Some (very sketchy) class notes on the Cyclotomic Euler System.

1. Cyclotomic fields. Fix a positive integer m and put F = Q(μ_{m})^{+}. Let F̄/F be an algebraic closure, and put G_{F} := Gal(F̄/F). For any postive integer r relatively prime to m, put F (r) := F (μ_{r}). If r = ∏_{j=1}^{ν} p_{j}^{e_{j}} is the prime power decomposition of r, we have that the field extensions F (p_{j}^{e_{j}})/F are totally ramified at the primes of F above p_{j} and they are unramified elsewhere. The composite of the fields F (p_{j}^{e_{j}})/F for j = 1, … ν is equal to F (r), and these fields are linearly disjoint subfield extensions of F (r)/F in the strong sense that the natural mapping

⊗_{F} F (p_{j}^{e_{j}}) → F (r)

is an isomorphism (here the tensor product runs through the factors F (p_{j}^{e_{j}}) for j = 1, … ν).

If G_{r} := Gal(F (r)/F), we have the natural isomorphism

G_{r} ≅ (Z/rZ)^{∗},

and the natural “coordinatization”

G_{r} = ∏_{j=1}^{ν} G_{p_{j}^{e_{j}}} .

If ℓ is a prime number not dividing r then the Frobenius element Frob_{ℓ} ∈ G_{r} corresponds to the image of the integer ℓ in (Z/rZ)^{∗} under the natural isomorphism, while if ℓ | r, writing ℓ = p_{1} we consider the factorization

G_{r} = G_{p_{1}^{e_{1}}} × ∏_{j=2}^{ν} G_{p_{j}^{e_{j}}} ,

and make the convention that Frob_{ℓ} ∈ G_{r} will denote the element which is trivial in the first factor, and Frob_{ℓ} (in the old sense) in the second factor.

Denote by R = the set of squarefree positive integers divisible only by primes ℓ ≡ ±1 modulo m (i.e., those that split completely in F/Q). We have, as above, that

G_{r} ≅ ∏_{ℓ | r} G_{ℓ},

and passing to group rings we have

Z[G_{r}] ≅ ⨂_{ℓ | r} Z[G_{ℓ}].

Let N_{ℓ} ∈ Z[G_{ℓ}] mean the sum of all the group elements of G_{ℓ}; that is

N_{ℓ} := ∑_{σ∈G_{ℓ}}[σ].

We view N_{ℓ} as an element in Z[G_{r}] in the evident way:

N_{ℓ} = ∑_{σ∈G_{ℓ}} 1_{Z[G_{r/ℓ}]} ⊗ [σ].

Note that G_{ℓ} is a cyclic group of order ℓ − 1, and let γ_{ℓ} ∈ G_{ℓ} denote a choice of generator. Define:

D_{ℓ} := ∑_{i=0}^{ℓ−2} i[γ_{ℓ}^{i}] ∈ Z[G_{ℓ}].

We have:

(γ_{ℓ} − 1)D_{ℓ} = (ℓ − 1) − N_{ℓ}.

Of course, the element D_{ℓ} depends upon the choice of generator γ_{ℓ}. If r ∈ R, making a choice of generator γ_{ℓ} for all prime numbers ℓ dividing r, we put

D_{r} := ∏_{ℓ | r} D_{ℓ} ∈ Z[G_{r}] ≅ ⨂_{ℓ | r} Z[G_{ℓ}].

2. Augmentation filtrations in groups rings. If R is a commutative ring with unit, and G is a finite cyclic group, let R[G] be the group ring of G with coefficients drawn from R. If g ∈ G, denote by [g] ∈ R[G] the corresponding element of the group-ring. Denote by I_{R} (or simply I) the kernel of the ring homomorphism R[G] → R which sends the subgroup G of the group of units in R[G] to 1 ∈ Z. The additive group of the R[G]-module I/I^{2} is a cyclic abelian group and more to the point, the mapping G → I/I^{2} which sends a group element g ∈ G to the coset of [g] − 1 ∈ I modulo I^{2} is surjective, and if R = Z, then it is an isomorphism of groups. If n is a positive integer which is a divisor of the order of G, and if R = Z/nZ, then I/I^{2} is cyclic of order n and the mapping G → I/I^{2} may be thought of as identifying I/I^{2} with G ⊗_{Z} R.

