An extension of the Fukaya-Kato method

In a groundbreaking paper, T. Fukaya and K. Kato proved a slight weakening of a conjecture of the author's relating modular symbols and cup products of cyclotomic units under an assumption that a Kubota-Leopoldt p-adic L-function has no multiple zeros. This article describes a refinement of their method that sheds light on the role of the p-adic L-function.


Introduction
In this paper, we explore the conjectural relationship between • modular symbols in the quotient P of the real part of the first homology group of a modular curve of level N by the action of an Eisenstein ideal, and • cup products of cyclotomic units in a second Galois cohomology group Y of the cyclotomic field Q(µ N ) with restricted ramification, More precisely, we consider maximal quotients of p-parts of the latter groups for an odd prime p dividing N on which (Z/NZ) × acts through a given even character θ via diamond operators and Galois elements, respectively. In [Sh1], we constructed two maps ϖ : P → Y and ϒ : Y → P and conjectured them to be inverse to each other, up to a canonical unit suspected to be 1 (see Conjecture 3.1.9). The map ϖ was defined explicitly to take a modular symbol to a cyclotomic unit, while ϒ was defined through the Galois action on the homology of a modular curve, or a tower thereof, in the spirit of the Mazur-Wiles method of proof of the main conjecture. By the main conjecture, both the homology group and the Galois cohomology group in question are annihilated by a power series ξ corresponding to a p-adic L-function. This power series ξ is (roughly) both a generator of the characteristic ideal of inverse limit of Galois cohomology groups up the p-cyclotomic tower and the constant term from an ordinary family of Eisenstein series for θ . Fukaya and Kato showed in [FK] that ξ ϒ • ϖ = ξ modulo torsion in P, where ξ is essentially the derivative of ξ . In Theorem 5.3.8, we show that this identity holds in P itself, employing joint work from [FKS2]. At least up to finite torsion in P, the conjecture then follows if ξ happens to be relatively prime to ξ in the relevant Iwasawa algebra.
Considerable progress has been made in the study of ϒ by Wake and Wang-Erickson [WWE] and Ohta [Oh3], by different methods. In cases that ϒ is known to be an isomorphism and Y is pseudo-cyclic, the identity of Fukaya and Kato implies the original conjecture, i.e., up to unit. This pseudo-cyclicity was related to the question of localizations of Hecke algebras being Gorenstein in the work of Wake and Wang-Erickson, as well as to the question of ϒ being a pseudo-isomorphism. Ohta shows that ϒ is in fact an isomorphism under an assumption on the revelant Dirichlet character that holds in the case of trivial tame level. We note that this implies in particular that P has no torsion in such eigenspaces, as Y does not.
The pseudo-cyclicity of Y is expected to hold as a consequence of a well-known and widely believed conjecture of Greenberg's on the finiteness of the plus part of the unramified Iwasawa module. Moreover, since the p-adic L-functions in question are unlikely to ever have multiple zeros, one would expect the unit in our conjecture to always be 1, as in its stronger form. Nevertheless, this might appear to reduce the conjecture to chance, which is less than desirable. This motivates us to attempt a finer study.
Our aim in this paper is to study the role of ξ in the work of Fukaya-Kato and ask whether it is possible to remove it in the method. As we shall see, this would be possible but for a global obstruction that stands in the way. We make this explicit by deducing an equivalent form of our conjecture in Theorem 5.5.1. The key idea is to consider cohomology groups which are intermediate between the restricted Galois cohomology of Q and Iwasawa cohomology over the cyclotomic Z p -extension of Q in an atypical sense. That is, via Shapiro's lemma, we may view Iwasawa cohomology as a cohomology group over Q with coefficients in an induced module. We consider the cohomology of a quotient of these induced coefficients by an arithmetically relevant two-variable power series. In particular, the cohomology of this intermediate quotient (see Definition 4.2.1) is not the cohomology of any intermediate extension. Crucial to this work is a rather peculiar, but surprisingly clean and quite general, construction of intermediate Coleman maps in Section 4.2.
We also show that the global obstruction would vanish under a divisibility of Beilinson-Kato elements by one minus the pth Hecke operator at an intermediate stage between Iwasawa cohomology and cohomology at the ground level: see Question 5.5.2. This "intermediate global divisibility" can be rephrased as the existence of a certain intermediate zeta map. The global obstruction to our conjecture corresponds to a weaker statement of existence of what would be a reduction of this map modulo the Eisenstein ideal. This reduced map can be characterized by properties of compatibility with a zeta map at the ground level of Q and with a p-adically local version of the intermediate zeta map which we show does indeed exist.
Of course, this leaves us with the question of whether these intermediate zeta maps are likely to exist. As such, we perform a feasibility check for an analogue of the conditions of Theorem 5.5.1 in a simpler setting, with cyclotomic units in place of Beilinson-Kato elements. That is, in Section 6, we explore the analogues of global obstruction and divisibility for cohomology with coefficients in a Tate module, rather than theétale homology of a tower of modular curves. We show that the global obstruction in the cohomology of the intermediate quotient does in fact vanish in this setting, while verifying intermediate global divisibility only under an assumption of vanishing of a p-part of a class group of a totally real abelian field. This is in line with our suspicions that intermediate global divisibility may be too much to hope for in general, while still lending some credence to the conjecture that ϒ and ϖ are indeed inverse maps, and not just by chance. b. We let T ord denote the inverse limit T ord = lim ← − r H 1 et (X 1 (N p r ) /Q , Z p (1)) ord of ordinary parts of firstétale cohomology groups of the closed modular curves X 1 (N p r ).
Remark 2.1.3. By viewing Q as the algebraic numbers in C, we have an identification T ord,+ ∼ = S ord of h ord -modules induced by the usual (i.e., complex) Eichler-Shimura isomorphisms at each stage of the modular tower. We note that Hecke actions on inverse limits of cohomology (as opposed to homology) groups are via the dual, or adjoint, operators.
Similarly but less crucially for our purposes, we have the following H ord -modules. Similarly, we letT ord c denote the inverse limit of the ordinary parts of the compactly supportedétale cohomology groups H 1 c,ét (Y 1 (N p r )/ Q , Z p (1)).
As in the cuspidal case, the H ord -modules M ord andT ord,+ are isomorphic.
Remark 2.1.5. Since signs are quite subtle in this work, we mention some conventions of algebraic topology used here and in [FK] (cf. [Ka,2.7]), as well as some calculations which follow from them. Consider the compatible G Q -equivariant Poincaré duality pairings onétale cohomology: Viewing Q as the algebraic numbers in C, these are compatible with the usual pairings of Poincaré duality for the isomorphic Betti cohomology groups of the C-points of the modular curves, which are given by evaluation of the cup product on a fundamental class given by the standard orientation of the Riemann surface X 1 (N p r )(C). These cup products induce identifications that take a class g to the unique homology class γ such that the map h → g ∪ h agrees with evaluating the cohomology class g on γ. Now, any unit g on Y 1 (N p r ) /Q gives rise via Kummer theory to a similarly denoted class in H 1 et (Y 1 (N p r ) /Q , Z p (1)). The order ord x g of the zero of g at a cusp x satisfies is the image of x under the canonical connecting map, and where ∂ x g is the boundary at x in H 0 ({x}, Z p ) ∼ = Z p of the relative homology class corresponding to g. These identities can be verified by comparison with de Rham cohomology: for a smooth function η x that is 1 on a small closed disk D x about x and 0 outside of a larger one in Y 1 (N p r ). On the other hand, if g is sent to the class of γ then where the sum is taken over all cusps y of X 1 (N p r ).

