of w in K | Q . We have e w | p = e σw | p for all σ ∈ G , soit follows that n ord w ( a ) = ord w p n ord w ( a ) /e w | p = ord σw p n ord w ( a ) /e w | p = ord σw (cid:0) X σ ∈G n σ σa (cid:1) = n σ ord w ( a )for all σ ∈ G . We deduce that all n σ agree, and so log p ( ι w ( P σ ∈G σa )) = 0 as well. Thiscombines with (32) to imply that X σ ∈G\{ } ( χ ( σ ) − · log p ( ι w ( σ − a )) = 0 . By assumption, χ = 1 and so we can apply the Theorem of Brumer-Baker yet again toobtain integers m σ , not all of them zero, such that x = ι w (cid:0) P σ ∈G\{ } m σ σa (cid:1) lies in thekernel of the p -adic logarithm. Now, x is integral at w and therefore we must have that x is a root of unity. However, the set { σa | σ ∈ G} is Z -linearly independent and so thiscan only happen if all m σ are zero, which is a contradiction. (4.11) Theorem. The equivalent assertions of Theorem (4.7) hold true if(i) W contains no prime that ramifies in k ∞ | k ,(ii) W contains exactly one prime which ramifies in k ∞ | k and p is completely split in k | Q . 29 The Gross-Kuz’min conjecture and condition (F): Proof of condition (F) in specialcases (4.12) Remark. (a) The proof of Theorem (4.11) can be adapted to the followingrelated, but more technical, variant of condition (ii): K | Q is normal, K ∞ | K is thecyclotomic Z p -extension of K , W contains exactly one prime v above p , and thedecomposition group G w ⊆ Gal( K | Q ) is a normal subgroup of Gal( K | Q ). The proofof this is essentially the same as the proof of Theorem (4.11) if one takes into accountRemark (4.4) (c).(b) To give a concrete example for a situation in which Theorem (4.11) can be applied,suppose that k is an imaginary quadratic field in which p splits completely. If wefix a prime ideal p of k above p , then there is a unique Z p -extension k ∞ of k that isunramified outside p . Given this, Theorem (4.11) implies that Condition (F) holdsfor all abelian extensions K | k with respect to the Z p -extension K ∞ = K · k ∞ of K .We remark, however, that this fact is already known, see the proof of [Rub88, Thm.1.4] where it is deduced from the known validity of Leopoldt’s Conjecture in thissetting (the latter is of course also derived from the Theorem of Brumer-Baker). Proof.
Let χ = 1 be a non-trivial character of G . By the argument of [HK20, Lem. 3.2]we may assume that K is the field cut out by the character χ . Due to Lemma (4.5)(b) we may moreover assume that S = S ∞ ( k ), i.e. Σ = S ∞ ( k ) ∪ S ram ( K ∞ | k ) and hence W ⊆ S ram ( K ∞ | K ). If W is empty, there is nothing to prove, so we may assume that W = { v } for a single place v ∈ S p ( k ). Given this, we have r Σ ( χ ) = r + 1 and so may applyTheorem (4.7). We shall now show that statement (ii) in (4.7) holds true in this situation.The codomain of Rec v,χ is of Q p ( χ )-dimension one, hence the map Rec v,χ is surjective assoon as it is non-zero.Let Γ w ⊆ Γ be the decomposition group at w and write d for the index (Γ w : I w ) of theinertia group I w at w . Let H ′ be the unique unramified extension of Q p of degree d andwrite H ′∞ for the maximal extension of H ′ that is totally ramified and abelian over Q p .This extension can be explicitly described using relative Lubin-Tate theory. In particular,we have an isomorphism 1 + p Z p ≃ −→ Gal( H ′∞ | H ) (33)that coincides with the composition of the inclusion Q × p ֒ → ( H ′ ) × and the local reciprocitymap ( H ′ ) × → Gal( H ′∞ | H ), see [Sha87, Ch. I, Prop. 1.8]. We deduce that Gal( H ′∞ | H ′ )splits as the direct sum of I w and a finite part. Thus, we have an isomorphismΓ dw = I w ≃ −→ Z p , σ log p χ ell ( σ ) , where χ ell denotes the inverse map of (33). We can therefore identify the map d Rec v,χ with the map e χ Q p ( χ ) U K, Σ ,T −→ e χ Q p ( χ ) Y K,W ⊗ Z p Γ dw ∼ = e χ Q p ( χ ) , a
7→ − de · X σ ∈G χ ( σ ) · log p ( ι w ( σ − a )) . Now, Lemma (4.10) implies that this map is non-zero, as desired.If p does not split completely in k , the situation is much more complicated. We are howeverable to prove the following result concerning the case of k being an imaginary quadraticfield. 30 The Gross-Kuz’min conjecture and condition (F): Proof of condition (F) in specialcases (4.13) Theorem.
Assume that k is an imaginary quadratic field such that p does notsplit in k | Q . There are infinitely many Z p -extensions k ∞ of k such that all of the followingconditions are satisfied:(a) A Σ ,T ( K ∞ ) Γ is finite,(b) at most two finite places of k split completely in k ∞ | k , neither of them contained in S ( K ) ∪ S p ( k ),(c) if the µ -invariant of A ( K cyc ∞ ) for the cyclotomic Z p -extension K cyc ∞ of K vanishes (asconjectured by Iwasawa), then also the µ -invariant of A ( K ∞ ) vanishes. (4.14) Remark. The techniques that are typically used to prove an equivariant MainConjecture (such as Conjecture (5.6)) require the vanishing of a µ -invariant (cf. [BKS17,Prop. 3.15]). Part (c) of Theorem (4.13) is therefore motivated by the possibility ofapplying such techniques to the situation at hand, namely a Z p -extension provided byTheorem (4.13). Proof.
