A NOTE ON CYCLOTOMIC EULER SYSTEMS AND THE DOUBLE COMPLEX METHOD

GREG W. ANDERSON AND YI OUYANG

Abstract. Let 𝔽 be a finite real abelian extension of ℚ. Let M be an odd positive integer. For every squarefree positive integer r the prime factors of which are congruent to 1 modulo M and split completely in 𝔽, the corresponding Kolyvagin class κ_{r} ∈ 𝔽^{×}/𝔽^{× M} satisfies a remarkable and crucial recursion which for each prime number ℓ dividing r determines the order of vanishing of κ_{r} at each place of 𝔽 above ℓ in terms of κ_{r/ℓ}. In this note we give the recursion a new and universal interpretation with the help of the double complex method introduced by Anderson and further developed by Das and Ouyang. Namely, we show that the recursion satisfied by Kolyvagin classes is the specialization of a universal recursion independent of 𝔽 satisfied by universal Kolyvagin classes in the group cohomology of the universal ordinary distribution à la Kubert tensored with ℤ/Mℤ. Further, we show by a method involving a variant of the diagonal shift operation introduced by Das that certain group cohomology classes belonging (up to sign) to a basis previously constructed by Ouyang also satisfy the universal recursion.

1. Introduction

Let 𝔽 be a finite real abelian extension of ℚ. Let M be an odd positive integer. For every squarefree positive integer r the prime factors of which are congruent to 1 modulo M and split completely in 𝔽, the corresponding Kolyvagin class κ_{r} ∈ 𝔽^{×}/𝔽^{× M} satisfies a remarkable and crucial recursion which for each prime number ℓ dividing r determines the order of vanishing of κ_{r} at each place of 𝔽 above ℓ in terms of κ_{r/ℓ}. See Proposition 2.4 of the Rubin appendix to Lang’s text [4] for a formulation of this recursion commonly employed in the literature. In this note we actually work with a formulation of the recursion slightly different from but equivalent to Rubin’s formulation (see Proposition 2.5 below).

The purpose of this note is to give the recursion satisfied by Kolyvagin classes a new and universal interpretation with the help of the double complex method introduced by Anderson [1] and further developed by Das [2] and Ouyang [5]. We show that the recursion is the specialization of a universal recursion independent of 𝔽 satisfied by universal Kolyvagin classes in the group cohomology of the universal ordinary distribution à la Kubert tensored with ℤ/Mℤ (see Proposition 4.5). Further, we show by a method involving a variant of the diagonal shift operation introduced by Das [2] that certain group cohomology classes belonging (up to sign) to a basis previously constructed by Ouyang [5] also satisfy the universal recursion (see Corollary 5.9). Taken together, our results show that it is possible to construct classes in 𝔽^{×}/𝔽^{×M} satisfying a useful recursion of Kolyvagin type by methods somewhat more conceptual than have heretofore been employed.

Date: October 28, 2001; revised April 14, 2002.

1991 Mathematics Subject Classification. Primary 11R18, 11R23; Secondary 11R34.

It is natural to expect that results similar to those presented in this note hold for general universal Euler systems. Indeed, the many beautiful results proved in Chapter 4 of Rubin’s book [6] strongly suggest the existence of a general theory of universal Kolyvagin recursions. But since there are significant technical difficulties to deal with before the double complex method can be brought to bear on the general theory of Euler systems, we can for now but affirm our hope to generalize the results of this note. In any case, we hope that our point of view might prove helpful to others in the search for new applications and new examples of Euler systems.

2. A brief review of cyclotomic Euler systems

2.1. Notation and setting. Let 𝔽 be a finite real abelian extension of the field ℚ of rational numbers and let \overline{𝔽} be an algebraic closure of 𝔽. Let M be an odd positive integer, and for every abelian group A let the M-torsion subgroup be denoted A_{M}. Let r be the formal product of all odd primes ℓ ≡ 1 mod M that split completely in 𝔽. In the sequel we refer to formal products of prime numbers as supernatural numbers. Let G be the Galois group over 𝔽 of the field generated over 𝔽 by all roots of unity in \overline{𝔽} of order dividing r. For each prime number ℓ dividing r:

• Let G_{ℓ} ⊂ G be the inertia subgroup at some place (hence all places) above

ℓ.

– Note that G_{ℓ} is cyclic of order ℓ − 1.

• Let σ_{ℓ} be a generator of G_{ℓ}.

• Let N_{ℓ} := ∑_{i=0}^{ℓ−2} σ_{ℓ}^{i} ∈ ℤ[G_{ℓ}] and N′_{ℓ} := ∑_{i=1}^{ℓ−2} iσ_{ℓ}^{i} ∈ ℤ[G_{ℓ}]. – Note the crucial identity N′_{ℓ}(σ_{ℓ} − 1) = ℓ − 1 − N_{ℓ}.

• Let Frob_{ℓ} ∈ G/G_{ℓ} be the arithmetic Frobenius automorphism at some place (hence all places) above ℓ.

• Let ord_{ℓ} be the normalized additive valuation of ℚ corresponding to ℓ. For each positive integer r dividing r:

• Let G_{r} ⊂ G be the subgroup generated by ⋃_{ℓ|r} G_{ℓ}.

– Note that the evident homomorphism ∏_{ℓ|r} G_{ℓ} → G_{r} is bijective.

• Let N′_{r} := ∏_{ℓ|r} N′_{ℓ} ∈ ℤ[G_{r}].

• Let 𝔽_{r} be the extension of 𝔽 generated by the r^{th} roots of unity in \overline{𝔽}. Put 𝔽_{r} := ⋃_{r|r} 𝔽_{r}.

• Let O_{r} be the ring of algebraic integers of 𝔽_{r}. Put O := O_{1} and O_{r} :=

⋃_{r|r} O_{r}. For each positive integer r dividing r and prime number ℓ:

• Let O_{r,(ℓ)} be the localization of O_{r} by the multiplicative system of elements

prime to ℓ. Put O_{(ℓ)} := O_{1,(ℓ)} and O_{r,(ℓ)} := ⋃_{r|r} O_{r,(ℓ)}. We fix a collection

{ξ_{r} ∈ O_{r}^{×}}_{r|r}

of global units such that for all positive integers r dividing r and prime numbers ℓ dividing r the following relations hold:

• ξ_{r}^{N_{ℓ}} = ξ_{r/ℓ}^{Frob_{ℓ} −1}.

• ξ_{r} ≡ ξ_{r/ℓ} modulo the radical of the ideal of O_{r} generated by ℓ.