Iwasawa modules
Note that we have a canonical decomposition Z × Definition 2.2.2. Set Λ = Z p 1 + pZ p , let χ denote the isomorphism let t ∈ 1 + pZ p be such that χ(t) = 1, let γ ∈ Λ be the group element defined by t, and set X = γ − 1 ∈ Λ.
Note that these definitions allow us to considerΛ as the Λ = Z p X -algebra Λ[∆/ −1 ].
b. Let Q ∞ denote the cyclotomic Z p -extension of Q, and set Γ = Gal(Q ∞ /Q).
We have an isomorphismΓ ∼ − → Z × p,N / −1 given by the cyclotomic character, which we use to identifyΛ with Z p Γ . We similarly identify Λ with Z p Γ . We also use this isomorphism to identify ∆/ −1 with a subgroup (and quotient) ofΓ.
Remark 2.2.4. Note that h ord is aΛ-algebra on which group elements act as inverses of diamond operators. At times, we may work withΛ-modules with distinct actions of inverse diamond operators and Galois elements. The action that we are considering should be discernable from context.
a. Let Z ∞ denote the integer ring of Q ∞ , and let Z r be the ring of integers of the extension c. Let Q p,∞ denote the cyclotomic Z p -extension of Q p , and let Q p,r denote the unique degree p r−1 extension of Q p in Q p,∞ .
Definition 2.2.6. For any algebraic extension F of Q, we consider the set S of primes dividing N p∞. We let G F,S denote the Galois group of the maximal S-ramified extension of F.
We may viewΓ as a quotient of G Q,S .
Definition 2.2.7. For aΛ-module M, we consider M as aΛ G Q,S -module M ι by letting σ ∈ G Q,S act by multiplication by the inverse of its image inΓ.
In particular, by taking completed tensor products with Λ ι , we may define Iwasawa cohomology groups. (Under our conventions, σ ∈ G Q,S acts on Λ ι by multiplication by the inverse of its image in Γ.) Definition 2.2.8. For a pro-pétale O-sheaf (or compact Z p G Q,S -module) F , the ith Sramified Iwasawa cohomology group of F is where we omit the subscript "ét" from ourétale (or, really, continuous Galois) cohomology groups.
We will frequently omit the word "compact" (and "Hausdorff") when considering compact (Hausdorff) modules over a completed group ring with coefficients in a compact (Hausdorff) Z p -alge.bra.
Remark 2.2.9. a. We similarly have compactly supported Iwasawa cohomology groups and local-at-p Iwasawa cohomology groups for pro-pétale O and Q p -sheaves F , respectively. We will also consider Iwasawa cohomology for O ∞ [µ N p ], defined usingΛ in place of Λ.
b. Via Shapiro's lemma, we may make the identification with the inverse limit taken with respect to corestriction maps, and similarly for the other types of Iwasawa cohomology groups, where in the local setting, the isomorphism is with a product of inverse limits of cohomology groups over primes over p.

Local actions at p
Definition 2.3.1. We fix an even p-adic Dirichlet character θ : ∆ → Q p × .
Definition 2.3.2. For any Q p × -valued character α of a group, let R α denote the Z p -algebra generated by the values of α.
We consider R θ as a quotient of Z p [∆] via the Z p -linear map to R θ induced by θ .
Remark 2.3.4. Given aΛ-module M, we view M θ as a module over the complete local ring Λ θ := R θ Γ = R θ X . We will most typically think of Λ θ as the θ -part of the algebra of inverse diamond operators, whereas Λ will often (but not as consistently) be viewed as an algebra of Galois elements.
a. Let T ord quo (resp.,T ord quo ) denote the maximal unramified H θ G Q p -quotient of T ord θ (resp., T ord θ ).
b. Let T ord sub denote the kernel of the quotient map T ord θ → T ord quo , which is also the kernel ofT ord θ →T ord quo .
Ohta [Oh1, Section 4] constructed a perfect "twisted Poincaré duality" pairing Ty) for all x, y ∈ T θ and T ∈ h ord θ . This is compatible with an analogously defined pairing i.e., the fixed part of the completed tensor product for the diagonal action of the Frobenius Fr p , where W is the completion of the valuation ring of Q ur p .
Remark 2.3.8. In the notation of Definition 2.3.7, the following hold.
a. There is a (noncanonical) natural isomorphism between the forgetful functor from compact unramified R G Q p -modules to compact R-modules and D and under which each b. Endowing D(U) for each U with the additional action of ϕ = 1 ⊗ Fr p , any choice of natural isomorphism as above induces canonical isomorphisms The following Λ-adic Eichler-Shimura isomorphisms can be found in [FK,1.7.9] and extend work of Ohta from [Oh1].

Eisenstein parts and quotients
For an H ord -module M, we let M m denote its Eisenstein part: the product of its localizations at the maximal ideals containing T − 1 − for primes N p and U − 1 for primes | N p.
a. We define the cuspidal Hecke algebra h as the Eisenstein part h ord m of Hida's ordinary cuspidal Hecke algebra h ord .
b. The Eisenstein ideal I of h is the ideal generated by T − 1 − for primes N p and U − 1 for primes | N p in h m .
We also set H = H ord m and in general use the following notational convention. By applying this convention, we obtain H-modules S, M, S , M , T ,T , T quo ,T quo , and T sub . (Note that T sub and T quo are a submodule and a quotient of T θ , rather than just T .) It is only these Eisenstein parts that will be of use to us in the rest of the paper, so we focus solely on them, eschewing greater generality, but obtaining somewhat finer results in the later consideration of zeta elements.
We make the following assumptions on our even character θ .
Hypothesis 2.4.3. We suppose that the following conditions on θ hold: a. p divides the generalized Bernoulli number B 2,θ −1 .
b. θ has conductor N or N p, is canonically split as a sequence of h θ -modules.
We consider the following power series corresponding to the Kubota-Leopoldt p-adic Lfunction of interest.
Definition 2.4.6. Let ξ = ξ θ ∈ Λ θ be the element characterized by the property that Remark 2.4.7. The Mazur-Wiles proof of the main conjecture implies that (h/I) Definition 2.4.8. We let T = T θ /IT θ .
Proof. Consider the Manin-Drinfeld modification of the inverse limit of the first homology groups of X 1 (N p r ) relative to the cusps, which is isomorphic toT ⊗ H h by [Sh1, Lemma 4.1]. Its quotient by T is isomorphic to h/I, generated by the image e ∞ of the compatible sequence of relative homology classes {0 → ∞} r of the geodesic paths from 0 to ∞ in the upper half-plane [Sh1, Lemma 4.8]. The Λ θ -moduleT ⊗ H h is free, as it has no X-torsion and its quotient by X is R θ -free as the Manin-Drinfeld modification of the Eisenstein part of the relative homology of X 1 (N p) (cf. [FK, (6.2.9)]). By Remarks 2.3.10 and 2.4.7, we then see that ξ e ∞ must be an element of a Λ θ -basis of T θ (cf. [FK, (6.2.10)]). The desired surjection is given by y → ξ e ∞ , y on y ∈ T θ , using the nondegeneracy of Ohta's pairing (2.2).
Remark 2.4.10. We have made a sign change here from our original map and that of [FK,6.3.18]. That is, we pair with ξ e ∞ on the left, rather than the right.
We define P as the kernel of the quotient map T → Q, yielding an exact sequence of (h/I) θ G Q,S -modules. We recall the following from the main results of [FK, Section 6.3].
Proposition 2.4.11. The canonical maps P → T quo /IT quo and T sub /IT sub → Q are isomorphisms of (h/I) θ G Q p -modules. Moreover, the action of G Q,S on P is trivial, and P can be identified with the fixed part of T under any complex conjugation.
Proof. The cokernel of the map π : T sub /IT sub → Q is an (h/I) θ G Q p -module quotient of T quo /IT quo . The ∆ p -action on T quo /IT quo is trivial, while the ∆ p -action on Q is via ωθ −1 , so by Hypothesis 2.4.3d, we have that π is surjective. Moreover, T sub /IT sub and Q are both free of rank one over (h/I) θ , so π must also be injective. This forces the other map to be an isomorphism as well.
Next, let us briefly outline the argument of Kurihara and Harder-Pink yielding the triviality of the action on P, as in [FK,6.3.15]. By Lemma 2.4.5, we have a direct sum decomposition T = P ⊕ Q as (h/I) θ -modules, with P being G Q,S -stable. The character defining the determinant of the action of G Q,S on the modular representation in which T θ is a lattice reduces exactly to the character defining the action on Q. Consequently, G Q,S must act trivially on P. Since complex conjugation then acts trivially on P and as −1 on the quotient Q, we have the final claim.
Proof. The maps T θ /T + θ →T θ /T + θ and T sub →T sub are isomorphisms, so it suffices to show that T sub → T θ /T + θ is an isomorphism. We know that it is surjective by Proposition 2.4.11 and Nakayama's lemma. But T sub is a free h θ -module of rank 1, and T θ /T + θ is an h θ -module of rank 1, so the surjectivity forces the map to be an isomorphism.
As in [FK,6.3.4], we see that our sequence (2.4) is uniquely locally split.
Proposition 2.4.13. The sequence (2.4) is uniquely split as a sequence of (h/I) θ G Q -modules for every | N p.
a. The G Q,S -action on P is trivial, and we have a canonical isomorphism P ∼ = S θ /IS θ of h θ -modules. For this, note that U p acts as an arithmetic Frobenius on T quo by [FK,1.8.1] and that D(T quo ) ∼ = S θ , and apply Proposition 2.4.11 and Remark 2.3.8(b).
b. The p-adic L-function ξ divides the Λ θ -characteristic ideal of P (for the action of inverse diamond operators) by an argument of Mazur-Wiles and Ohta (see [FK,7.1.3]).
Putting these isomorphisms together with Remark 2.3.8a and Proposition 2.3.9, we have isomorphisms Note that the first of each of these pairs of isomorphisms is noncanonical, only becoming canonical upon reduction modulo U p − 1, but we can and do fix compatible choices.