Now let m and r ∈ R be as before, and fix n an odd positive number relatively prime to r. Put R := Z/nZ. We shall be working in group rings with coefficients in R. We will work with the subset R_{n} ⊂ R consisting of those squarefree integers r ∈ R all of whose prime divisors q have the property that q ≡ 1 modulo n. Given an integer r = q_{1} · q_{2} … q_{ν} ∈ R_{n} we have the standard product decomposition G(r) = ∏_{j=1}^{ν} G(q_{j}) and

R[G(r)] = ⊗_{j=1}^{ν}R[G(q_{j})].

Let I_{q_{j}} ⊂ R[G(q_{j})] be the augmentation ideals for j = 1, …, ν. The basic ideals we will be dealing with are

I_{r} := ⊗_{j=1}^{ν} I_{q_{j}} ⊂ R[G_{r}],

and

I′_{r} := ∑_{j=1}^{ν} I_{q_{1}} ⊗ I_{q_{2}} ⊗ … ⊗ I_{q_{j}}^{2} … ⊗ I_{q_{ν}} ⊂ I_{r}.

Lemma 1.

1. I_{r} = ⋂_{q | r} I_{q} ⊗ R[G_{r/q}] ⊂ R[G_{r}],

2. I_{r}/I′_{r}  ≅ ⊗_{j=1}^{ν} I_{q_{j}}/I_{q_{j}}^{2}.

and

3. \frac{I_{r} + ∑_{j} I_{q_{j}}^{2} ⊗ R[G_{r/q_{j}}]}{∑_{j} I_{q_{j}}^{2} ⊗ R[G_{r/q_{j}}]} ≅ I_{r}/I′_{r}.

Proof. …

For short, put W_{q} := I_{q}/I_{q}^{2} and put
W_{r} := I_{q_{1}}/I_{q_{1}}^{2} ⊗ I_{q_{2}}/I_{q_{2}}^{2} ⊗ … ⊗ I_{q_{ν}} /I_{q_{ν}}^{2} = W_{q_{1}} ⊗ W_{q_{2}} ⊗ … ⊗ W_{q_{ν}} ,

noting that all the W_{q_{j}} are cyclic groups of order n, as is W_{r}. Let us refer to these W_{q_{j}}’s and W_{r} as twisting modules. You can“get rid” of the twisting module W_{r} by making a choice of generator, but then the formulas will depend upon this choice of generator. Alternatively, one can just make a choice of generator γ_{q_{j}} ∈ G(q_{j}) for j = 1 …, ν and note that the image of γ_{q_{j}} in W_{q_{j}} = I_{q_{j}}/I_{q_{j}}^{2} gives a choice of generator in each W_{q_{j}}; the tensor product of these generators gives a choice of generator in W_{r}.

We have the natural projection

η_{r} : I_{r} + ∑_{j} I_{q_{j}}^{2} ⊗ R[G_{r/q_{j}}] ⟶ \frac{I_{r} + ∑_{j} I_{q_{j}}^{2} ⊗ R[G_{r/q_{j}}]}{∑_{j} I_{q_{j}}^{2} ⊗ R[G_{r/q_{j}}]} ≅ W_{r}

which restricts to the projection which we will denote by the same letter:

η_{r} : I_{r} ⟶ I_{r}/I′_{r} ≅ W_{r}.

3. Euler Systems. If r ∈ R, by the universal r- Euler system X_{r} let us mean the Z[G_{r}]-module with the following presentation. The module X_{r} has a set of generators labelled x_{s} where s runs through all positive divisors of r with relations given as follows:

1. For all positive divisors s of r, the element x_{s} is fixed under multiplication by G(r/s) ⊂ G(r).

2. For all prime numbers ℓ and positive integers s such that ℓs divides r, we have

N_{ℓ}x_{ℓs} = (Frob_{ℓ} − 1)x_{s}.

Note that any G(r)-equivariant homomorphism

X_{r} → F (r)^{∗}

(or at least any G(r)-equivariant homomorphism to the subgroup of units in F (r)^{∗}) is the beginning of an Euler system of the type we have already studied.