By the argument of [HK20, Lem. 3.2] the property (a) is satisfied if, for everycharacter χ ∈ b G , the module e χ Q p ( χ ) A Σ ,T ( K χ, ∞ ) vanishes, where K χ, ∞ = K χ · k ∞ and K χ denotes the subfield of K cut out by the character χ . By Remark (4.4) (a) thisholds for χ = 1 because k contains only prime above p , so it suffices to consider non-trivial characters. By Lemma (4.5) it is enough to check if e χ Q p ( χ ) A Σ χ ,T ( K χ, ∞ ) vanishes,where Σ χ = S ram ( K χ, ∞ | k ) ∪ S ∞ ( k ). In this situation we may apply Theorem (4.7) whichasserts that the aforementioned vanishing is equivalent to the surjectivity of the map L v ∈ W χ Rec v,χ defined in (ii) of Theorem (4.7) as e χ Q p ( χ ) O × K χ , Σ χ ,T → e χ Q p ( χ ) Y K χ ,W χ ⊗ Z p Γ χ , a X w ∈ W χ X σ ∈G χ χ ( σ ) w ⊗ (rec w ( σ − a ) − , where G χ = Gal( K χ | k ), Γ χ = Gal( K χ, ∞ | K χ ), and W χ = { v ∈ Σ χ \ S ∞ ( k ) | χ ( v ) = 1 } .Observe that we must have W χ ⊆ S ram ( k ∞ | k ) = { v } for the unique place v of k above p .If W χ = ∅ , there is nothing to show. We may therefore assume that W χ = { v } , and welet b G W be the subset of b G comprising all non-trivial characters χ such that W χ = ∅ .Note that we have a commutative diagram e χ Q p ( χ ) O × K, Σ χ ,T e χ Q p ( χ ) ⊗ Z p Γ e χ Q p ( χ ) O K χ , Σ χ ,T e χ Q p ( χ ) ⊗ Z p Γ χ , e χ Rec v N K | Kχ ≃ Rec v,χ where the isomorphism on the right is induced by the inclusion Γ = Gal( K ∞ | K ) ⊆ Γ χ (which has finite index). It is therefore sufficient to check if the map e χ Rec v is surjective(or, equivalently, non-zero) for all χ ∈ b G W .The basic strategy of the remainder of this proof is now to show that this holds if oneavoids, if necessary, certain ’bad’ Z p -extensions. As a first step towards this, we will nowfirst give a more explicit description of the map e χ Rec v .Let F ∞ be the compositum of all Z p extensions of k , which is a Z p -extension as a con-sequence of the known validity of Leopoldt’s Conjecture for this setting. In fact, we know31 The Gross-Kuz’min conjecture and condition (F): Proof of condition (F) in specialcases that Gal( F ∞ | k ) = Z p γ cyc ⊕ Z p γ anti , where γ cyc , γ anti ∈ Gal( F ∞ | k ) are such that the fixedfields F h γ cyc i∞ and F h γ anti i∞ are the cyclotomic and anti-cyclotomic Z p -extensions of k , re-spectively. Write Gal( F ∞ | k ) v = Gal( F ∞ K | K ) w for a choice of decomposition group at v inside F ∞ | k and w inside Gal( F ∞ K | K ), respectively. If I v denotes the inertia subgroupof Gal( F ∞ | k ) v , then explicit local class field theory [Sha87, Ch. I, Prop. 1.8] gives thatthe inverse of the local reciprocity map identifies I v with a quotient of 1 + p v , where p v isthe maximal ideal of the valuation ring O k v ⊆ k v of k v . Since 1 + p v and I v are both of Z p -rank two, I v must agree with the torsion-free part of 1 + p v . We can therefore find aninteger s ≥ F ∞ | k ) p s dv ⊆ I p s v ≃ −→ (1 + p v ) p s ⊆ p sv ≃ −→ p sv , (34)where d = (Gal( F ∞ | k ) : I v ), the first arrow is the inverse of the local reciprocity mapArt v : k × v → Gal( F ∞ | k ) v , and the second arrow is the p -adic logarithm. The cokernel of(34) is finite, hence, for all characters χ ∈ b G W , it induces an isomorphism ω χ : e χ Q p ( χ ) ⊗ Z p Gal( F ∞ · K | K ) = e χ Q p ( χ ) ⊗ Z p Gal( F ∞ | k ) ≃ −→ e χ Q p ( χ ) ⊗ Z p p v . Given this, we can identify the map e χ Q p ( χ ) · U K,W,T → e χ Q p ( χ ) ⊗ Z p Gal( F ∞ K | K ) , a p s d X σ ∈G χ ( σ ) ⊗ Art v ( ι w ( σ − a )) , where ι w : K × ֒ → K × w denotes the canonical embedding, with the map f ρ χ : e χ Q p ( χ ) ⊗ Z O × K,W,T → e χ Q p ( χ ) ⊗ Z p p v , a