Cohomological study
In this section, we first introduce known results on the cohomology of the reduced lattice that is the quotient T of T θ by the Eisenstein ideal. We recall the work of Fukaya and Kato [FK] in which the derivative ξ of a Kubota-Leopoldt p-adic L-function ξ appears in the study of certain connecting homomorphisms in the cohomology of subquotients of T (1). We then perform an analogous study, replacing T by a certain "intermediate" quotient T † of Λ ι⊗ Z p T , and we show that in this setting the role of ξ is played more simply by 1.

Cohomology of the reduced lattice
, Z p (2)) θ and consider it as a Λ θ -module for the action of inverse diamond operators.
Remark 3.1.2. Let Y denote the θ -eigenspace of the Tate twist of the minus part of the unramified Iwasawa module over Q(µ N p ∞ ). Then the canonical maps are isomorphisms by our hypotheses on θ . In particular, the characteristic ideal of Y is generated by ξ by the Iwasawa main conjecture.
Lemma 3.1.4 (Fukaya-Kato). The cohomology groups H i (O, Q(1)) are zero for i / ∈ {1, 2} and are isomorphic to Y otherwise, More precisely, the connecting map in the long exact sequence ) vanishes since it is isomorphic to the Tate twist of the group of norm compatible systems of p-completions of p-units in the cyclotomic Z p -extension of Q(µ N p ), its θ -eigenspace is zero since θ is even, not equal to ω 2 , and Hypothesis 2.4.3d holds. Since G Q,S has p-cohomological dimension 2, we have an exact sequence in which the middle map is zero by Stickelberger theory (or the main conjecture and the fact that Y has no p-torsion).
We also note the following simple lemma on the compactly-supported cohomology of P.
Lemma 3.1.5. The compactly supported cohomology groups H i c (O, P(1)) are zero for i / ∈ {2, 3} and are isomorphic to P otherwise. For i = 3, the isomorphism is given by the invariant map, whereas for i = 2, we have a canonical isomorphism H 2 c (O, P(1)) ∼ = Γ⊗ Z p P of Poitou-Tate duality that we compose with the map induced by −χ : Γ → Z p . Moreover, the natural maps Proof. This is well-known for i = 3, since the compactly supported cohomology of O has p-cohomological dimension 3 and P has trivial Galois action. That is, we have canonical isomorphisms For i = 1, we similarly have Since the above arguments work for any compact Z p -module M with trivial G Q,S -action, the functors M → H i c (O, M(1)) are exact for i = 2, 3 and are trivial for all other i. The maximal pro-p, abelian, S-ramified extension extension of Q is Q ∞ in that no prime dividing N is 1 modulo p, so we have and we apply the isomorphism −χ : Γ → Z p to obtain the result. A similar argument gives the analogous statements for Z[ 1 p ] and through it the isomorphisms.
We can define a cocycle b : G Q,S → Hom h (Q, P) using the exact sequence (2.4) by Then b restricts to an everywhere unramified homomorphism on the absolute Galois group of Q(µ N p ∞ ) by Proposition 2.4.13, which we can view as having domain Y by Remark 3.1.2. Through the isomorphism of Proposition 2.4.9, we have moreover a canonical isomorphism Hom h (Q, P) ∼ = P of Λ θ -modules. The result is the desired map ϒ (see [Sh1,Section 4.4], though note that we have not multiplied by any additional unit here).
Definition 3.1.6. Let ϒ : Y → P denote the homomorphism of Λ θ -modules induced by b and Proposition 2.4.9.
We also have a map in the other direction that takes a trace-compatible system of Manin symbols to a corestriction compatible system of cup products of cyclotomic units.
Definition 3.1.7. Let ϖ : S θ → Y denote the map constructed in [Sh1, Proposition 5.7], with reference to [FK,5.2.3], where the latter is shown to factor through P.
We also use ϖ to denote the induced map ϖ : P → Y .
Remark 3.1.8. We recall that ϖ is the restriction of the inverse limit under trace and corestriction of maps We briefly define the symbols that appear.
On the right hand side of (3.1), the symbol ( , ) r denotes the pairing on cyclotomic N punits induced by the cup product and Kummer theory, noting that the image of the pairing lands in the plus part of H 2 .
On the left hand side of (3.1), we have where w r is the Atkin-Lehner involution of level N p r and the matrix a b We project the resulting element to the plus part after the operations, denoting this with ( ) We recall the conjecture of [Sh1].
Conjecture 3.1.9. The maps ϖ : P → Y and ϒ : Y → P are inverse maps.
Actually, Conjecture 3.1.9 was originally conjectured by the author up to a canonical unit. There were indications that this unit might be 1 (if sign conventions were correct), but while the author advertised this suspicion rather widely and included it in preprint versions of the paper, he opted not to conjecture it in the final published version. It was the work of Fukaya and Kato in [FK] that finally made it clear that the unit should indeed be 1, not least because one would expect that the hypotheses under which they can prove it should hold without exception. Nevertheless, one does not actually know how to prove that their hypotheses always hold. Indeed, this paper is motivated by a desire to explore where the difficulty lies in removing them.
Remark 3.1.10. Hida theory tells us that the Λ θ -characteristic ideal of P is divisible by (ξ θ ), and the main conjecture of Iwasawa theory as proven by Mazur-Wiles tells us that the Λ θcharacteristic ideal of Y is equal to (ξ θ ). As Y is well-known to be p-torsion free (i.e., by results of Iwasawa and Ferrero-Washington), Conjecture 3.1.9 is reduced to showing that ϒ • ϖ = 1 on P.
Consider the complex where "C" here is used to denote the standard cochain complexes and the map in the cone uses the local splitting T → P. We have an exact sequence of complexes where C c is the complex defining compactly supported cohomology, and which has connecting homomorphisms for i ≥ 0. For i = 1, let us denote this connecting homomorphism by Θ. The connecting homomorphism for i = 2 can be identified with ϒ: see [FK,9.4.3], though note that we obtain that they obtain the opposite sign. One can simply take this as the definition of ϒ for the purposes of this article. Nevertheless, we give a fairly detailed sketch of the proof using the results of [Sh2], as it is by now an old result due independently to the author.
Lemma 3.1.11. Under the identifications of Lemmas 3.1.4 and 3.1.5, the connecting homomorphism Proof. We consider a diagram where the connecting homomorphism that is the lower map is given by left cup product with b : The left vertical map employs the surjection Λ ι θ (1) → Q determined by Proposition 2.4.9, and the right vertical map uses the quotient mapΛ ι → Z p , which is to say it becomes corestriction via Shapiro's lemma. The diagram is then commutative taking the upper horizontal map to be given by left cup product with the cocycle G Q(µ N p ∞ ),S → P given by following the restriction of b with evaluation at the canonical generator of Q. Recall that this cocycle is a homomorphism that by definition factors through ϒ : Y → P. That the upper horizontal map then agrees with ϒ via the identifications of the groups with Y and P is seen by noting that it is Pontryagin dual via Poitou-Tate duality to the Pontryagin dual of ϒ, via an argument mimicking the proof of [Sh2, Proposition 3.1.3] (noting Proposition 2.4.3 therein, which in particular implies that the signs agree).
Remark 3.1.12. The connecting map H 2 (O, Q(1)) → H 3 c (O, P(1)) that we use is the negative of the corresponding map in [FK], since the identification of Q with (h/I) ι θ (1) of Proposition 2.4.9, and hence of Y with H 2 (O, Q(1)), is of opposite sign to that of [FK,6.4.3].
Definition 3.1.13. For a Z p G Q,S -module M, let ∂ M denote a connecting homomorphism in a long exact sequence in cohomology attached to the Tate twist of the exact sequence Remark 3.1.14. The maps ∂ M for any Z p G Q,S -module M agree with left cup product by the cocycle −χ defining the extension class (3.2) (cf. [Sh2, Proposition 2.3.3]). As pointed out in [FK,9.3.4], the sign in −χ occurs as G Q acts on Λ ι through left multiplication by the inverse of its quotient map to Γ ⊂ Λ.
Lemma 3.1.15. Let M be a compact or discrete Z p G Q,S -module. Then the diagram anticommutes.
Proof. Recall that which is to say that takes a class φ to the image of the compactly-supported cocycle (∂ M (φ ), 0), whereas the composition , 0) in that the differential used to compute the connecting homomorphism restricts to the negative of the local differential.
We also have the following lemma.
Lemma 3.1.16. The connecting homomorphism ∂ P : H 2 c (O, P(1)) → H 3 c (O, P(1)) is identified with the identity map on P via the isomorphisms of Lemma 3.1.5.
Proof. As noted in Remark 3.1.14, the connecting map ∂ P is given by left cup product with −χ ∈ H 1 (O, Z p ). By the commutativity (with elements of the even degree cohomology group H 2 c (O, P(1))) and associativity of cup products, ∂ P is Poitou-Tate dual to the map The following exercise in Galois cohomology encapsulates a key aspect of the work of Fukaya-Kato [FK, Sections 9.3-9.5]. We omit the proof, as the reader will find its key ideas contained in the refined study that follows (cf. Proposition 3.3.9 for the commutativity of the lefthand square and Lemma 3.3.6 for the middle square on the right). In particular, note that Θ is identified with −ξ ϒ as a map Y → P. In this section, we aim to remove the derivative by modifying the diagram.