Proposition. ([9]) If p does not divide r, the module X_{r} has no p-torsion. Moreover X_{r} ⊗ Q is free of rank 1 over Q[G_{r}].

Let X_{r} denote the r-universal Euler system, and define

Θ_{r} := ∑_{σ∈G_{r}} x_{r}^{σ} ⊗ [σ] ∈ X_{r} ⊗_{R} R[G_{r}].

We have a natural action of G_{r} ×G_{r} on X_{r} ⊗_{R}R[G_{r}]. Denote the action of the element (g, h) ∈ G_{r} × G_{r} by g ⊗ h. We have:

Fr_{q} ⊗ Fr_{q} · Θ_{r} = Θ_{r},

or equivalently

Fr_{q} ⊗ 1 · Θ_{r} = 1 ⊗ Fr_{q}^{−1}Θ_{r}.

For primes q dividing r consider the natural projection π_{r,r/q} : R[G_{r}] → R[G_{r/q}] with kernel I_{q} ⊗ R[G_{r/q}] ⊂ R[G_{r}]. Forming the homomorphism

1 ⊗ π_{r,r/q} : X_{r} ⊗_{R} R[G_{r}] → X_{r} ⊗_{R} R[G_{r/q}],

note that the norm axiom for Euler Systems gives

1 ⊗ π_{r,r/q}(Θ_{r}) = (Fr_{q} ⊗ 1 − 1)Θ_{r/q}.

Remark. The Θ_{r}’s do not lie in I_{r}, but we have the weaker inclusion:

Lemma 2.

Θ_{r} ∈ X_{r} ⊗ (I_{r} + ∑_{j} I_{q_{j}}^{2} ⊗ R[G_{r/q_{j}}]).

Define

Θ̃_{r} := (1 ⊗ η_{r})Θ_{r} ∈ X_{r} ⊗ W_{r}.

Lemma 3. The element Θ̃_{r} ∈ X_{r} ⊗ W_{r} is fixed under the natural action of G(r). That is,

Θ̃_{r} ∈ (X_{r} ⊗_{Z} R)^{G(r)} ⊗ W_{r}.

4. The Cyclotomic Euler System. As proved in class (by Stephanie Yang and Brian Osserman) we have

Proposition. There is a Z[G_{r}]-module homomorphism

E : X_{r} → O_{F (r)}^{∗} ⊂ F (r)^{∗}

defined on the generators x_{r} by the rule:

x_{r} ↦ ξ_{r} := (ζ_{m}ζ_{r} − 1)(ζ_{m}^{−1}ζ_{r} − 1),

where for any positive integer d, ζ_{d} denotes a choice of primitive d-th root of unity.

5. Kolyvagin classes coming from the cyclotomic Euler System. We have fixed a positive integer m. Now fix a positive odd integer n and consider, as before, the set R_{n} of positive squarefree integers whose prime divisors are all congruent to ±1 modulo m and congruent to 1 modulo n. For r ∈ R_{n} the homomorphism E induces a homomorphism

E : (X_{r} ⊗_{Z} R)^{G(r)} ⊗ W_{r} → (F (r)^{∗} ⊗_{Z} R)^{G(r)} ⊗ W_{r} = (F (r)^{∗}/F (r)^{∗n})^{G(r)} ⊗ W_{r}.

We have a canonical identification F (r)^{∗}/F (r)^{∗n} = H^{1}(G_{F} , μ_{n}) coming from the Kummer exact sequence,

0 → μ_{n} → F̄^{∗} → F̄^{∗} → 0,

the mapping F̄^{∗} → F̄^{∗} being given by taking n-th powers. Consider now the exact sequence coming from the Hochschild-Serre Spectral Sequence,

H^{1}(G(r), H^{0}(G_{F (r)}, μ_{n})) → H^{1}(G_{F} , μ_{n}) → H^{1}(G_{F (r)}, μ_{n})^{G(r)} → H^{2}(G(r), H^{0}(G_{F (r)}, μ_{n})),

and note that (p is an odd prime) F (r) does not contain a primitive p-th root of unity, and therefore the flanking groups in the above exact sequence are zero, giving an isomorphism

H^{1}(G_{F} , μ_{n}) ≅ H^{1}(G_{F (r)}, μ_{n})^{G(r)},

or equivalently, we have that the natural mapping

F^{∗}/F^{∗n} → (F (r)^{∗}/F (r)^{∗n})^{G(r)}

is an isomorphism. So, the natural mapping

ι : F^{∗}/F^{∗n} ⊗ W_{r} → (F (r)^{∗}/F (r)^{∗n})^{G(r)} ⊗ W_{r}

is also.