Such a collection {ξ_{r}} is called an Euler system.

Lemma 2.2. Let r be any positive integer dividing r. The sequence

1 → 𝔽_{r}^{×} ⟶^{x↦x^{M}} 𝔽_{r}^{×} → 𝔽_{r}^{×}/𝔽_{r}^{×M} → 1

is exact and so is the sequence

1 → 𝔽^{×} ⟶^{x↦x^{M}} 𝔽^{×M} → H^{0} (G_{r}, 𝔽_{r}^{×}/𝔽_{r}^{×M}) → 1

of G_{r}-invariants. Via the latter sequence make now the identification

H^{0} (G_{r}, 𝔽_{r}^{×}/𝔽_{r}^{×M}) = 𝔽^{×}/𝔽^{×M}.

We have

(image of H^{0} (G_{r}, O_{r}^{×}/O_{r}^{×M}) in H^{0} (G_{r}, 𝔽_{r}^{×}/𝔽_{r}^{×M}) ) ⊂ O_{(ℓ)}^{×}/O_{(ℓ)}^{×M}

for all prime numbers ℓ not dividing r.

Proof. The first sequence is exact because under our hypotheses the field 𝔽_{r} contains no nontrivial M^{th} roots of unity. The second sequence is exact by Satz 90. We turn to the proof of the last assertion. Fix ξ ∈ O_{r}^{×} representing a class in H^{0}(G_{r}, O_{r}^{×}/O_{r}^{×M}) and write

ξ = αβ^{M} (α ∈ 𝔽^{×}, β ∈ 𝔽_{r}^{×M}) .

It suffices to verify that α up to a factor in 𝔽^{×M} belongs to O_{(ℓ)}^{×}. Since O_{(ℓ)} is a principal ideal domain, it suffices to verify that for each prime P of O dividing ℓ the order with which P divides α is divisible by M. But the latter is obvious because any prime of O dividing ℓ cannot divide r and hence is unramified in O_{r}. □

2.3. Kolyvagin classes. Fix a positive integer r dividing r. For each prime number ℓ dividing r one has

ξ_{r}^{N′_{r}(1−σ_{ℓ})} ≡ ξ_{r}^{N′_{r/ℓ}N_{ℓ}} ≡ ξ_{r/ℓ}^{N′_{r/ℓ}(Frob_{ℓ} −1)} ≡ 1 mod O_{r}^{×M}

by induction on the number of prime divisors of r and hence

ξ_{r}^{N′_{r}} mod O_{r}^{×M} ∈ H^{0} (G_{r}, O_{r}^{×}/O_{r}^{×M}) .

By Lemma 2.2 there exists a unique class

κ_{r} ∈ 𝔽^{×}/𝔽^{×M}

such that

ξ_{r}^{N′_{r}} ≡ κ_{r} mod 𝔽_{r}^{×M}

and moreover we have

(ℓ, r) = 1 ⇒ κ_{r} ∈ O_{(ℓ)}^{×}/O_{(ℓ)}^{×M}

for all prime numbers ℓ. We call κ_{r} the Kolyvagin class indexed by r.

2.4. The operations ν_{ℓ}, [·]_{ℓ} and exp_{ℓ}. Let a prime number ℓ dividing r be given. Let

ν_{ℓ} : O_{(ℓ)}^{×}/O_{(ℓ)}^{×M} → (O/ℓ)_{M}^{×}

be the unique homomorphism such that

ν_{ℓ} (x mod O_{(ℓ)}^{×M}) ≡ x^{\frac{ℓ−1}{M}} mod ℓO_{(ℓ)}

for all x ∈ O_{(ℓ)}^{×}. Let

[·]_{ℓ} : 𝔽^{×}/𝔽^{×M} → ( \begin{matrix}group of fractional \\ O_{(ℓ)}-ideals \end{matrix} ) ⊗ (ℤ/Mℤ)

be the unique homomorphism such that

[x mod 𝔽^{×M}]_{ℓ} = ( \begin{matrix}fractional O_{(ℓ)}-ideal \\ generated by x \end{matrix} ) mod ( \begin{matrix}M^{th} powers of \\ fractional O_{(ℓ)}-ideals \end{matrix} )

for all x ∈ 𝔽^{×}. We claim that there exists a unique isomorphism

exp_{ℓ} : ( \begin{matrix}group of fractional \\ O_{(ℓ)}-ideals \end{matrix} ) ⊗ (ℤ/Mℤ) →^{∼} (O/ℓ)_{M}^{×}

such that

exp_{ℓ} (( \begin{matrix}fractional O_{(ℓ)}-ideal \\ generated by x^{N_{ℓ}} \end{matrix} ) ⊗ (1 mod M)) ≡ (x^{1−σ_{ℓ}})^{\frac{ℓ−1}{M}} mod \sqrt{ℓO_{ℓ,(ℓ)}}

for all x ∈ 𝔽_{ℓ}^{×} where \sqrt{ℓO_{ℓ,(ℓ)}} denotes the radical of the ideal of O_{ℓ,(ℓ)} generated by ℓ. Now each maximal ideal of O_{(ℓ)} is totally ramified in O_{ℓ,(ℓ)}, hence every fractional O_{(ℓ)}-ideal is generated by x^{N_{ℓ}} for some x ∈ 𝔽_{ℓ}^{×} unique up to a factor in O_{ℓ,(ℓ)}^{×} and hence exp_{ℓ} is well defined. Upon completing the extension 𝔽_{ℓ}/𝔽 at any place of 𝔽_{ℓ} above ℓ, one obtains a Kummer extension with Galois group G_{ℓ} and hence exp_{ℓ} is an isomorphism. The claim is proved.

Proposition 2.5. For all positive integers r dividing r and prime numbers ℓ dividing r the identity

exp_{ℓ}[κ_{r}]_{ℓ} ≡ \sqrt[M]{ξ_{r}^{N′_{r}(σ_{ℓ}−1)}} ≡ \sqrt[M]{ξ_{r/ℓ}^{N′_{r/ℓ}(ℓ−Frob_{ℓ})}} ≡ ν_{ℓ}κ_{r/ℓ} mod \sqrt{ℓO_{r,(ℓ)}}

holds, where the M^{th} roots are chosen to be the unique such existing in O_{r}^{×} and \sqrt{ℓO_{r,(ℓ)}} denotes the radical of the ideal of O_{r,(ℓ)} generated by ℓ. (This is a reformulation of Proposition 2.4 of the Rubin appendix to Lang’s text [4].)