Intermediate quotients
Let⊗ Z p denote the completed tensor product over Z p . We use it consistently even in cases for which the usual tensor product gives the same module (in part, to indicate that our modules carry a compact topology). In the following, when we write Λ (as opposed to Λ ι ), we shall consider it as carrying a trivial G Q,S -action.
Let a i ∈ R θ be such that ξ = ∑ ∞ i=0 a i X i . We then have thatξ is given by multiplication by the identically denoted element Definition 3.2.3. For n ≥ 0, define ξ (n) : Λ ι θ → Λ ι θ to be the continuous Λ θ G Q,S -module homomorphism given by multiplication by Remark 3.2.4. Note that ξ (1) is ξ of Proposition 3.1.17.
We make the identification and frequently refer to X ⊗ 1 ∈ Λ⊗ Z p Λ θ more simply by X.
While not used later, the following description ofξ gives one some insight into its form.
Proof. For n ≥ 1, setξ which we aim to prove lie in Λ⊗ Z p Λ θ , so X nξ n tends to zero. It suffices to show that First, noting the simple identity (x − y) i = (x − y) ∑ i j=1 x j−1 y i− j , we have that Suppose then that (3.3) holds for some n ≥ 1. Since Xξ n+1 =ξ n − X n ⊗ ξ (n) , we have completing the induction.