7→ − p s d X σ ∈G χ ( σ ) ⊗ log p ( ι w ( σ − a )) . Let γ, δ ∈ Gal( F ∞ | k ) be a Z p -basis and write k δ = F h δ i∞ for the Z p -extension of k that iscut out by δ . Observe that all Z p -extensions of k are of this form. We also set K δ = K · k δ .Recall that the map rec w is the composite of ι w : K × ֒ → K × w and the local reciprocity map K × w → Gal( K δ | K ), and note that the latter map can be described as the compositionof Art v and the restriction map on decomposition groups Gal( F ∞ K | K ) w → Gal( K δ | K ) w .More explicitly, if γ x δ y is an element of Gal( F ∞ K | K ) w , then its restriction to K δ coincideswith γ x .Observe that it is sufficient to check the non-vanishing of the map e χ Rec v after multiplic-ation by p s d . The above discussion implies that p s d · e χ Rec v can be identified with themap ρ χ = π χ,γ ◦ f ρ χ , where π χ,γ denotes the projection map π χ,γ : e χ Q p ( χ ) ⊗ Z p p v → e χ Q p ( χ ) , xω χ ( γ p s d ) + yω χ ( δ p s d ) x. By Lemma (4.10) the map f ρ χ is non-zero. It follows that the map ρ χ can only be zeroif the image of f ρ χ is contained in, and hence coincides with, the kernel of the projectionmap π χ,γ , i.e. the submodule of e χ Q p ( χ ) ⊗ Z p p v generated by ω χ ( δ ). Thus, the map ρ χ isnon-zero for all χ = 1 if δ is not a Z × p -multiple of an element in the set { δ χ | χ ∈ b G W } ,where δ χ denotes a topological generator of ω − χ (cid:16)f ρ χ ( e χ Q p ( χ ) U K,W,T ) ∩ ω χ (Gal( F ∞ | k )) (cid:17) . Abelian extensions of imaginary quadratic fields
We now claim that we can choose an integer N ≥ Z p -extensions in the setΩ( N ) = { k δ | δ = γ p n anti · γ cyc for some n ≥ N } satisfy all of the conditions (a) – (c). Indeed, if N is big enough such that for n ≥ N noneof the elements γ p n anti · γ cyc is a Z × p -multiple of an element in { δ χ | χ ∈ b G W } , then each Z p -extension in Ω( N ) will have property (a). Note that k δ ∩ k cyc = k cyc n if δ = γ p n anti · γ cyc .Since no finite place splits completely in k cyc | k , we may therefore choose N such that thesecond part of (b) is satisfied for each element of Ω( N ). The first part of (b), in turn,follows from a result of Emsalem [Ems87] which, as a particular case, asserts that in any Z p -extension of k that is not the anticyclotomic extension at most two finite primes cansplit completely.Finally, Kleine has proved in [Kle17, Thm. 3.10 (i)] that there is an integer n ≥ µ -invariant of A ( M ∞ ) for a Z p -extension M ∞ of K vanishes if the µ -invariantof A ( K cyc ∞ ) vanishes and the n -th layers of M ∞ | K and K cyc ∞ | K agree. Consequently, N can be chosen big enough such that property (c) is satisfied by K ∞ = k ∞ · K for every k ∞ ∈ Ω( N ). In this section we shall concern ourselves with the special case where the base field k isimaginary quadratic. Fix an imaginary quadratic field k . We will often distinguish between two cases: • (split case) The rational prime p splits in k . In this case we fix a choice of primeideal p ⊆ O k above p , i.e. we then have p O k = pp with p = p . • (non-split case) The prime p is either inert in k , i.e. p O k = p is prime, or ramified,i.e. p O k = p .Let K be the maximal abelian extension of k and fix a field K ∈ Ω( K| k ). Define k ∞ to be • the unique Z p -extension of k unramified outside p , in the split case, • any Z p -extension of k in which only finitely many finite places split completely, noneof them ramified in K | k .As in § K ∞ = K · k ∞ and write K n for the n -th layer of the Z p -extension K ∞ | K . As before, we shorten the notations G K , Γ K,n , Γ nK and V K introduced in § G , G n , Γ n , Γ n and V , respectively. We also note that, in the split case, no finite place splitscompletely in k ∞ | k , see [Sha87, Ch. II, Prop. 1.9].Let moreover χ ∈ b G be a character. We also introduce the following notation: • K χ = K ker χ the field cut out by the character χ , and G χ = G K χ its Galois group, • K χ, ∞ = K χ · k ∞ the composite of K χ with the Z p -extension k ∞ of k , and Γ χ =Gal( K χ, ∞ | K χ ) ∼ = Z p its Galois group,33 Abelian extensions of imaginary quadratic fields: The conjecture of Mazur-Rubin andSano for elliptic units • K χ,n the n -th layer of K χ, ∞ | K χ , and Γ χ,n = Gal( K χ,n | K χ ) and Γ nχ = Gal( K χ, ∞ | K χ,n )the relevant Galois groups.Fix