We may view the image of Θ̃_{r} under the homomorphism E as lying in the submodule of H^{1}(G_{F (r)}, μ_{n}) ⊗ W_{r} fixed under the action of G(r).

Define

κ_{r} ∈ F^{∗}/F^{∗n} ⊗ W_{r}

by the formula

ι(κ_{r}) = E(Θ̃_{r}).

Summary Remarks. For each r ∈ R_{n} we have defined a Kolyvagin class

κ_{r} ∈ F^{∗}/F^{∗n} ⊗ W_{r} = H^{1}(G_{F} , μ_{n}) ⊗ W_{r}.

One easily sees that the element κ_{r} is represented by an r-unit in O_{F}; that is,

κ_{r} ∈ O_{F} [1/r]^{∗}/O_{F} [1/r]^{∗n} ⊗ W_{r}.

Equivalently, for the “natural” f/s structure on H^{1}(G_{F} , μ_{n}) we have that the restriction res_{v} of κ_{r} to any prime v of F not dividing r lies in H_{f}^{1}(G_{F_{v}}, μ_{n})⊗W_{r} ⊂ H^{1}(G_{F_{v}}, μ_{n})⊗W_{r}. It is in fact, the case that for any v dividing r, the restriction res_{v}(κ_{r}) ∈ H^{1}(G_{F_{v}}, μ_{n})⊗W_{r} can be given by an explicit (“universal”) formula in terms of the data {res_{v}(κ_{d})} ⊗ W_{d} for all proper divisors d of r (this is standard for the singular part of res_{v}(κ_{r}) but is also true for the full classes (joint work with Karl Rubin) . A system of classes κ_{r} for r ∈ R_{n} as above satisfying such universal formulas will be called a Kolyvagin system.

6. The basic 1-cocycle in the universal r-Euler system. Making a choice of generator γ_{ℓ} ∈ G(ℓ) for all prime numbers ℓ, we have a chosen generator ϵ_{r} ∈ W_{r} of each of the cyclic modules W_{r} for r ∈ R_{n} so we can “get rid of the twist modules”, i.e., we can

identify all the W_{r}’s with Z/nZ. With the same choice of generators γ_{ℓ} ∈ G(ℓ) (for prime numbers ℓ dividing r) we have the operator D_{r} ∈ Z[G(r)].

Lemma 4.

D_{r}x_{r} ⊗ ⨂_{ℓ | r} ϵ_{ℓ} = Θ̃_{r} ∈ X_{r} ⊗ W_{r}.

Proof. I will do this for r = ℓ using that ∑_{j=0}^{ℓ−2} γ_{ℓ}^{j}x_{ℓ} = 0, and therefore

Θ̃_{ℓ} = ∑_{j=0}^{ℓ−2} γ_{ℓ}^{j}x_{ℓ} ⊗ [γ_{ℓ}^{j}] = ∑_{j=0}^{ℓ−2} γ_{ℓ}^{j}x_{ℓ} ⊗ ([γ_{ℓ}^{j}] − 1) = ∑_{j=0}^{ℓ−2} γ_{ℓ}^{j}x_{ℓ} ⊗ {[ϵ_{ℓ}](∑_{k=0}^{j−1}[γ_{ℓ}^{k}])},

and since the last factor may be replace by the integer j since the expression we are calculating is being considered modulo the square of the augmentation ideal, we get:

Θ̃_{ℓ} = ∑_{j=0}^{ℓ−2} jγ_{ℓ}^{j}x_{ℓ} ⊗ [ϵ_{ℓ}].

That is,

Θ̃_{ℓ} = D_{ℓ}x_{ℓ} ⊗ [ϵ_{ℓ}].

So, by Lemma 3, or by simply redoing it in this language, we have that

Lemma 5. The image of D_{r}x_{r} in X_{r}/nX_{r} is fixed under the action of G(r).