Refined cohomological study
Let us set and similarly for P and Q. We shall give this dagger notation a more general definition in Section 4. We first consider Q † .
Definition 3.3.1. Given a Λ θ -module M for which we consider Λ⊗ Z p M as a Λ⊗ Z p Λ θmodule whereby f ∈ Λ⊗ Z p Λ θ acts by multiplication by w( f ), we letξ denote the Λ⊗ Z p Λ θmodule endomorphism of Λ⊗ Z p M induced by the action of 1 ⊗ ξ .
Remark 3.3.2. Viewed as an element of the ring Proposition 3.3.3. We have isomorphisms Proof. We have a commutative diagram S -modules with exact rows and columns, where the maps In it, we view f ∈ Λ⊗ Z p Λ θ as acting on the leftmost two columns as w( f ) and on the rightmost column by f (and in that sense 1 ⊗ ξ should be understood as the endomorphism induced byξ ). In particular, this provides a canonical isomorphism of (Λ⊗ Z p Λ θ ) G Q,S -modules, again understanding that f ∈ Λ⊗ Z p Λ θ acts on the right by w( f ). As Λ is Z p -free with trivial G Q,S -action, we have for all i. So, we have exact sequences of Λ⊗ Z p Λ θ -modules for all i. Asξ is a unit times a distinguished polynomial in Λ θ X and the leading coefficient of a distinguished polynomial is 1, the endomorphismξ has no kernel on the modules Λ⊗ Z p H i (O, Q(1)). Hence, the exact sequence provides the result.
Consider the exact sequences of Λ G Q,S -modules, with the action of Λ induced from the action on Q. For the latter sequence, note that multiplication byξ induces an isomorphism We similarly have exact sequences where Xξ 1 =ξ − 1 ⊗ ξ . The following refines Proposition 9.3.3 of [FK] in our case of interest.
of the isomorphisms of Lemma 3.1.4. Then ∂ † Q fits in a commutative diagram in which the horizontal maps are connecting homomorphisms of (3.5) and (3.4) and the vertical maps are induced from maps in (3.6) and (3.7) and are surjective and injective, respectively.
Proof. The commutativity of the outside square follows from the morphism of exact sequences By Proposition 3.3.3, the left-hand vertical map is identified with the reduction modulo X map and the right-hand vertical map is identified with the canonical injection given by w(ξ 1 ).
It is sufficient to verify the commutativity of the upper triangle in the diagram of the proposition. The commutative diagram with exact rows gives rise to a commutative diagram Noting (3.8), the upper triangle commutes.
Proposition 3.3.5. We have a commutative square of isomorphisms between Λ-modules canonically isomorphic to P, in which every vertical and horizontal map is identified with the identity map on P. The same holds with O replaced with Z[ 1 p ].
Proof. First, note that the diagram commutes as in the proof of Proposition 3.3.4. We have as compactly supported cohomology has p-cohomological dimension 3, and an exact sequence Note that H 2 c (O, Λ ι⊗ Z p P(1)) is isomorphic to the tensor product with P of the Galois group of the maximal abelian pro-p, S-ramified extension of Q ∞ , which is trivial (since no prime dividing N is 1 modulo p), so the first two terms are zero. The last map is also zero since multiplication byξ is trivial on P. Thus, we have We choose the identification of H 2 c (O, P † (1)) with P which makes this the identity map, and the right-hand vertical map is identified with the identity map on P via invariant maps. As for the upper map, note that it factors as where the first map is the connecting homomorphism, which is seen to be the identity map by using Poitou-Tate duality as in Lemma 3.1.16, and the second map is again clearly identified with the identity map on P. The same argument works with O replaced by Z[ 1 p ].
The following is result is a special case of the anticommutativity of connecting homomorphisms for a commutative square of short exact sequences of complexes.
By Proposition 3.3.4, the upper horizontal map in the diagram of Lemma 3.3.6 factors as and these maps are identified with where the first map is the quotient map. Together with Proposition 3.3.5, it then follows that Θ † factors as for some map Φ satisfying −∂ † P • Φ = ϒ • ∂ † Q , which we may also view as a map Φ : Y → P. Proof. Consider the commutative diagram with exact columns. The right-hand column is isomorphic to the quotient byξ of the middle terms of the short exact sequence where the first map is injective since H 2 c,Iw (O ∞ , P(1)) = 0. In the second sum, since any prime dividing N is inert in Q ∞ and p is totally ramified, we have H 2 Iw (Q ,∞ , P(1)) ∼ = P for every | N p. The third term is also isomorphic to P via the invariant map. As these groups are killed byξ , the sequence remains exact upon taking the quotient by the action ofξ , and the map is an isomorphism. By the diagram (3.9), it therefore suffices to show that H 1 (Q , Q † (1)) = 0 for all primes | N. We verify this claim.
Let K = Q (µ N p ∞ ), and set Γ = Gal(K /Q ), Inflation-restriction provides an exact sequence We have H 1 (K , Q † (1)) ∼ = Q † by Kummer theory and the valuation map (since all roots of unity are infinitely divisible by p in K × ). As ∆ acts on Q † through the restriction of θ −1 , the ∆ -invariants of Q † are trivial by Hypothesis 2.4.3b. So, we have H 1 (K , Q † (1)) Γ = 0. Moreover, since ∆ has prime-to-p order, inflation provides an isomorphism and again the inertia subgroup of ∆ acts nontrivially on Q † (1) by assumption.
Proof. Applying Lemma 3.3.7, we have a diagram H 2 c (O, P † (1)) with exact rows and columns. The snake lemma map from the diagram is then the negative of the connecting homomorphism Θ † by a standard lemma.
We now have that all squares in the diagram are commutative.

Local study
In this section, we let R denote a complete Noetherian semi-local Z p -algebra. We let A denote an unramified R G Q p -module. Exactly when discussing this general setting, we shall allow p to be any prime.

Coleman maps
Let U ur ∞ (resp., K ur ∞ ) denote the p-completion of the group of norm compatible sequences of units (resp., of nonzero elements) in the tower given by the cyclotomic Z p -extension Q ur p,∞ of Q ur p .
Definition 4.1.1. The Coleman map Col : K ur ∞ → X −1 W X is the unique map of Λ-modules restricting to a map U ur ∞ → W X = W 1 + pZ p defined on (u r ) r≥1 ∈ U ur ∞ with u r ∈ Q ur p,r by Here, W X acts continuously and W -linearly on W x with the result of h ∈ W X acting on x denoted by [h](x), via the action determined by [a](x) = x a ∈ W x for a ∈ 1 + pZ p . Also, f (x) ∈ W x − 1 is the Coleman power series with f (ζ p r ) = Fr r p (u r ) for all r, and ψ is defined on g(x) ∈ W x by ψ(g)(x) = Fr p (g)(x p ).
We can extend this definition as follows.
Definition 4.1.2. The Coleman map for A is the map where Fr p acts diagonally on the tensor products.
The following is a slight extension, allowing A Fr p =1 to be nonzero, of the restriction of [FK, 4.2.7] to invariants for ∆ ∼ = Gal(Q p (µ p ∞ )/Q p,∞ ). Note that Col A agrees with the map denoted Col in [FK] on the fixed part under Gal(Q p (µ p )/Q p ). Proof. Since A(1) has no G Q ur p,∞ -fixed part and Gal(Q ur p,∞ /Q p,∞ ) ∼ =Ẑ has cohomological dimension 1, the inflation map Inf in the definition of Col A is an isomorphism. It is well-known that the Coleman map Col is injective and, as follows for instance from the proof of [FK,4.2.7], it restricts to an isomorphism U ur ∞ ∼ − → W X . In particular, Col A is injective. It follows that we have an exact sequence with the first map the inverse of 1 ⊗ Col and the second determined by the valuation map on the norm to Q ur p of an element of K ur ∞ . The kernel of 1 − Fr p applied to this sequence gives the surjectivity since K ur ∞ contains the Frobenius fixed sequence that is the projection of (1 − ζ p n ) n to the ∆-invariant group. By the injectivity of Col A , this forces the induced map A Fr p =1 → X −1 D(A) X /D(A) X to have image X −1 A Fr p =1 . Since the image of Col A contains D(A) X , it must then equal C(A).
In addition to Col A , we also have a homomorphism at the level of Q p that can be defined as follows, following [FK,4.2

.2].
Definition 4.1.5. We let denote the composition where the first map is induced by restriction and the map is given by projection to the second coordinate.
Remark 4.1.6. The map Col A is in general only split surjective, with a canonical splitting given by the valuation map  A(1)).
Proof. By replacing A by A/(Fr p − 1)A, we may suppose that A has trivial Galois action, and it then suffices to consider A = Z p . The connecting homomorphism ∂ A is given by left cup product with −χ by Remark 3.1.14. Note that for a ∈ Q × p , we have χ ∪ a = χ(ρ(a)), where ρ : Q × p → G ab Q p is the local reciprocity map (cf. [Se, Chapter XIV, Propositions 1.3 and 2.5]). But ρ(u)(ζ p n ) = ζ u −1 p n for u ∈ 1 + pZ p and ρ(p)(ζ p n ) = ζ p n . Then −χ(ρ(p)) = 0 and −χ(ρ(u)) = (1 − p −1 ) log(u) for u ∈ 1 + pZ p . Thus, The relationship between Col and Col is given by the following [FK,4.2.9].
Proposition 4.1.8. Let ev 0 : D(A) X → D(A) denote evaluation at 0, and let cor be the core- A(1)). Then we have