Proof. Write

ξ_{r}^{N′_{r}} = α_{r}β_{r}^{M} (α_{r} ∈ 𝔽^{×}, β_{r} ∈ 𝔽_{r}^{×})

and

ξ_{r/ℓ}^{N′_{r/ℓ}} = α_{r/ℓ}β_{r/ℓ}^{M} (α_{r/ℓ} ∈ O_{(ℓ)}^{×}, β_{r/ℓ} ∈ O_{r/ℓ,(ℓ)}^{×}) .

Choose γ_{r} ∈ 𝔽_{ℓ}^{×} such that

γ_{r}^{N_{ℓ}} and α_{r} generate the same fractional O_{(ℓ)}-ideal.

One then has

γr^{\frac{ℓ−1}{M}} β_{r} ∈ O_{r,(ℓ)}^{×}.

Further, one has

ξ_{r}^{N′_{r}(σ_{ℓ}−1)} = β_{r}^{M(σ_{ℓ}−1)} = ξ_{r}^{N′_{r/ℓ}(ℓ−1)}β_{r/ℓ}^{M(1−Frob_{ℓ})}

and hence

β_{r}^{σ_{ℓ}−1} = (ξ_{r}^{N′_{r/ℓ}})^{\frac{ℓ−1}{M}} β_{r/ℓ}^{1−Frob_{ℓ}}

because there are no nontrivial M^{th} roots of unity in 𝔽_{r}. Finally, one has

(γ_{r}^{1−σ_{ℓ}})^{\frac{ℓ−1}{M}} ≡ β_{r}^{σ_{ℓ}−1} ≡ αr/ℓ^{\frac{ℓ−1}{M}} β_{r/ℓ}^{ℓ−Frob_{ℓ}} ≡ αr/ℓ^{\frac{ℓ−1}{M}} mod \sqrt{ℓO_{r,(ℓ)}}

which by the definitions proves the result. □

2.6. The Kolyvagin recursion. We say that a system of classes

{λ_{r} ∈ 𝔽^{×}/𝔽^{×M}}_{r|r}

indexed by the positive integers dividing r satisfies the Kolyvagin recursion if for all positive integers r dividing r and prime numbers ℓ the following hold:

• (ℓ, r) = 1 ⇒ λ_{r} ∈ O_{(ℓ)}^{×}/O_{(ℓ)}^{×M}.

• ℓ | r ⇒ exp_{ℓ}[λ_{r}]_{ℓ} = ν_{ℓ}λ_{r/ℓ}.

In this language Proposition 2.5 and the discussion leading up to it can be condensed to the assertion that the system of Kolyvagin classes satisfies the Kolyvagin recursion.

3. Universal constructions

3.1. The free abelian group A. For each supernatural number s put

\frac{1}{s}ℤ := ⋃_{s|s} \frac{1}{s}ℤ,

the union being extended over all positive integers s dividing s. Let A be the free abelian group generated by the family of symbols of the form

[a] (a ∈ \frac{1}{r}ℤ/ℤ) .

We equip A with an action of G by the rule

σ[a] = [b] ⇔ σϕ(a) = ϕ(b)

for all a, b ∈ \frac{1}{r}ℤ/ℤ, injective homomorphisms ϕ : \frac{1}{r}ℤ/ℤ → 𝔽_{r}^{×}, and σ ∈ G. For each supernatural number s dividing r, let A(s) be the subgroup of A generated by symbols of the form

[a] (a ∈ \frac{1}{s}ℤ/ℤ) .

Note that A(s) is stable under the action of G. Note that for each prime number ℓ dividing r the group A(r/ℓ) can be viewed as a G/G_{ℓ}-module. Note that

A = ⋃_{r|r} A(r)

where r ranges over the positive integers dividing r.

3.2. The universal ordinary distribution. Given any supernatural number s dividing r, let U_{s} be the quotient of A(s) by the subgroup generated by all elements of the form

[a] − ∑_{ℓb=a}[b] (ℓ: a prime number dividing s, a ∈ \frac{ℓ}{s}ℤ/ℤ) .

Notice that the action of G on A(s) descends to U_{s}. Note that for every prime number ℓ dividing r the group U_{r/ℓ} can be viewed as a G/G_{ℓ}-module. The map

(a ↦ (class in U_{s} represented by [a])) : \frac{1}{s}ℤ/ℤ → U_{s}

is the universal example of a one-dimensional ordinary distribution of level s à la Kubert. Put

U := U_{r}.

Abusing language slightly, we call U the universal ordinary distribution. See Kubert [3], Lang [4] or Anderson’s appendix to Ouyang’s paper [5] for background on the theory of the universal ordinary distribution. By the classical results of Kubert [3], for any supernatural number s dividing r, the map

U_{s} → U

induced by the inclusion A(s) ⊆ A is an injective homomorphism of free abelian groups with free cokernel and hence the induced map

H^{0}(G, U_{s}/MU_{s}) → H^{0}(G, U/MU)

is also injective. Thus we may and we do henceforth identify U_{s} (resp. H^{0}(G, U_{s}/MU_{s})) with a subgroup of U (resp. H^{0}(G, U/MU)). Note that we have

U_{s} = ⋃_{s|s} U_{s}, H^{0}(G, U_{s}/MU_{s}) = ⋃_{s|s} H^{0}(G, U_{s}/MU_{s})

where the index s in both unions ranges over the positive integers dividing s.

Lemma 3.3. For every prime number ℓ dividing r the equation

(ℓ − Frob_{ℓ})x = 0

has no nonzero solution x ∈ U_{r/ℓ}.

Proof. Fix a solution x ∈ U_{r/ℓ} of the equation in question. Choose a positive integer r dividing r/ℓ such that x ∈ U_{r}. Choose ϕ ∈ G_{r} inducing the same automorphism of A(r) as does Frob_{ℓ}. Let m be the order of ϕ in the group G_{r}. Then one has an identity

(ℓ^{m} − 1)x = (ℓ^{m−1} + ℓ^{m−2}ϕ + ⋯ + ℓϕ^{m−2} + ϕ^{m−1})(ℓ − ϕ)x = 0.

It follows that x = 0 because U_{r/ℓ} is a torsion-free abelian group. □

3.4. The submodule I_{ℓ}. Let a prime number ℓ dividing r be given. We define

I_{ℓ} ⊂ U

to be the subgroup generated by all elements of U represented by expressions of the form

[a] − [b] (a, b ∈ \frac{1}{r}ℤ/ℤ, a − b ∈ \frac{1}{ℓ}ℤ/ℤ) .