Corollary. For g ∈ G(r) there is a unique element (denote it c_{r}(g)) in X_{r} such that n · c_{r}(g) = (g − 1)D_{r}x_{r}. The mapping g ↦ c_{r}(g) is a 1-cocycle of G(r) with values in the G(r)-module X_{r} (the basic 1-cocycle).

Proof. The first assertion follows from Lemma 5 and the Proposition above (which gives us that the abelian group X_{r} has no nontrivial n-torsion). The second assertion is then a straightforward calculation.

7. An alternative description of the Kolyvagin classes. Recall (section 4) the cyclotomic r-Euler System given as a G(r)-module homomorphism

E : X_{r} → F (r)^{∗}.

The mapping E takes the generator x_{r} to the cyclotomic unit ξ_{r}. (Recall that ξ_{r} is the cyclotomic unit defined by Stephanie and Brian.)

Convention. I will write the action of Z[G(r)] on F (r)^{∗} as exponentiation.

Let g ∈ G_{F}. Applying E to c_{r}(g) we see that ξ_{r}^{D_{r}·(g−1)} is an n-th power in F (r)^{∗}. (Moreover, any n-th power in F (r)^{∗} has a unique n-th root in F (r)^{∗} since F (r)^{∗} contains no nontrivial n-th root of unity.) So

Ec_{r}(g) = ξr^{\frac{Dr ·(g−1)}{n}}

is the unique n-th root of ξ_{r}^{D_{r}·(g−1)} in F (r)^{∗}. By Hilbert’s Theorem 90, H^{1}(G(r), F (r)^{∗}) = 0 and therefore the image of the basic 1-cocycle c_{r} under the G(r)-module homomorphism E is a coboundary. That is,

Ec_{r}(g) = ξr^{\frac{Dr ·(g−1)}{n}} = β^{(1−g)}

for some β ∈ F (r)^{∗} and for all g ∈ G_{F}. Clearly, β is not determined by the above equation but is only well-defined modulo the image of F^{∗}. Let us make a choice of such a β (remembering that it is a choice we have made). We have

β^{n(1−g)} = ξ_{r}^{D_{r}(g−1)}.

¿From the displayed formula above, the element κ′_{r} := β^{n}ξ_{r}^{D_{r}} ∈ F (r)^{∗} is fixed under the action of G(r) and therefore:

κ′_{r} ∈ F^{∗}.

One has

Lemma 6.

κ_{r} = κ′_{r} ⊗ ϵ_{r}.

8. Representing Kolyvagin classes by a 1-cocycle. The image of κ′_{r} in H^{1}(G_{F} , μ_{n}) is represented by a 1-cocycle on G_{F} with values in μ_{n} of the form g ↦ ((κ′_{r})^{\frac{1}{n}})^{g−1}. The conventions here are as follows: If g ∈ G_{F} and a ∈ F^{∗}, then (a^{\frac{1}{n}})^{g−1} means that you take an n-th root of a in F̄ and apply g − 1 to this n-th root. Of course, the 1-cocycle will depend upon the n-th root chosen, but its cohomology class will not. Let us pin down a specific 1-cocycle k_{r} by taking

k_{r}(g) = β^{g−1} · (ξr^{\frac{Dr}{n}})^{g−1}

for all g ∈ G_{F}. I personally find this particularly confusing because the second factor on

the right-hand side, (ξr^{\frac{Dr}{n}})^{g−1}, is not necessarily the same as ξr^{\frac{Dr (g−1)}{n}}, i.e. the unique n-th root of ξ_{r}^{D_{r}(g−1)} in F (r)^{∗} as introduced above.

8. The singular part of the Kolyvagin classes. Let us work with the 1-cocycle g ↦ k_{r}(g) constructed above. If v is a prime of F, consider the natural mappings

H^{1}(G_{F} , μ_{n}) → H^{1}(G_{F_{v}}, μ_{n}) → H_{s}^{1}(G_{F_{v}}, μ_{n}),

and denote the restriction of k_{r} to G_{F_{v}}-cohomology and to the singular part of G_{F_{v}}- cohomology by k_{r,v} and k_{r,v,s} respectively.