a prime ideal a ( O k that is coprime to 6 pfm K , where m = m K denotes the conductorof K and f ⊆ O k is a non-zero ideal coprime to p which does not split completely in k ∞ andis such that O × k → ( O k / f ) × is injective. Then T = { a } has the property that U E,S ( E ) ,T is Z p -torsion free for every subfield E of K ∞ | k .We set S = S ∞ ( k ) ∪ { q | f } , V = V χ = S ∞ ( k ) and note that Σ = S ∞ ( k ) ∪ { q | fpm } forthe set Σ introduced at the end of § V ′ χ = ( S split ( K χ | k ) ∩ Σ if χ = 1 , Σ \ { p } if χ = 1and set W χ = V ′ χ \ V . Write η Ell for an Euler system that belongs to ES ( K| k, K ∞ , S, T ) and is such that forevery n ∈ N η Ell k ( fmp n ) = ε Vk ( fmp n ) | K,S ( k ( fmp n )) ,T = ψ fmp n , a is the elliptic unit defined in Example (2.8) (c). Such an Euler system exists by Corol-lary (2.7). (5.1) Theorem. Assume that χ = 1 is a character of G . Then Conjecture (3.10) holdsfor the Euler system η Ell and the data ( K χ, ∞ | k, K χ , S, T ) fixed above. (5.2) Remark. In the split case our methods only allow to prove Conjecture (3.10) forthe unique Z p -extension k ∞ | k which is unramified outside p . However, this is sufficientto establish the relevant case of the equivariant Tamagawa Number Conjecture in thissetting (see Theorem (5.9)) and this, in turn, implies Conjecture (3.10) for any choice of Z p -extension in which no finite place contained in Σ splits completely via the argumentin the proof of Theorem (3.12).For the proof of Theorem (5.1) we shall appeal to relative Lubin-Tate theory [Sha87, Ch. I].In order to do this, we first need to establish a little more notation.Let H be a finite extension of Q p and denote the cardinality of its residue field O H / p H by q . We fix an integer d > H ′ be the unramified extension of H of degree d . Wewrite ϕ ∈ Gal( H ′ | H ) for the arithmetic Frobenius automorphism.Fix an element ξ ∈ H × such that ord H ( ξ ) = d . For each power series f satisfyingFrobenius-like properties (for details see [Sha87, Ch. I]) there exists a unique one-dimensionalcommutative formal group law F f ∈ O H ′ J X, Y K satisfying F ϕf ◦ f = f ◦ F f called a relativeLubin-Tate group (relative to the extension H ′ /H ). We let W nf be the group of divisionpoints of level n of F f and set g W nf = W nf \ W n − f for every n ∈ N . Then H ′ n = H ′ ( W n +1 f ) isa totally ramified extension of H ′ of degree q n ( q −
1) and H ′∞ = S n ∈ N H ′ n is the maximaltotally ramified extension of H ′ that is abelian over H .Fix ω i ∈ f W iϕ − i ( f ) such that ( ϕ − i f )( ω i ) = ω i − and let u ∈ lim ←− n ( H ′ n ) × be a norm-coherent34 Abelian extensions of imaginary quadratic fields: The conjecture of Mazur-Rubin andSano for elliptic units sequence. There is a unique integer ν ( u ) such that u n O H ′ n = p ν ( u ) H ′ n for all n ≥
0. By[Sha87, Ch. I, Thm. 2.2] there is a unique power series Col u ∈ t ν ( u ) O H ′ J t K × such that( ϕ − ( i +1) Col u )( ω i +1 ) = u i for all i ≥
0. This power series Col u is called the Coleman power series associated to u .Let ρ : Gal( H ′∞ | H ) → Q / Z be a character of finite order. Write H ρ = ( H ′∞ ) ker ρ for thefield cut out by ρ and choose m minimal with the property that H ρ ⊆ H ′ m .If u ∈ lim ←− n O × H ′ n is a norm-coherent sequence, then class field theory implies that N H ′ | H ( u ) =1. Hilbert’s Theorem 90 therefore ensures the existence of an element β σ,ρ ∈ H × ρ satisfying( σ − · β σ,ρ = N H ′ m | H ρ ( u m ), where σ denotes a generator of Gal( H ρ | H ).The following is proved in [BH20, Cor. 3.17]. (5.3) Proposition. Using the notation introduced above, assume that ρ ( σ ) = H ρ : H ] + Z .Then we have ord H ρ ( β σ,ρ ) e H ρ | H = − ρ (rec H (N H ′ | H (Col u (0)))) in Q (cid:30) Z , where we write e H ρ | H for the ramification degree of the extension H ρ | H and rec H denotesthe local reciprocity map H × → Gal( H ′∞ | H ). Proof of Theorem (5.1):