Note that since I_{ℓ} is G-stable and

(σ_{ℓ} − 1)U ⊂ I_{ℓ},

the quotient U/I_{ℓ} can be viewed as a G/G_{ℓ}-module.

3.5. The resolution L. Let L be the free abelian group generated by symbols of the form

[a, g] (g: positive integer dividing r, a ∈ \frac{g}{r} ℤ/ℤ) .

We equip the abelian group L with an action of G by the rule

σ[a, g] = [a′, g′] ⇔ σϕ(a) = ϕ(a′) and g = g′

for all symbols [a, g] and [a′, g′] in the canonical basis of L, injective homomorphisms ϕ : \frac{1}{r}ℤ/ℤ → 𝔽_{r}^{×} and σ ∈ G. We equip the group L with a G-stable grading by declaring that

(degree of [a, g]) := − ∑_{ℓ} ord_{ℓ} g = −(number of prime divisors of g).

Here and in analogous summations below the indices ℓ, ℓ′ and p are understood to range over prime numbers dividing r, subject to further restrictions as noted. We equip L with a G-equivariant differential d of degree 1 by the rule

d[a, g] := ∑_{ℓ|g}(−1)^{∑_{ℓ′<ℓ} ord_{ℓ′} g} ([a, g/ℓ] − ∑_{ℓb=a}[b, g/ℓ]) .

Now let s be any supernatural number dividing r. Let L(s) be the subgroup of L generated by symbols of the form

[a, g] (g : positive integer dividing s, a ∈ \frac{g}{s} ℤ/ℤ) .

Then L(s) is a G- and d-stable graded subgroup of L. It is known that

H^{0}(L(s), d) = U_{s}

via the isomorphism induced by the G-equivariant mapping

([a, 1] ↦ [a]) : (degree zero component of L(s)) → A(s)

and that

H^{n}(L(s), d) = 0 for n ≠ 0.

In other words, (L(s), d) is a resolution of U_{s} in the category of G-modules. See Anderson’s appendix to Ouyang’s paper [5] for details and further discussion.

Proposition 3.6. The sequence

0 ⟶ U_{r/ℓ} ⟶^{ℓ−Frob_{ℓ}} U_{r/ℓ} ⟶ U/I_{ℓ} ⟶ 0

is exact where the map U_{r/ℓ} → U/I_{ℓ} is that induced by the inclusion U_{r/ℓ} ⊂ U.

Proof. Let

s_{ℓ} : L → L(r/ℓ)

be the unique homomorphism such that

s_{ℓ}[a, g] ≡ \begin{cases}(−1)^{∑_{ℓ′<ℓ} ord_{ℓ′} g}[a, g/ℓ] & if ℓ | g \\ 0 & otherwise \end{cases}

for all symbols [a, g] in the canonical basis of L. The homomorphism s_{ℓ} is of degree 1 and satisfies the relation

s_{ℓ}d = −ds_{ℓ}

as can be verified by a straightforward calculation. Now consider the sequence

Σ : 0 → L(r/ℓ) ⟶ L/L′ →^{s_{ℓ}} L(r/ℓ) → 0

where L′ is the subgroup of L generated by all elements of the form

[a, g] − [b, g] (g | \frac{r}{ℓ}, a, b ∈ \frac{g}{r} ℤ/ℤ, a − b ∈ \frac{1}{ℓ}ℤ/ℤ)

and the map L(r/ℓ) → L/L′ is that induced by the inclusion L(r/ℓ) ⊂ L. It is easy to verify that Σ is short exact. Since L′ is a graded d- and G-stable subgroup of L and (σ_{ℓ} − 1)L ⊂ L′, it follows that Σ can be viewed as a short exact sequence of complexes of G/G_{ℓ}-modules. Because H^{∗}(L(r/ℓ), d) is concentrated in degree 0, the long exact sequence of G/G_{ℓ}-modules deduced from Σ by taking d-cohomology has at most four nonzero terms and after making the evident identifications takes the form

⋯ → 0 → H^{−1}(L/L′, d) → U_{r/ℓ} ⟶^{1−Frob_{ℓ}^{−1} ℓ} U_{r/ℓ} ⟶ U/I_{ℓ} → 0 → …

where the map U_{r/ℓ} → U/I_{ℓ} is that induced by the inclusion U_{r/ℓ} ⊂ U. By Lemma 3.3 we have

H^{−1}(L/L′, d) = ker (U_{r/ℓ} ⟶^{− Frob_{ℓ}^{−1}(ℓ−Frob_{ℓ})} U_{r/ℓ}) = 0,

whence the result. □

Proposition/Definition 3.7. For every prime number ℓ dividing r there exists a unique homomorphism

D_{ℓ} : H^{0}(G, U/MU) → H^{0}(G, U_{r/ℓ}/MU_{r/ℓ})

such that

\frac{(σ_{ℓ} − 1)x}{M} ≡ \frac{(ℓ − Frob_{ℓ})y}{M} mod I_{ℓ} ⇔ D_{ℓ}(x mod MU) = y mod MU_{r/ℓ}

for all x ∈ U representing a class in H^{0}(G, U/MU) and y ∈ U_{r/ℓ} representing a class in H^{0}(G, U_{r/ℓ}/MU_{r/ℓ}). Moreover one has

D_{ℓ}H^{0}(G, U_{r}/MU_{r}) ⊂ H^{0}(G, U_{r/ℓ}/MU_{r/ℓ})

for all positive integers r dividing r and divisible by ℓ.

Proof. Put

X := {x ∈ U | x represents a class in H^{0}(G, U/MU)} ,
Y := {y ∈ U_{r/ℓ} | (ℓ − Frob_{ℓ})y ∈ MU_{r/ℓ}} ,
Z := \{(x, y) ∈ X × Y |\frac{(σ_{ℓ}−1)x}{M} ≡ \frac{(ℓ−Frob_{ℓ})y}{M} mod I_{ℓ} \} .

Fix a positive integer r dividing r and divisible by ℓ. To prove the proposition it is enough to prove the following three claims:

(1) Z ∩ (MU × Y) = MU × MU_{r/ℓ}.

(2) (σ − 1)Z ⊂ MU × MU_{r/ℓ} for all σ ∈ G.

(3) For all x ∈ X ∩ U_{r} there exists y ∈ Y ∩ U_{r/ℓ} such that (x, y) ∈ Z.