Intermediate Coleman maps
In this subsection, we aim to construct a map Col † A that plays an analogous role to Col A for a certain quotient of Λ ι⊗ Z p A. We suppose that R is local to simplify the discussion and fix an element with nonzero image in k X for k the residue field of R. The multiplication-by-α map is then injective on Λ⊗ Z p A.
By Weierstrass preparation, A † is a finite direct sum of copies of A as an R-module.
where the first map sends a ∈ D(A) to α(1 ⊗ a) ∈ C † (A).
The pushout C (A) has a relatively simple explicit description in the case A Fr p =1 = 0, noting that 1 − ϕ −1 is then injective on D(A). Moreover, the injective pushout map from to C (A) is given by multiplication by 1 − ϕ −1 .
The following defines an intermediate Coleman map from H 1 (Q p , A † (1)) to C (A).
Theorem 4.2.4. There is an isomorphism fitting in an isomorphism of exact sequences of Col A arising from Remark 4.1.6 and the map H 1 (Q p , A(1)) → H 1 (Q p , A † (1)) induced by α : A → A † . We claim that the two compositions agree. Given the claim, we define Col † A as the inverse of the map given by universal property of the pushout C (A), and the left-hand square in the diagram of the proposition commutes.
To see the claim, consider the diagram in which the two compositions are found by tracing its perimeter. The two right-hand squares clearly commute. Since the multiplication-by-α maps in this diagram are all injective, we are reduced to the commutativity of the left part of the diagram (aside from the dashed arrow). This commutativity follows from Proposition 4.1.8, which is equivalent to the statement that the two compositions C(A)/XC(A) → H 1 (Q p , A(1)) agree on the image of D(A) in The commutativity of the right-hand square in the map of exact sequences is seen as follows: we have the diagram Remark 4.2.5. The middle square of the commutative diagram (4.2) gives a comparison between Col A and Col † A . Note that in the case α = 1, the map Col A is defined as a split surjection (as we have kept the conventions of [FK]), whereas Col † A is an isomorphism to A Fr p =1 ⊕ D(A). Remark 4.2.6. In [FK,Section 4], Coleman maps Col are defined on the Iwasawa cohomology of A(1) for the extension Q p (µ p ∞ ) of Q p , as opposed to just Q p,∞ . The second Iwasawa cohomology groups of A(1) for each of these extensions are isomorphic via corestriction. Outside of the trivial eigenspace for Gal(Q(µ p ∞ )/Q p,∞ ) that we consider here, analogously defined intermediate Coleman maps would simply amount to reductions of the original Coleman maps.
Recall that the action of ϕ −1 on D(T quo ) agrees with the action of U p on S θ . Given the identifications of Lemma 4.2.3, Theorem 4.2.4 then has the following corollary.

There is an isomorphism
Col † : H 1 (Q p , T † quo (1)) → S θ fitting in an isomorphism of exact sequences where ψ factors through the inverse to the map induced by multiplication by α on the cokernel of multiplication by 1 −U p on S θ .
We make the following definition for later use.

Local zeta maps
In this subsection, we use an ad hoc local version of the global zeta map of Fukaya-Kato. We shall see how it ties in with global elements in Section 5. Fix an isomorphism M θ ∼ − → M θ of H θ that reduces to the canonical isomorphism We use it, in particular, to identify S θ with S θ in the remainder of the paper. We then have isomorphisms the second being the inverse of the map that takes an element to the endomorphism it defines. We will specify the following element α θ precisely in Section 4.
We may then define a local zeta map. Its significance lies in that is induced by the restriction of a zeta map of Fukaya and Kato for our later good choice of α θ . Definition 4.3.2. Let z quo denote the unique map of Λ⊗ Z p h θ -modules such that Col •z quo is identified with multiplication by α θ ∈ Λ⊗ Z p h θ .

Proposition 4.3.3 (Fukaya-Kato).
There exists a unique h θ -module homomorphism on Λ⊗ Z p S θ , and such that Col •z quo is multiplication by ξ modulo I.
Proof. Since z quo is defined so that Col •z quo is multiplication by α and (1 − U p ) ev 0 • Col = Col • cor by Proposition 4.1.8 (noting [FK,1.8.1] to see that ϕ −1 acts as U p on T quo ), we have that Since Col is an isomorphism for T quo , we can define z quo to be the unique map satisfying Col •z quo = α(0). As α(0) modulo I isξ 1 (0) = ξ by definition, we are done.
We prove an analogue of Proposition 4.3.3 not involving the derivative ξ for the intermediate quotient T † quo .
Proposition 4.3.4. There exists a unique map of Λ⊗ Z p h θ -modules with the property that the square commutes, and the composition Col † • z † quo : S θ → S θ is reduction modulo (U p − 1).

Proof. Consider the composition
By definition of α and Col † , this map is induced by multiplication by (U p − 1)α. In particular, it factors through S θ since it lands kernel of multiplication by X in S θ . So, we have the existence and uniqueness of z † quo making the square commute and such that the composition is induced by multiplication by α. The composition of this map with ψ of Corollary 4.2.7 is reduction modulo (U p − 1), which gives the final statement.
5 Global study