First we observe that by [BKS17, Prop. 4.4 (iv)] we may reduceto the case W χ ⊆ S ram ( K χ, ∞ | K χ ) = { p } and Σ = S ∞ ( k ) ∪ { q | f χ m χ p } , where m χ isthe conductor of K χ , and f χ ⊆ O K is a non-zero ideal coprime to p such that O × K → ( O K / f χ m χ ) × is injective and no prime divisor of f χ splits completely in K χ | k and k ∞ | k .Since Conjecture (3.10) is trivial if W χ = ∅ , we may assume that W χ = { p } and shallhenceforth simply write W to denote this set.Let n = (Γ χ : Γ χ, p ) be the index of the decomposition group at p inside Γ χ , i.e. n isminimal such that p does not split in K χ, ∞ | K χ,n . We shall now first demonstrate that itsuffices to prove Conjecture (3.10) for the field K χ,n .By Proposition (3.5) (a) the Darmon derivatives κ and κ ′ n of η Ell with respect to thetopological generators γ ∈ Γ χ and γ p n ∈ Γ nχ , respectively, exist. By definition these arethe bottom values of norm-coherent sequences ( κ m ) m and ( κ ′ n + m ) m which satisfy( γ − · κ m = η Ell K χ,m and ( γ p n − · κ ′ n + m = η Ell K χ,n + m for all m big enough. It follows that we have( γ − · N Γ χ,n · κ ′ n + m = ( γ p n − · κ ′ n + m = η Ell K χ,n + m , hence, by uniqueness, we must have ( κ n + m ) m = (N Γ χ,n κ ′ n + m ) m and it follows that κ =N χ,n · κ ′ n = p n N Γ χ,n κ ′ n . This implies that κ ⊗ ( γ −
1) = p n N Γ χ,n κ ′ n ⊗ ( γ −
1) = (N Γ χ,n κ ′ n ) ⊗ ( γ p n −
1) (35)inside U K χ , Σ ,T ⊗ Z p I (Γ χ ) /I (Γ χ ) . As I (Γ χ,n ) /I (Γ χ,n ) is Z p -torsion free, the inclusion U K χ , Σ ,T ֒ → U K χ,n , Σ ,T induces an injection U K χ , Σ ,T ⊗ Z p I (Γ χ,n ) (cid:30) I (Γ χ,n ) ֒ → U K χ,n ⊗ Z p I (Γ χ,n ) (cid:30) I (Γ χ,n ) (36)35 Abelian extensions of imaginary quadratic fields: The conjecture of Mazur-Rubin andSano for elliptic units that allows us to view (35) as an equality inside the right hand side of (36). Assumingthe validity of the conjecture for K χ,n , we may therefore continue the calculation in (35)as follows: N Γ χ,n ( κ ′ n ⊗ ( γ p n − Γ χ,n (Rec W ◦ Ord − W )( η K χ,n )= (Rec W ◦ Ord − W )(N Γ χ,n η K χ,n )= (Rec W ◦ Ord − W )( η K χ ) , (37)where the last equality holds since by assumption S ( K χ ) = S ( K χ,n ). Observe that wehave a commutative diagram U K χ,n , Σ ,T Q p · U K χ,n , Σ ,T ⊗ Z p I (Γ χ,n ) (cid:30) I (Γ χ,n ) U K χ , Σ ,T Q p · U K χ , Σ ,T ⊗ Z p I (Γ χ,n ) (cid:30) I (Γ χ,n ) , Rec W ◦ Ord − W Rec W ◦ Ord − W where the right hand vertical arrow is induced by (36). We caution the reader that thetwo horizontal arrows, although both labelled Rec W ◦ Ord − W , do not coincide but thatinducing the bottom arrow from G χ to G χ,n gives the top arrow.Given this commutative diagram, the equations (35) and (37) taken together finish theproof of the claim. We therefore may, and will, assume without loss of generality that p has full decomposition group in K χ, ∞ | K χ .Let w be a place of K χ above p and choose an embedding ι w : Q ֒ → Q p that restricts to w on K χ . In the following, we will denote the completion of a finite abelian extensionfield F of k at the place induced by ι w by e F . Put H = f K χ and H ′ = ^ k ( fm χ ). Using that H ′ n = ^ k ( fm χ p n +1 ), we can then define a norm-coherent sequence u = ( u n ) n ∈ lim ←− n O H ′ n bysetting u n = ι w ( ψ fm χ p n +1 , a ) for all n > . (5.4) Lemma. We have Col u (0) = ι w ( ψ fm χ , a ) . Proof.
In the split case this is [Sha87, Chp. II, Sec 4.9, Prop.] combined with the evaluationof the power series at zero and an application of the monogeneity relation of Robert’s ψ -function. A more detailed proof of the split case is given in [OV16, Prop. 4.5] followingthe same strategy as [Sha87].We claim that essentially the same proof works in the non-split case. First observe that inthe proof of part (i) of the cited Proposition in [Sha87] the fact that the prime is split in k is not used. In part (ii) the condition that p is split is used to obtain a certain generatorof the Tate module ( ω n ) of the underlying formal group b E (because in this case the formalgroup b E is