We turn to the proof of the first claim. Only the containment ⊂ requires proof; the containment ⊃ is trivial. Suppose we are given (x, y) ∈ Z ∩ (MU × Y). Then \frac{(ℓ−Frob_{ℓ})y}{M} ∈ I_{ℓ} ∩ U_{r/ℓ} and hence by Proposition 3.6 there exists z ∈ U_{r/ℓ} such that (ℓ − Frob_{ℓ})y = M(ℓ − Frob_{ℓ})z. By Lemma 3.3 it follows that y = Mz. Thus the first claim is proved. The second claim follows immediately from the first.

We turn finally to the proof of the third claim. Let

ρ_{ℓ} : A(r) → A(r/ℓ)

be the unique homomorphism such that

ρ_{ℓ}[a + b] := [a]

for all a ∈ \frac{ℓ}{r} ℤ/ℤ and b ∈ \frac{1}{ℓ}ℤ/ℤ. For each prime number p dividing r, let

γ_{p} : A(r/p) → A(r)

be the unique homomorphism such that

γ_{p}[a] := [a] − ∑_{pb=a}[b]

for all a ∈ \frac{p}{r}ℤ/ℤ. Note that ρ_{ℓ} commutes with γ_{p} for p ≠ ℓ and that the composite homomorphism ρ_{ℓ}γ_{ℓ} induces the endomorphism (1 − Frob_{ℓ}^{−1} ℓ) of A(r/ℓ). Choose a lifting a ∈ A(r) of x. By hypothesis there exists an identity

(σ_{ℓ} − 1)a = Mb + ∑_{p|r} γ_{p}b_{p} (b ∈ A(r), b_{p} ∈ A(r/p)),

and hence also an identity

0 = Mρ_{ℓ}b − (ℓ − Frob_{ℓ})(Frob_{ℓ}^{−1} b_{ℓ}) + ∑_{p| \frac{r}{ℓ}} γ_{p}ρ_{ℓ}b_{p}.

Then the element y ∈ U_{r/ℓ} represented by Frob_{ℓ}^{−1} b_{ℓ} has the desired property, namely that (x, y) ∈ Z. Thus the third claim is proved and with it the result. □

3.8. The universal Kolyvagin recursion. We say that a family of classes

{c_{r} ∈ H^{0}(G, U/MU)}_{r|r}

indexed by the positive integers r dividing r satisfies the universal Kolyvagin recursion if the following conditions hold for all positive integers r dividing r and prime numbers ℓ:

• c_{r} ∈ H^{0}(G_{r}, U_{r}/MU_{r}) = H^{0}(G, U_{r}/MU_{r}) ⊂ H^{0}(G, U/MU).

• ℓ | r ⇒ D_{ℓ}c_{r} = c_{r/ℓ}.

The terminology is justified by the next result.

Proposition 3.9. Let

ξ : U → O_{r}^{×}

be any G-equivariant homomorphism such that

ξI_{ℓ} ⊂ 1 + \sqrt{ℓO_{r,(ℓ)}}

for all primes ℓ dividing r where \sqrt{ℓO_{r,(ℓ)}} denotes the radical of the ideal of O_{r,(ℓ)} generated by ℓ. Let

κ : H^{0}(G, U/MU) → H^{0}(G, 𝔽_{r}^{×}/𝔽_{r}^{×M}) =^{Satz 90} 𝔽^{×}/𝔽^{×M}

be the homomorphism induced by ξ. Let

{c_{r} ∈ H^{0}(G, U/MU)}_{r|r}

be any system of classes satisfying the universal Kolyvagin recursion. Then the corresponding system of classes

{κc_{r} ∈ 𝔽^{×}/𝔽^{×M}}_{r|r}

satisfies the Kolyvagin recursion.

Proof. Fix a positive integer r dividing r and a prime number ℓ. It suffices to prove the following two assertions:

(1) (ℓ, r) = 1 ⇒ κc_{r} ∈ O_{(ℓ)}^{×}/O_{(ℓ)}^{×M}.

(2) ℓ | r ⇒ exp_{ℓ}[κc_{r}]_{ℓ} = ν_{ℓ}κc_{r/ℓ}.

We have ξU_{r} ⊂ O_{r}^{×} by the G-equivariance of ξ, whence assertion 1 via Lemma 2.2. We turn to the proof of assertion 2. By hypothesis ℓ divides r. Fix

c̃_{r} ∈ U_{r}, c̃_{r/ℓ} ∈ U_{r/ℓ}

representing the classes c_{r} and c_{r/ℓ}, respectively. Write

ξc̃_{r} := α_{r}β_{r}^{M} (α_{r} ∈ 𝔽^{×}, β_{r} ∈ 𝔽_{r}^{×})

and

ξc̃_{r/ℓ} = α_{r/ℓ}β_{r/ℓ}^{M} (α_{r/ℓ} ∈ O_{(ℓ)}^{×}, β_{r/ℓ} ∈ O_{r,(ℓ)}^{×}) .

One then has

ξ (\frac{(σ_{ℓ} − 1)c̃_{r}}{M} ) = β_{r}^{(σ_{ℓ}−1)}, ξ (\frac{(ℓ − Frob_{ℓ})c̃_{r/ℓ}}{M} ) = αr/ℓ^{\frac{(ℓ−1)}{M}} β_{r/ℓ}^{ℓ−Frob_{ℓ}}

since there are no nontrivial M^{th} roots of unity in O_{r}^{×}. Choose γ_{r} ∈ 𝔽_{ℓ}^{×} such that

γ_{r}^{N_{ℓ}} and α_{r} generate the same fractional O_{(ℓ)}-ideal.

One then has

γr^{\frac{ℓ−1}{M}} β_{r} ∈ O_{r,(ℓ)}^{×}.

Finally, one has

(γ_{r}^{1−σ_{ℓ}})^{\frac{ℓ−1}{M}} ≡ β_{r}^{σ_{ℓ}−1} ≡ αr/ℓ^{\frac{ℓ−1}{M}} β_{r/ℓ}^{ℓ−Frob_{ℓ}} ≡ αr/ℓ^{\frac{ℓ−1}{M}} mod \sqrt{ℓO_{r,(ℓ)}}

where the crucial middle congruence holds by Proposition/Definition 3.7 and hypothesis. Therefore assertion 2 holds and the proposition is proved. □

4. Comparisons

4.1. The universal Euler system. For each positive integer r dividing r, let

x_{r} ∈ U_{r} ⊂ U

be the class represented by

[∑_{p|r} \frac{1}{p}] ∈ A(r)

where the interior sum is extended over all primes p dividing r. For all positive integers r dividing r and prime numbers ℓ dividing r the following clearly hold:

• N_{ℓ}x_{r} = (Frob_{ℓ} −1)x_{r/ℓ}.
• x_{r} ≡ x_{r/ℓ} mod I_{ℓ}.