Global cohomology
We first consider torsion in global cohomology groups. As we are working only with the needed eigenspace of the Eisenstein part of cohomology, we can obtain finer results than [FK, Section 3] in our case of interest.
Lemma 5.1.1. We have two exact sequences of Λ⊗ Z p H θ -modules. In the first, the terms have no nonzero Λ⊗ Z p Λ θ -torsion, and in the second, they have no Λ θ -torsion.
Proof. The first sequence is automatically exact, as zeroth Iwasawa cohomology groups are trivial. Note thatT θ /T θ has trivial G Q(µ N ) -action by [FK,3.2.4]. (Alternatively, one can see this by observing that the action factors through the Galois group of the totally ramified at p extension Q(µ N p ∞ )/Q(µ N ), since all cusps of Y 1 (N p r ) are defined over Q(µ N p r ), and then that the G Q p -action onT θ /T θ ∼ =T quo /T quo is unramified.) So, the second sequence is exact as H 0 (O,T θ /T θ (1)) = 0.
We can filter any h θ [G Q,S ]-subquotient M of T θ by the powers of I, and we clearly have H 0 (O, M(1)) = 0 if H 0 (O, I k M/I k+1 M(1)) = 0 for all k ≥ 0. Let µ ∈ Λ θ be nonzero, and set M = T θ /µT θ . As T θ is Λ θ -free, we have an exact sequence so H 0 (O, M(1)) surjects onto (in fact, is isomorphic to) the µ-torsion in H 1 (O, T θ (1)). Set T k = I k T θ /I k+1 T θ . Let P k denote the h θ [G Q,S ]-module that is the image of the multiplication map I k ⊗ h θ P → T k . The G Q -action on P k is then trivial, and on the quotient Q k = T k /P k , the G Q -action factors through Z × p,N with ∆ acting as ωθ −1 . As a nonzero h θ [G Q ]subquotient of T k (1), it then follows (since θ = ω 2 by Hypothesis 2.4.3c) that I k M/I k+1 M(1) has no nonzero G Q -fixed elements. Thus, H 1 (O, T θ (1)) has no µ-torsion. Replacing M with Λ ι⊗ Z p M and µ with a nonzero element λ ∈ Λ⊗ Z p Λ θ , a similar argument applies to show that H 1 (O, Λ ι⊗ Z p T θ (1)) has no nonzero λ -torsion (as the ∆-action on Λ ι is trivial). It remains to deal with theT θ /T θ -terms. Via the restriction and Coleman maps, we have an injection the latter isomorphism using [Oh2, Proposition 3.1.2] and Hypothesis 2.4.3d (though in the case said hypothesis fails, we have (Λ⊗ Z p Λ θ ) 2 instead, and the result is the same). Clearly the latter module is Λ⊗ Z p Λ θ -torsion free. Also, Kummer theory provides us with the isomorphism in the injection being a consequence of Theorem 2.3.9, [Oh2, Proposition 3.1.2], and Hypothesis 2.4.3d (the latter again being unnecessary for the result) and again the latter module is Λ θtorsion free.
Proof. Since multiplication by 1 − U p is injective on T θ , showing that 1 − U p is injective on H 1 (O, T θ (1)) amounts to showing that the Tate twist of T θ /(U p − 1)T θ has trivial G Qinvariants. Note that the G Q p -action T quo is unramified, and therefore, the action of G Q ur p on T quo (1) is given by multiplication by the cyclotomic character. Therefore, we have H 0 (Q p , (T quo /(U p − 1)T quo )(1)) = 0 and the statement for H 1 (Q p , T quo (1)). Since T θ = T sub ⊕ T quo as h θ -modules, it therefore suffices to show that no nontrivial element of (T sub /(U p − 1)T sub )(1) is fixed by G Q in (T θ /(U p − 1)T θ )(1). Now, T sub is isomorphic to h θ as an h θ -module, and T sub /IT sub is isomorphic to the h θ [G Q ]-quotient Q of T θ /IT θ . For m = I + (p, X)h, we have as G Q -modules (where G Q acts on R θ through θ −1 ), so has no fixed elements since θ = ω 2 . If x ∈ (T sub /(U p − 1)T sub )(1) is nonzero and fixed by G Q inside (T θ /(U p − 1)T θ )(1), then it is also fixed in xT sub /(xm + (U p − 1))T sub (1) by the maximality of mh θ . This is isomorphic to a nonzero quotient of T sub /mT sub (1) under multiplication by x, so it has no fixed elements, which contradicts x = 0.
Proof. Fukaya and Kato showed that the inflation map is an isomorphism [FK,9.5.2], and their argument works with the degree p r (unramified) extensions Q ,r and F ,r replacing Q and F , respectively. Note that lim where T ,sub and T ,quo have rank 1 over h θ , and the quotient T ,quo has an unramified action of G Q . Inertia at acts on T ,sub by the restriction of the character θ −1 that is primitive at , so H 0 (Q ur , T ,sub (1)) is trivial. It thus remains only to show that H 1 (F , T ,quo (1)) is trivial. As U acts T ,quo as a geometric Frobenius Φ with eigenvalues congruent to 1 modulo I, the G F -action on T quo becomes trivial upon restriction to the Galois group of the unramified Z p -extension F ,∞ of F . Since G F ,∞ has no nontrivial p-quotient, inflation proivdes an isomorphism As F ,∞ does not contain a primitive pth root of unity, the group is zero. It follows that H 1 (F , H 0 (Q ur , T θ (1))) is trivial, as required.
The quotient maps Note that (u : v) θ depends upon u only modulo N p. By [FK,3.2.5], the elements (u : v) θ generate M θ , and under Hypothesis 2.4.3, the group S θ is generated by the symbols (u : v) θ with u ≡ 0 mod N p by [FK,6.2.6].
Definition 5.2.6. Let and let We define symbols attached to elements of these sets.
where κ : Z × p → Λ sends a unit to the group element of its projection to 1 + pZ p .

Zeta elements
We first very briefly recall the Kato-Beilinson elements (or zeta elements) of [FK, Section 2]. We then, in the form we shall require, slightly refine the resulting maps of Fukaya and Kato [FK, Section 3] and describe the properties of them that we need.
The following definition is from [FK,2.4.2].
Definition 5.3.1. For r, s ≥ 0 and u, v ∈ Z with (u, v, N p) = (1), and supposing that u, v ≡ 0 mod N p r if s = 0, we define c,d z r,s (u : v) to be the image under the norm and Hochschild-Serre maps of the cup product c g a p s , c Remark 5.3.2. As a consequence of [FK,2.4.4, 3.1.9], the elements c,d z r,s (u : v) are for r, s ≥ 1 compatible with the maps induced by quotients of modular curves and corestriction maps for the ring extensions. Moreover, the corestriction map Let us use c,d z r,s (u : v) θ to denote the projection of c,d z r,s (u : v) to the Eisenstein component for θ .
for all s ∈ Z p , where ζ p denotes the p-adic Riemann zeta function.
The following result, constructing a zeta map, is a refinement of a result of Fukaya and Kato [FK,3.3.3]. It is in essence a consequence of [FKS2,Theorem 3.15].
Note thatT θ /T θ ∼ = (H/I) ι θ (1) as h[G Q ]-modules, which in turn are canonically isomorphic tõ T quo /T quo as h[G Q p ]-modules. This kernel is then trivial as a consequence of weak Leopoldt. By the exactness of the first sequence in Lemma 5.1.1, we have the claim.
From now on, we take α = α θ to be as given in Theorem 5.3.5. We prove the following slight refinement of [FK,3.3.9] on a zeta map at the level of Q as a consequence of [FKS2, Theorem 3.17].
Theorem 5.3.6. There exists a unique map of h θ -modules with the property that for the map cor : The composition of z with equals the map z quo of Proposition 4.3.3 for α as in Theorem 5.3.5.
Proof. In [FKS2, Theorem 3.17] (noting [FK,3.3.14]), we prove (using Lemma 5.1.1 of this paper) the existence of an H θ -module homomorphism The comparison with z is [FK, 3.3.9(ii)], the uniqueness being Lemma 5.1.2. The comparison with z quo follows from Proposition 4.3.3, the comparison with z, and Theorem 5.3.5.
Fukaya and Kato prove the following in [FK, 5.2.10-11 and 9.2.1]. We sketch their proof primarily to make clear how to obtain the sign in its comparison. That is, there are two sign differences from their proof which effectively cancel each other, and the sign of the second map in the composition in its statement is the opposite of that of [FK,6.3.9]. The main result [FK,0.14] in the work of Fukaya and Kato states that ξ ϒ • ϖ and ξ induce the same endomorphism of P ⊗ Z p Q p . As P is not known to be p-torsion free, this is slightly weaker than equality as endomorphisms of P. With the results of [FKS2] in hand, it is now a relatively straightforward matter to show that the stronger statement holds by following the argument of [FK].