isomorphic to b G m and hence of height one). It is then shown that there existtorsion points u n which can be used to give an explicit description of the elements ω n at each level [Sha87, Chp. II, Sec. 4.4, (12)]. In the non-split case one can now invertthe strategy: Indeed, it is easy to see that there exist torsion points u n such that theexplicit description given in (12) is a generator of the Tate module of b E . Using this as thedefinition of ( ω n ), the remaining steps in the proof are exactly as in the split case.36 Abelian extensions of imaginary quadratic fields: The higher-rank equivariant IwasawaMain Conjecture in the split case
Fix a topological generator γ of Γ χ . We define the isomorphisms s γ : Γ χ −→ Z p , s γ,n : Γ χ,n −→ Z (cid:30) p n Z γ a a γ a a mod p n Z and the character ρ γ,n : Gal( H ′∞ | H ) π n −→ Gal( ] K χ,n | H ) s γ,n −−→ Z (cid:30) p n Z ∼ = p n Z (cid:30) Z , (38)where π n is the natural projection map induced by restriction. By definition, ρ γ,n is acharacter of finite order with kernel Gal( H ′∞ | ] K χ,n ), hence Proposition (5.3) combines withLemma (5.4) to reveal thatord ^ K χ,n ( β γ n ,ρ γ,n ) e ^ K χ,n | H ≡ − s γ,n ( π n (rec H (N H ′ | H ( ψ fm χ , a ))) p n mod Z (39)for all n ≥
0. By definition the Darmon derivative κ of η Ell with respect to γ is thebottom value of a norm-coherent sequence κ = ( κ n ) ∈ lim ←− n U K χ,n , Σ that satisfies( γ − ι w ( κ n ) = ι w ( η Ell K χ,n ) = N H ′ m | H ργ,n ( u m )for n big enough, thus we may take β γ n ,ρ γ,n ≡ ι w ( κ n ) mod f K χ × . We now obtain from(39) thatord ^ K χ,n ( ι w ( κ n )) = p n e ^ Kχ,n | H · ord H (N ^ K χ,n | H ( ι w ( κ n ))) = p n e ^ Kχ,n | H · ord H ( ι w ( κ )) ≡ − s γ,n ( π n (rec H (N H ′ | H ( ψ fm χ , a ))) mod p n Z . Taking the limit over n then givesord w ( κ ) = − s γ (rec w ( η Ell K χ )) (40)as an equality in Z p . By repeating the argument we also obtain equation (40) for theplaces σw , where σ ∈ G χ . Collating these equations, we find thatOrd W ( κ ) ⊗ ( γ −
1) = X σ ∈G χ ord σw ( κ ) σ ⊗ ( γ − − X σ ∈G χ s γ (rec σw ( η Ell K χ )) σ ⊗ ( γ − − Rec W ( η Ell K χ ) . By Lemma (3.14) this concludes the proof of Theorem (5.1).
The aim of this subsection is to prove the equivariant Iwasawa Main Conjecture for thedata ( K ∞ | k, Σ , T ) as defined in Section (5.1). This will be deduced from the classicalone-variable Iwasawa Main Conjecture in this setting.37 Abelian extensions of imaginary quadratic fields: The higher-rank equivariant IwasawaMain Conjecture in the split case
One-variable Main Conjecture
For any E ∈ Ω( K| k ) we let Ψ E be the Z p [ G E ]-modulegenerated by the roots of unity µ ( E ) contained in E and all norms N k ( n ) | k ( n ) ∩ E ( ψ n , b ) forall non-zero ideals n ( O k and all O k -ideals b which are coprime to 6 n . The group of elliptic units of E is then defined as C E = Ψ E ∩ U E .Furthermore, we set U E ∞ = lim ←− n U E n and C E ∞ = lim ←− n C E n . As in § K ∞ | k ) ∼ = ∆ × Z p for some finite abelian group ∆, thereby fixing a field L ∈ Ω( K| k ) such that ∆ = G L . If χ ∈ b ∆ is a character, we write Λ χ = Z p [im χ ] J Γ L K . (5.5) Theorem (Rubin, Vigui´e) . Assume that p splits in k . Let χ ∈ b ∆ be a character,then char Λ χ ( A ( L ∞ ) χ ) = char Λ χ (cid:0) U L ∞ (cid:30) C L ∞ (cid:1) χ . Proof. If p >
3, this is exactly the statement of [Vig14, Thm. 1.1]. However, the assump-tion p > loc. cit. and we explain how to adapt the last stepsof the proof to also cover the case of p ∈ { , } . Firstly, class field theory gives the exactsequence in (7.11). For the validity of (7.12) we may appeal to [OV16] (instead of [Gil85]),and to prove (7.13) we use [Vig12, Thm. 1.1] (instead of [Sha87, Chp. III, 2.1, Thm.]).The remaining part of the proof does not make use of the hypothesis p > Equivariant Iwasawa Main Conjecture
We first recall the (higher-rank) equivariantIwasawa Main Conjecture in this setting as proposed in [BKS17, Conj. 3.1 and Rk. 3.3]. (5.6) Conjecture.