We call the family

{x_{r} ∈ U}_{r|r}

the universal Euler system.

4.2. Recovery of the Euler system by specialization. One can easily verify the existence of a unique G-equivariant homomorphism

ξ : U → O_{r}^{×}

such that

ξx_{r} = ξ_{r}

for all positive integers r dividing r. Thus the given Euler system {ξ_{r}} is recovered by specialization via the homomorphism ξ from the universal Euler system {x_{r}} . Note that

ξI_{ℓ} ⊂ 1 + \sqrt{ℓO_{r,(ℓ)}}

for all primes ℓ dividing r.

4.3. Universal Kolyvagin classes. Fix a positive integer r dividing r. We claim that

N′_{r}x_{r} ∈ U_{r} ⊂ U

represents a class

c_{r} ∈ H^{0}(G_{r}, U_{r}/MU_{r}) = H^{0}(G, U_{r}/MU_{r}) ⊂ H^{0}(G, U/MU).

For each prime ℓ dividing r one has

(σ_{ℓ} − 1)N′_{r}x_{r} ≡ −N_{ℓ}N′_{r/ℓ}x_{r} ≡ −(Frob_{ℓ} −1)N′_{r/ℓ}x_{r/ℓ} ≡ 0 mod MU_{r}

by induction on the number of prime divisors of r. Therefore c_{r} is indeed G_{r}-invariant. We call c_{r} the universal Kolyvagin class indexed by r.

4.4. Recovery of the Kolyvagin classes by specialization. Let

κ : H^{0}(G, U/MU) → H^{0}(G, 𝔽_{r}^{×}/𝔽_{r}^{×M}) =^{Satz 90} 𝔽^{×}/𝔽^{×M}

be the homomorphism induced by ξ. For all positive integers r dividing r one has

ξN′_{r}x_{r} = ξ_{r}^{N′_{r}}

and hence

κc_{r} = κ_{r}.

Thus the system {κ_{r}} of Kolyvagin classes is recovered by specialization via the homomorphism κ from the system {c_{r}} of universal Kolyvagin classes.

Proposition 4.5. The universal Kolyvagin classes satisfy the universal Kolyvagin recursion.

Proof. Fix a positive integer r dividing r. By definition the universal Kolyvagin class c_{r} is represented by N′_{r}x_{r} ∈ U_{r} and hence c_{r} ∈ H^{0}(G, U_{r}/MU_{r}). One has an identity

\frac{(σ_{ℓ} − 1)N′_{r}x_{r}}{M} = \frac{ℓ − 1}{M} · N′_{r/ℓ}(x_{r} − x_{r/ℓ}) + \frac{(ℓ − Frob_{ℓ})N′_{r/ℓ}x_{r/ℓ}}{M}

and hence D_{ℓ}c_{r} = c_{r/ℓ} by Proposition/Definition 3.7. □

4.6. Remark. From Propositions 3.9 and 4.5 one recovers the Proposition 2.5 (the latter being a reformulation of the well known Proposition 2.4 of the Rubin appendix to Lang’s text [4]) by somewhat more conceptual means. We wonder if various well known generalizations of Proposition 2.5 (we have uppermost in mind Theorem 4.5.4 on p. 91 of Rubin’s book [6]) could be analogously recovered.

5. The action of D_{ℓ} on the canonical basis for H^{0}(G, U/MU)

5.1. The bigraded ℤ[G]-module K. Let K be the free abelian group on symbols of the form

[a, g, h] ( \begin{matrix}g: positive integer dividing r \\ h: positive integer dividing some power of r \\ a ∈ \frac{g}{r} ℤ/ℤ \end{matrix} ) .

We equip the group K with a bigrading and associated total grading by declaring that

(bidegree of [a, g, h]) := (− ∑_{ℓ} ord_{ℓ} g, ∑_{ℓ} ord_{ℓ} h) ,

(total degree of [a, g, h]) := − ∑_{ℓ} ord_{ℓ} g + ∑_{ℓ} ord_{ℓ} h.

We equip the group K with the unique structure of bigraded ℤ[G]-module such that

σ[a, g, h] = [a′, g′, h′] ⇔ (σϕ(a) = ϕ(b) and g = g′ and h = h′)

for all symbols [a, g, h] and [a′, g′, h′] in the canonical basis of K, injective homomorphisms ϕ : \frac{1}{r}ℤ/ℤ → 𝔽_{r}^{×} and σ ∈ G.

5.2. The differentials d and δ. For each prime number ℓ dividing r we define a G-equivariant differential d_{ℓ} : K → K of bidegree (1, 0) by the rule

d_{ℓ}[a, g, h] := \{ \begin{matrix}(−1)^{∑_{ℓ′<ℓ} ord_{ℓ′} gh} ([a, g/ℓ, h] − ∑_{ℓb=a}[b, g/ℓ, h]) & if ℓ | g, \\ 0 & otherwise, \end{matrix}

and a G-equivariant differential δ_{ℓ} : K → K of bidegree (0, 1) by the rule

δ_{ℓ}[a, g, h] := (−1)^{ord_{ℓ} g+∑_{ℓ′<ℓ} ord_{ℓ} gh} \{ \begin{matrix}(1 − σ_{ℓ})[a, g, hℓ] & if ord_{ℓ} h ≡ 0 mod 2, \\ N_{ℓ}[a, g, hℓ] & if ord_{ℓ} h ≡ 1 mod 2. \end{matrix}

One can verify by a straightforward calculation that any two distinct operators in the family {d_{ℓ}}∪{δ_{ℓ}} anticommute. We equip K with anti-commuting differentials d and δ of bidegree (1, 0) and (0, 1), respectively by the rules

d[a, g, h] := ∑_{ℓ} d_{ℓ}[a, g, h], δ[a, g, h] := ∑_{ℓ} δ_{ℓ}[a, g, h].

Since the sums above contain but finitely many nonzero terms, in fact d and δ are well-defined. Thus we have defined a double complex (K, d, δ) in the category of G-modules.

5.3. Comparison with Ouyang’s definitions. We define an involutive G-equivariant bigraded automorphism ϵ of K by the rule

ϵ[a, g, h] := (−1)^{∑_{ℓ′<ℓ}(ord_{ℓ} g)·(ord_{ℓ′} h)}[a, g, h].