Refined global cohomology
We prove analogues for intermediate cohomology of earlier results on global cohomology. We begin with an extension of Lemma 5.1.4. Let use Col † P : H 1 (Q p , P † (1)) → P to denote the composition ψ • Col † P .
Proof. The anticommutativity of the square is proven by the analogous argument to Lemma 3.1.15, and the identifications of H 2 (Q p , P(1)) and H 3 c (O, P(1)) with P agree as before. By Proposition 3.3.5, the latter identification agrees via ∂ † P with the identification of H 2 c (O, P † (1)) with P. Finally, Col † P = inv •∂ † P by the commutativity of (4.2).
Next, we have an analogue of Lemma 5.1.5.
Proposition 5.4.2. The exact sequences are canonically split, compatibly with the map from the former sequence to the latter. The splitting of the surjection in the latter sequence takes image in H 1 (Q p , P † (1)) inside the direct sum. Moreover, the splittings are compatible with the maps of these sequences to (via the quotient map P † → P) and from (via α : P → P † ) the corresponding split sequences of Lemma 5.1.5.
Proof. Since Fr p acts trivially on P, the exact sequence of Theorem 4.2.4 is canonically split, with the first term identified with H 1 Iw (Q p,∞ , P(1))/Xα and the third identified with H 2 Iw (Q p,∞ , P(1)) where the latter map is the inclusion of the summand for = p.
The composition then gives a canonical splitting of the first exact sequence which is compatible with the map between the two. The final statement follows easily from the fact that the splitting of Col † : H 1 (Q p , P † (1)) → Via the splitting of Lemma 5.1.5, we have an isomorphism and we letz † P be the projection ofz † quo • ev 0 to the first component. We next check commutativity of the the first diagram. By Proposition 5.4.2, we may do this after projection to the summands corresponding to P and Q, respectively. For the Pcomponents, note thatz † P is the projection ofz † quo • ev 0 to H 1 (O, P † (1)). The composition of this map with the surjection to H 1 (O, P(1)) is the projection ofz quo • ev 0 to H 1 (O, P(1)). This equalsz P • ev 0 in that the restriction ofz P to H 1 (Q , P(1)) is trivial for primes | N. That is, z P is a reduction of z : S θ → H 1 (O, T θ (1)), and H 1 (Q , T θ (1)) is trivial by Lemma 5.1.3.
For the Q-components, we need only remark that the composition ofz † Q = −1 ⊗ ϖ with the map to H 1 (O, Q(1)) isz Q = −ϖ (see Proposition 3.3.4), so we see that the first diagram commutes on the summands corresponding to Q.
For the second diagram, we have by definition thatz † P equals the projection ofz † quo • ev 0 to , the latter group being identified with P, factors through ϒ•ϖ = 1. As the composition of −z † quo with H 1 (Q p , P † (1)) → H 2 c (O, P † (1)) is also identified with 1, the commutativity holds.
Note that the data ofz † is equivalent to the data of its restriction to an h θ /I-module homomorphism P → H 1 (O, T † (1)) sending x ∈ P toz † (1 ⊗ x) and fitting in the corresponding commutative diagrams arising from restriction to P.
The above discussion can be summarized by the diagram which fully commutes if we know the existence of the conjectural mapz † in Theorem 5.5.1. The equality ϒ • ϖ = 1 is then seen by tracing the outside of the diagram. This begs the following question, which would be in analogy with the construction of z by Fukaya and is (1 −U p )z . By Lemma 5.1.2, we see that the composition of z † with the latter map is z . In particular, the first diagram in Theorem 5.5.1 commutes.
and 1 −U p has trivial kernel on S θ , we have that the composition of z † with the map is z † quo . That is, the second diagram in Theorem 5.5.1 commutes.

Test case
We explore the feasibility of the equivalent conditions to Conjecture 3.1.9 found in Theorem 5.5.1, working with cyclotomic elements in place of Beilinson-Kato elements. We find, somewhat reassuringly, that an analogue of the conditions of Theorem 5.5.1 holds in this setting. On the other hand, an analogue of the stronger Question 5.5.2, which amounts to a norm relation for a good choice of z † , has a potential obstruction. We show that this norm relation does hold if an even eigenspace of the completely split Iwasawa module vanishes.

Notation
Let us first introduce changes to our notation from the previous sections. Most importantly, we now allow our prime p to divide ϕ(N). That is, we let p be an odd prime, and we let N ≥ 3 be a positive integer with p N. Let ∆ = (Z/N pZ) × as before, which we identify with Gal(Q(µ N p ∞ )/Q ∞ ) ∼ = Gal(Q(µ N p )/Q).
a. Let ∆ p and ∆ be the Sylow p-subgroup of ∆ and its prime-to-p order complement, respectively.
b. Let θ : ∆ → Q p × be a nontrivial even character of ∆ which is trivial on decomposition at p and primitive at all primes dividing N.
c. Let R θ be the Z p [∆ p ]-algebra of values of θ , which we then view as a Z p [∆]-module with ∆ acting through θ .
e. Let R = R ι θ be the Z p [∆]-module that is R θ endowed with the inverse of the Galois action described above.
Remark 6.1.2. Our choice of R is made so that for any i ≥ 0 by Shapiro's lemma, and similarly for Iwasawa cohomology.
We shall also use the following.
a. Let σ denote the image of the Frobenius Fr p at p in ∆ p .
b. Let R σ =1 denote the maximal quotient and R σ =1 the maximal submodule of R θ on which σ acts trivially.
c. Let Y (resp., X) denote the Galois group of the maximal completely locally split (resp., unramified) abelian pro-p extension of Q(µ N p ∞ ).
d. Let X denote the Galois group of the maximal abelian, unramified outside N p, pro-p extension of Q(µ N p ∞ ).
e. Let E (resp., C ) denote the group of norm compaitible systems of p-completions of global units (resp., cyclotomic units) in the tower Q(µ N p ∞ )/Q.

Zeta and Coleman maps
We now take our zeta map as having image the θ -eigenspace of the cyclotomic units. as the unique such map taking 1 to the projection of 1 − ζ N .
We use z quo and z quo to denote the restrictions of z and z to the cohomology of G Q p .
Remark 6.2.2. The zeta map and its ground level analogue satisfy the well-known norm relation among cyclotomic units.
Definition 6.2.3. We let ξ ∈ Λ θ be the unique element satisfying ρ(ξ (u 1−s − 1)) = L p (θ ρ, s) for all s ∈ Z p and p-adic characters ρ of ∆ p , where we useρ to denote the map R θ → Q p induced by ρ.
We note the following equivariant formulation of Iwasawa's theorem.
Remark 6.2.4. As an R θ -module, D(R) is free of rank 1, and it can be identified with R θ as a Z p -algebra after a choice of normal basis of the valuation ring of the unramified extension of Q(µ p ) defined by the decomposition group of ∆ p . We can and do choose this identification such that the Coleman map Col = Col R : H 1 Iw (Q p,∞ , R(1)) → X −1 Λ θ satisfies Col •z = ξ .
To shorten notation, let us write C for the image C(R) of Col and similarly with superscripts adorning C. Consider the Coleman map Col † : H 1 (Q p , R † (1)) → C for A = R and α = ξ of Theorem 4.2.4. The analogous argument to that of Proposition 4.3.4 yields the following.
Proof. Identifying H 1 (Q p , R † (1)) with C via Col † , we define z † quo to be as the pushout map R θ ∼ = D(R) → C . By definition of C , following this by 1 − σ −1 , we get the composition which is to say, recalling Remark 6.2.4, the composition of z quo with H 1 Iw (Q p,∞ , R(1)) → H 1 (Q p , R † (1)).

Brief cohomological study
We describe the structure of some relevant cohomology groups. Lemma 6.3.1. For each prime N, the cohomology groups H i (Q , R(1)), H i (Q , R † (1)), and H i Iw (Q , Λ ι θ (1)) for i ∈ {1, 2} are all trivial. Proof. By Shapiro's lemma, the group H i (Q , R(1)) is isomorphic to the θ -eigenspace of the product of the groups H i (Q (µ N p ), A θ (1)) over primes over in the field Q(µ N p ), where A θ is the Z p -algebra of θ -values with the trivial action of Galois. Since the pro-p completion of Q (µ N p ) × is generated by a uniformizer, each of these first cohomology groups is isomorphic to A θ via the Kummer isomorphism. The second cohomology groups are also isomorphic to A θ via the invariant map. Since inertia at in Gal(Q (µ N p )/Q ) acts trivially on this product and θ is primitive at , the θ -eigenspace of the product is zero.