There exists a V -basis L K ∞ | k, Σ ,T of Det V ( D • K ∞ , Σ ,T ) such that, forevery character χ ∈ b ∆ and every integer n ≥
0, the image of L K ∞ | k, Σ ,T under the mapDet V ( D • K ∞ , Σ ,T ) Det Z p [ G χ,n ] ( D • K χ,n , Σ ,T ) U K χ,n , Σ ,T , Θ Kχ,n | k, Σ ,T where the last arrow is the projection map defined in § ε VK χ,n | k, Σ ,T . (5.7) Remark. In [BKS17] the authors assume that S contains S p ( k ) and that no finiteplace splits completely in the Z p extension under consideration. Formally, our Conjecture(5.6) is therefore slightly more general than [BKS17, Conj. 3.1]. (5.8) Theorem. If p is split in k , then Conjecture (5.6) holds.Before turning to the proof of Theorem (5.8), we first briefly recall some terminology usefulwhen dealing with the ring V (cf. [BKS17, § p be the p -Sylow subgroup of ∆and write ∆ ′ for the coprime-to- p part of ∆. A height-one prime q of V is called regular if p q , and singular otherwise.For any regular prime q there is a unique character χ q ∈ b ∆ / ∼ Q p such that V q coincideswith the localisation of Λ χ q [ p ] at the ideal q Λ χ q [ p ], where χ ∼ Q p χ ′ if and only if thereexists σ ∈ Gal( Q p | Q p ) such that χ = σ ◦ χ ′ . Moreover, there is a one-to-one correspondencebetween the set of singular primes of V and c ∆ ′ / ∼ Q p , which to every singular q associatesa character χ q with the property that V q coincides with the localisation of Λ χ q [∆ p ] at q Λ χ q [∆ p ]. 38 Abelian extensions of imaginary quadratic fields: The higher-rank equivariant IwasawaMain Conjecture in the split case
For a prime ideal q ⊆ V of height one we define a subset Υ q ⊆ b ∆ / ∼ Q p byΥ q := ( { χ q } , if q is regular, { χ ∈ b ∆ / ∼ Q p | χ | ∆ ′ = χ q } if q is singular. Proof of Theorem (5.8):
This is essentially the same as the proof of [Ble06, Thm. 5.1]. Assuch we only indicate the main steps. We first remark that the µ -invariant of A ∞ Σ ,T ( K ∞ )vanishes on every character component. Indeed, by the argument of [BD20, Lem. 4.13]we may assume T = ∅ (note that in loc. cit. the notation Σ is used for the set T ) and sothe claim follows from the main result of [OV16]. Thus, [BKS17, Prop. 3.11] implies thatConjecture (5.6) holds if and only if there is an equality V q · ε q K ∞ | k, Σ ,T = Fitt V ( A Σ ,T ( K ∞ )) q · Fitt V ( X K ∞ , Σ \ V ) q · ( U K ∞ , Σ ,T ) q (41)for every prime ideal q ⊆ V of height one and ε q K ∞ | k, Σ ,T is the q -part of the Rubin-Starkelement as defined in [BKS17, § loc. cit. the assumption p > ∗ )”). However, at each place where the assumption p > k iscompletely split in any extension of k (since k is totally imaginary).Let us first assume that q is a regular prime and let χ = χ q be the character associatedto q . In this case (41) is equivalent toFitt χ (cid:0) U K χ, ∞ , Σ ,T (cid:30) h ε VK χ, ∞ | k, Σ ,T i Λ χ (cid:1) q = Fitt χ ( A Σ ,T ( K χ, ∞ )) q · Fitt χ ( X K χ, ∞ , Σ \ V ) q , (42)where we regard q as an ideal of Λ χ . From the exact sequence0 U K χ, ∞ , Σ ,T (cid:30) h ε VK χ, ∞ | k, Σ ,T i Λ χ U K χ, ∞ , Σ (cid:30) h ε VK χ, ∞ | k, Σ ,T i Λ χ lim ←− n M w ∈ T Kχ,n κ × K χ,n ,w A Σ ,T ( K χ, ∞ ) A Σ ( K χ, ∞ ) 0and (25) we deduce that (42) holds if and only ifFitt χ (cid:0) U K χ, ∞ , Σ (cid:30) h ε VK χ, ∞ | k, Σ ,T i Λ χ (cid:1) q = (Frob a − N a ) − · Fitt χ ( A Σ ( K χ, ∞ )) q · Fitt χ ( X K χ, ∞ , Σ \ V ) q (43)holds. By [Ble06, (21) and (22)] (see also [Fla04, (5.12)]) we have exact sequences0 U K χ, ∞ U K χ, ∞ , Σ Y K χ, ∞ , { p } A ∞ ( K χ, ∞ ) A ∞ Σ ( K χ, ∞ ) 0and 0 X K χ, ∞ , Σ \ ( V ∪{ p } ) X K χ, ∞ , Σ Y K χ, ∞ , { p } ⊕ Y K χ, ∞ ,V . Taking into account that X K χ, ∞ , Σ ∼ = X K χ, ∞ , Σ \ V ⊕ Y K χ, ∞ ,V , these two exact sequencescombine to imply that (43) can be reformulated asFitt χ (cid:0) U K χ, ∞ (cid:30) h ε VK χ, ∞ | k, Σ ,T i Λ χ (cid:1) q = (Frob a − N a ) − · Fitt χ ( A ( K χ, ∞ )) q · Fitt χ ( X K χ, ∞ , Σ \ ( V ∪{ p } ) ) q . (44)39 Abelian extensions of imaginary quadratic fields: Proof of Theorem B