By a straightforward calculation one finds that

ϵd_{ℓ}ϵ[a, g, h] = \{ \begin{matrix}(−1)^{∑_{ℓ′<ℓ} ord_{ℓ′} g} ([a, g/ℓ, h] − ∑_{ℓb=a}[b, g/ℓ, h]) & if ℓ | g, \\ 0 & otherwise, \end{matrix}

and

ϵδ_{ℓ}ϵ[a, g, h] = (−1)^{∑_{ℓ′} ord_{ℓ′} g+∑_{ℓ′<ℓ} ord_{ℓ′} h} \{ \begin{matrix}(1 − σ_{ℓ})[a, g, hℓ] & if ord_{ℓ} h ≡ 0 mod 2, \\ N_{ℓ}[a, g, hℓ] & if ord_{ℓ} h ≡ 1 mod 2. \end{matrix}

It follows that d (resp. δ) as defined in this paper is ϵ-conjugate to d (resp. δ) as defined by the rule appearing on p. 14 of Ouyang’s paper [5] and hence mutatis mutandis Ouyang’s theory applies to the double complex (K, d, δ).

5.4. Identification of H^{0}(K/MK, d + δ) with H^{0}(G, U/MU). For any positive integer r dividing r let K(r) be the subgroup of K generated by all symbols of the form

[a, g, h] ( \begin{matrix}g: positive integer dividing r \\ h: positive integer dividing some power of r \\ a ∈ \frac{g}{r} ℤ/ℤ \end{matrix} ) .

Then K(r) is G-, d-, and δ-stable. It is explained in detail in Ouyang’s paper [5] how to make the identification

H^{∗}(K(r)/MK(r), d + δ) = H^{∗}(G_{r}, U_{r}/MU_{r}).

For our purposes in this note it is enough simply to know that the G-equivariant homomorphism

((class represented by [a, 1, 1]) ↦ (class represented by [a]))

: (bidegree (0, 0) component of K(r)/MK(r)) → U_{r}/MU_{r} induces an isomorphism

H^{0}(K(r)/MK(r), d + δ)→^{∼}H^{0}(G_{r}, U_{r}/MU_{r}) = H^{0}(G, U_{r}/MU_{r}).

Then, passing to the limit over r, we find that the G-equivariant homomorphism

((class represented by [a, 1, 1]) ↦ (class represented by [a]))

: (bidegree (0, 0) component of K/MK) → U/MU

induces an isomorphism

H^{0}(K/MK, d + δ)→^{∼}H^{0}(G, U/MU).

The latter fact can be also be verified directly by a straightforward spectral sequence argument, the key observation being that the subcomplexes of K with fixed ordinate are direct sums of copies of the complex (L, d) discussed in §3.5.

5.5. The canonical basis for H^{0}(G_{r}, U_{r}/MU_{r}). Fix a positive integer r dividing r. Let S(r) be the bigraded G-, d- and δ-stable subgroup of K(r) generated by all symbols [a, g, h] of the canonical basis of K(r) of the form

[a, g, h] (if a = 0, then g does not divide h).

By Proposition 5.4 on p.20 of Ouyang’s paper [5], which is the main technical result of that paper, the quotient map

(K(r), d + δ) → (K(r)/S(r), d + δ)

is a quasi-isomorphism of complexes. Ouyang’s result is proved by verifying that the induced map of spectral sequences is an isomorphism at E_{2}. Clearly the family of symbols of the form

[0, g, gh] ( \begin{matrix}g: positive integer dividing r \\ h: positive integer dividing some power of r \end{matrix} )

forms a graded basis for K(r)/S(r); moreover, it is easy to check that

dK(r) + δK(r) ⊂ S(r) + MK(r).

The upshot is that there exists a unique ℤ/Mℤ-basis

{c̄_{g} ∈ H^{0}(G, U_{r}/MU_{r})}_{g|r}

indexed by the positive integers g dividing r such that the corresponding class c̄_{g} is represented by a 0-cocycle of the complex (K(r)/MK(r), d + δ) congruent modulo S(r)+MK(r) to the symbol [0, g, g]. Up to signs determined by the automorphism ϵ defined in §5.3, the canonical basis constructed here coincides with the canonical basis provided by Theorem 5.5 on p. 22 of Ouyang’s paper [5].

5.6. The canonical basis for H^{0}(G, U/MU). Let S be the bigraded G-, d- and δ-stable subgroup of K generated by all symbols [a, g, h] of the canonical basis of K of the form

[a, g, h] (if a = 0, then g does not divide h).

We have

K = ⋃_{r|r} K(r), S = ⋃_{r|r} S(r)

where in both unions r ranges over the positive integers dividing r. Passing to the limit over r in the obvious way, we find that there exists a unique ℤ/Mℤ-basis

{c̄_{r} ∈ H^{0}(G_{r}, U_{r}/MU_{r})}_{r|r}

for H^{0}(G, U/MU) indexed by the positive integers r dividing r such that the corresponding class c̄_{r} admits representation by a 0-cocycle of the complex (K(r)/MK(r), d + δ) congruent modulo S(r) + MK(r) to the symbol [0, r, r]. We call the family {c̄_{r}} the canonical basis for H^{0}(G, U/MU).

5.7. The diagonal shift operator Δ_{ℓ}. For each prime number ℓ dividing r, we define the corresponding diagonal shift operator Δ_{ℓ} on K of bidegree (1, −1) by the rule

Δ_{ℓ}[a, g, h] := \{ \begin{matrix}[a, g/ℓ, h/ℓ] & if ℓ | g and ℓ | h, \\ 0 & otherwise. \end{matrix}

One has

Δ_{ℓ}d_{p} = d_{p}Δ_{ℓ}, Δ_{ℓ}δ_{p} = δ_{p}Δ_{ℓ}

for all prime numbers p distinct from ℓ. One has

Δ_{ℓ}d_{ℓ} = d_{ℓ}Δ_{ℓ} = 0, (δ_{ℓ}Δ_{ℓ} − Δ_{ℓ}δ_{ℓ})K ⊂ MK.

For every positive integer r dividing r one has

Δ_{ℓ}K(r) ⊂ \{ \begin{matrix}K(r/ℓ) & if ℓ divides r, \\ \{0\} & otherwise. \end{matrix}

The action of Δ_{ℓ} therefore passes to

H^{0}(K(r)/MK(r), d + δ) = H^{0}(G_{r}, U(r)/MU(r))

and in the limit to

H^{0}(K/MK, d + δ) = H^{0}(G, U/MU).

Our definition of the diagonal shift operation Δ_{ℓ} is inspired by a very similar diagonal shift operation defined on p. 3564 of the paper of Das [2] and exploited there to great advantage.

Proposition 5.8. For each prime number ℓ dividing r the endomorphism of H^{0}(G, U/MU) induced by the diagonal shift operation Δ_{ℓ} coincides with D_{ℓ}.

Proof. Fix a positive integer r dividing r and divisible by ℓ. Fix a class

c ∈ H^{0}(G_{r}, U_{r}/MU_{r}).

It suffices to show that D_{ℓ} and the endomorphism of H^{0}(G, U/MU) induced by Δ_{ℓ} applied to c give the same result. Let c be a 0-chain in K(r) reducing modulo MK(r) to a 0-cycle representing c. Write

0 = (d + δ)c + Mb

where b is a 1-chain of K(r). For any positive integer g dividing r and positive integer h dividing some power of r let

(a ↦ a ⊗ [g, h]) : A(r/g) → K(r)

be the unique homomorphism such that

[a] ⊗ [g, h] := [a, g, h]

for all a ∈ \frac{g}{r} ℤ/ℤ. Write

c = ∑ c_{g,h} ⊗ [g, h], Δ_{ℓ}c = ∑ c_{gℓ,hℓ} ⊗ [g, h] (c_{g,h} ∈ A(r/g))

and

b = ∑ b_{g,h} ⊗ [g, h] (b_{g,h} ∈ A(r/g))

where all the sums are extended over pairs (g, h) consisting of a positive integer g dividing r and a positive integer h dividing a power of r. Let ρ_{ℓ} : A(r) → A(r/ℓ) and γ_{p} : A(r/p) → A(r) be as in the proof of Proposition/Definition 3.7. By hypothesis one has an identity

0 = (∑_{\begin{matrix}p|r \\ p<ℓ \end{matrix}} γ_{p}c_{p,ℓ}) + γ_{ℓ}c_{ℓ,ℓ} − (∑_{\begin{matrix}p|r \\ p>ℓ \end{matrix}} γ_{p}c_{p,ℓ}) + (1 − σ_{ℓ})c_{1,1} + Mb_{1,ℓ}

and hence also an identity

0 = (∑_{\begin{matrix}p|r \\ p<ℓ \end{matrix}} γ_{p}ρ_{ℓ}c_{p,ℓ}) − (ℓ − Frob_{ℓ})(Frob_{ℓ}^{−1} c_{ℓ,ℓ}) − (∑_{\begin{matrix}p|r \\ p>ℓ \end{matrix}} γ_{p}ρ_{ℓ}c_{p,ℓ}) + Mρ_{ℓ}b_{1,ℓ}.

Let x ∈ U_{r} be the element represented by c_{1,1} and let y ∈ U_{r/ℓ} be the element represented by c_{ℓ,ℓ}. One the one hand, the class of H^{0}(G_{r}, U_{r}/MU_{r}) represented by the 0-cocycle c mod M of the complex (K(r)/MK(r), d + δ) is x mod MU_{r} and the class of H^{0}(G_{r/ℓ}, U_{r/ℓ}/MU_{r/ℓ}) represented by the 0-cocycle Δ_{ℓ}c mod M of the complex (K(r/ℓ)/MK(r/ℓ), d + δ) is y mod MU_{r/ℓ}. But on the other hand, one has

\frac{(σ_{ℓ} − 1)x}{M} ≡ \frac{(ℓ − Frob_{ℓ})y}{M} mod I_{ℓ}

and hence

D_{ℓ}(x mod MU) ≡ y mod MU_{r/ℓ}

by Proposition/Definition 3.7. Therefore the results of applying D_{ℓ} and the endomorphism of H^{0}(G, U/MU) induced by Δ_{ℓ} to the class x mod MU indeed coincide. □

Corollary 5.9. The canonical basis {c̄_{r}} satisfies the universal Kolyvagin recursion.

Proof. Clear. □

Corollary 5.10. Any system of classes {b_{r}} satisfying the universal Kolyvagin recursion and the normalization b_{1} = c̄_{1} is a ℤ/Mℤ-basis of H^{0}(G, U/MU).

Proof. Fix a positive integer r dividing r and arbitrarily and let

r = ℓ_{1} ⋯ ℓ_{n}

be the prime factorization of r. One then has

D_{ℓ_{1}} ⋯ D_{ℓ_{n}}b_{r} = b_{1} = c̄_{1} = D_{ℓ_{1}} ⋯ D_{ℓ_{n}} c̄_{r}

and hence

b_{r} − c̄_{r} ∈ ker (H^{0}(G_{r}, U_{r}/MU_{r}) ⟶^{D_{ℓ_{1}} ···D_{ℓn}} H^{0}(G_{r}, U_{r}/MU_{r})) = ⨁_{\begin{matrix}r′|r \\ r′≠r \end{matrix}} ℤ/Mℤ · c̄_{r′},

whence the result. □

5.11. Remark. From Corollary 5.10 it follows in particular that the system {c_{r}} of universal Kolyvagin classes defined in §4.3 is a ℤ/Mℤ-basis for H^{0}(G, U/MU). More precisely, it follows that for every positive integer r dividing r, the family {c_{g}}_{g|r} is a ℤ/Mℤ-basis for H^{0}(G_{r}, U_{r}/MU_{r}). The latter fact is none other than Theorem B on p. 2 of Ouyang’s paper [5]; our way of proving Theorem B here is simpler than Ouyang’s original method.

References

[1] Anderson, G. W., A double complex for computing the sign-cohomology of the universal ordinary distribution, Contemp. Math. 224(1999), 1–27. 1

[2] Das, P., Algebraic Gamma monomials and double coverings of cyclotomic fields. Trans. Amer. Math. Soc. 352(2000), 3557–3594 1, 15

[3] Kubert, D. S., The universal ordinary distribution, Bull. Soc. Math. France 107(1979), 179– 202. 6

[4] Lang, S., Cyclotomic Fields I and II, combined 2^{nd} edition, Grad. Texts in Math. 121, Springer Verlag, 1990. 1, 4, 6, 12

[5] Ouyang, Y., Group cohomology of the universal ordinary distribution, J. reine angew. Math. 537(2001) 1–32. 1, 6, 7, 13, 14, 17

[6] Rubin, K., Euler systems, Annals of Math. Studies 147, Princeton University Press, 2000. 2, 12

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

  E-mail address: gwanders@math.umn.edu

Department of Mathematics, University of Toronto, 100 St. George St., Toronto, ON M5S 3G3, Canada

  E-mail address: youyang@math.toronto.edu