Let Σ χ = S n ≥ S ( K χ,n ), and put b = 1 if χ = 1 and b = 0 otherwise. It follows from theproperties of Rubin-Stark elements thatFitt χ (cid:16) h ( γ − b ε VK χ, ∞ , Σ χ,T i Λ χ (cid:30) h ε VK χ, ∞ , Σ ,T i Λ χ (cid:17) q = ( γ − − b · Y v ∈ Σ \ Σ χ (1 − Frob − v )= Fitt χ ( X K χ, ∞ , Σ \ Σ χ ) q , where γ ∈ Γ χ denotes a topological generator (see also [Fla04, Lem. 5.5] and [Ble06,Lem. 5.4]).Let C K ∞ ( a ) ⊆ C K ∞ be the submodule comprising only all norms of the elements ψ n , a forour fixed prime a (instead of all possible choices of auxiliary ideals b as in the definitionof C K ∞ ). The arguments in [Ble06, pp. 97-99] then show thatFitt V ( C K ∞ (cid:30) C K ∞ ( a )) q = (Frob a − N a ) V q (45)as well as C K ∞ ( a ) q = V q ( γ − b ε K χ, ∞ , Σ χ ,T . (46)We can reformulate Theorem (5.5) asFitt V ( U K ∞ (cid:30) C K ∞ ) q = Fitt V ( A ( K ∞ )) q = Fitt χ ( A ( K ∞ ,χ )) q , (47)hence combining the last four equations yields equation (44) as claimed, thereby finishingthe proof for regular primes.Let now q be a singular prime. Since the µ -invariant of A ( K ∞ ) vanishes, [BKS17,Prop. 3.15] shows that it is enough to demonstrate that for every character χ ∈ Υ q the µ -invariant of U K χ, ∞ , Σ ,T / h ε VK χ, ∞ , Σ ,T i Λ χ vanishes. To do this, we will show that in thechain of inclusionsΛ χ ε VK ∞ ,χ , Σ ,T ⊆ C K ∞ ,χ ( a ) ⊆ C K ∞ ,χ ⊆ U K ∞ ,χ ⊆ U K ∞ ,χ , Σ ,T (48)the quotients of any two subsequent modules have vanishing µ -invariant. Firstly, the quo-tient U K ∞ ,χ , Σ ,T /U K ∞ ,χ injects into the finitely generated Z p -module Y K ∞ ,χ , { p } so has van-ishing µ -invariant. From Theorem (5.5) and the vanishing of the µ -invariant of A ( K ∞ ) weknow that U K ∞ ,χ / C K ∞ ,χ has vanishing µ -invariant. As in [Ble06, p. 100-101] one can showthat the V q -units Frob a − N a and Q v ∈ Σ \ ( { p }∪ V ) (1 − Frob − v ) annihilate C K ∞ ,χ / C K ∞ ,χ ( a )and C K ∞ ,χ ( a ) / h ε VK ∞ ,χ , Σ ,T i Λ χ , respectively. This concludes the proof for singular height-oneprime ideals. We are finally in a position to prove Theorem B from the introduction. (5.9) Theorem.
Let p be a prime number, k an imaginary quadratic field, and K | k afinite abelian Galois extension with Galois group G .(a) If p splits in k , then eTNC( h (Spec( K )) , Z p [ G ]) holds.(b) If p does not split in k , then eTNC( h (Spec( K )) , Z p [ G ]) holds if the equivariantIwasawa Main Conjecture (5.6) holds for K ∞ = K · k ∞ , where k ∞ is one of the Z p -extensions provided by Theorem (4.13), and the data (Σ , T ).40 eferences Proof.
This will follow from [BKS17, Thm. 5.2]. Although our hypotheses are slightlyweaker than those utilised in loc. cit. (cf. Remark (5.7)), the proof of [BKS17, Thm. 5.2]does not require any changes if one replaces the complex C L,S introduced in [BKS17, § D • L,S we defined in § C L,S,T can be taken to be D • L,S,T .)We address each condition required for the application of said general result [BKS17,Thm. 5.2] separately: • The equivariant Iwasawa Main Conjecture for the split case is proven in Theorem (5.8). • In the non-split case, the Γ χ -invariants of A Σ ,T ( K χ, ∞ ) are finite for every character χ ∈ b G by Theorem (4.13). In the split case, this holds by Remark (4.12) (b). • The validity of the Iwasawa-theoretic Mazur-Rubin-Sano conjecture for all non-trivial characters χ ∈ b G is exactly Theorem (5.1). For the trivial character theset V ′ χ = Σ \ { p } consists only of places unramified in k ∞ | k , hence in this case theconjecture holds as a consequence of [BKS17, Prop. 4.4 (iv)]. (5.10) Corollary. If all prime factors of [ K : k ] are split in k , then eTNC( h (Spec( K )) , Z [ G ])holds. Proof. If p ∤ [ K : k ], then eTNC( h (Spec( K )) , Z p [ G ]) is equivalent to the p -part of theStrong Stark conjecture, see [Bur01, Lem. 2.2.7]. The Corollary therefore follows directlyfrom Theorem (5.9) and a slight extension (see Remark (5.11) (a) below) of the result ofBley [Ble98, Part II, Thm. 1.1] showing the validity of the p -part of the Strong StarkConjecture for primes p ∤ [ K : k ] in our setting. (5.11) Remark. (a) As we have already alluded to in the proof of Corollary (5.10),by taking into account the improvements of [Rub91, §
3] provided in [Rub94] onecan easily remove the condition p ∤ h k from Theorem 1.1 in [Ble98, Part II]. Con-sequently, the argument given in loc. cit. in fact proves that the p -part of the StrongStark Conjecture holds for abelian extension extensions of imaginary quadratic fieldsand primes p ∤ [ K : k ].(b) As illustrated by Corollary (5.10), the validity of the p -part of the eTNC for splitprimes p | h k allows for a significant improvement towards the integral eTNC. Pre-viously, one had to restrict to cases where k is one of only nine imaginary quadraticfields of class number one and all prime factors of [ K : k ] are split in k to get uncon-ditional results towards the validity of the eTNC for the pair ( h (Spec( K )) , Z [ G ]). References [Ble98] Werner Bley.
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Fonctions
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King’s College London, Department of Mathematics, London WC2R 2LS, UnitedKingdom
Email address: [email protected]
Ludwig-Maximilians-Universit¨at M¨unchen, Mathematisches Institut, Theresienstr.39, 80333 M¨unchen, Germany
Email address: [email protected]@math.lmu.de