Notation. In the computation we shall make in this section, we shall be considering v a prime of F dividing q which is a prime dividing r, and w a prime of F (r) lying over v. Let h_{w} : O_{F (r)_{w}} → F̄_{q} denote the reduction modulo the maximal ideal composed with an imbedding into the algebraically closed field F̄_{q} and let h_{v} : O_{F_{v}} → F̄_{q} denote the restriction of the reduction mapping h_{w} to O_{F_{v}}. (In effect, we are choosing a prime w̄ lying above w in the ring of integers Ow̄ of the algebraic closure of F_{v} and working there.) Also the congruence symbol ≡ in an equation will mean that both sides of the equation are units (e.g., in Ow̄) and we have an equality between their images in F̄_{q}.

We have that μ_{n} has trivial G_{F_{v}}-action, and so H_{s}^{1}(G_{F_{v}}, μ_{n}) may be identified with a subgroup of Hom(I_{F_{v}}^{τ}, μ_{n}) where I_{F_{v}}^{τ} is the tame inertia group of G_{F_{v}}. Fix Γ_{v} ∈ I_{F_{v}} where I_{F_{v}} is the inertia group in G_{F_{v}}, and let us suppose that Γ_{v} projects to a topological generator of I_{F_{v}}^{τ} and projects to γ_{q} ∈ G(q). Under the above identification the cohomology class k_{r,v,s} gives us a homomorphism also denoted

k_{r,v,s} : I_{F_{v}}^{τ} → μ_{n}(F_{v}) = μ_{n}(F_{q}).

Since I_{F_{v}}^{τ} is generated by the image our chosen element Γ_{v}, k_{r,v,s} is determined by the element h_{v}(k_{r}(Γ_{v})) ∈ μ_{n}(F̄_{q}) = μ_{n}(F_{q}). Let us compute this element. Note that β^{γ_{q}−1} is a unit in O_{F (r)_{w}}.

Lemma 8.

k_{r,v,s}(Γ_{v}) = h_{w}(β^{γ_{q}−1}) ∈ μ_{n}(F̄_{q}).

Proof. By definition, k_{r,v,s}(Γ_{v}) is the image of β^{Γ_{v}−1} · (ξ_{r}^{D_{r}/n})^{Γ_{v}−1} in μ_{n}(F̄_{q}). So we must show that the image of (ξ_{r}^{D_{r}/n})^{Γ_{v}−1} in μ_{n}(F_{q}) is 1. But this follows from the fact that ξ_{r}^{D_{r}/n} is a unit (e.g., in F (r)_{w}) and Γ_{v} is in the inertia group.

9. Calculating the singular part of the Kolyvagin classes from Kolyvagin classes of lower level. Let r ∈ R_{n} as before, and let q ∈ R_{n} be a prime number not dividing r. Let w be a prime of F (rq) dividing q and keep to the conventions of the previous section. We are going to relate the Kolvyvagin class at level rq with that of level r, using the Euler system relation between the corresponding ξ’s. Let the β’s of the last section be indexed by their level, so we rewrite the formula of Lemma 8 as:

k_{rq,v,s}(Γ_{v}) = h_{w}(β_{rq}^{γ_{q}−1}) ∈ μ_{n}(F_{q}),

and recall that β_{rq}^{1−γ_{q}} is characterized by the property that it lies in F (rq) and its n-th power is [ξ_{rq}^{D_{rq}·(γ_{q}−1)}]. By the relations (γ_{q} − 1)D_{q} = (q − 1) − N_{q} we have

\frac{D_{rq} · (Γ_{q} − 1)}{n} = \{(q − 1)/n − N_{q}/n\}D_{r},

so (taking n-th roots in F (rq)^{∗})

[ξ_{rq}^{D_{rq}·(Γ_{q}−1)}]^{\frac{1}{n}} = [ξ_{rq}^{D_{r}\{(q−1)−N_{q}\}}]^{\frac{1}{n}} = \frac{ξ_{rq}^{D_{r} \frac{(q−1)}{n}}}{[ξ_{r}^{D_{r}(Frob_{q}−1)}]^{\frac{1}{n}}} ∈ F (rq)^{∗}

≡ \frac{ξ_{r}^{D_{r} \frac{(q−1)}{n}}}{[ξ_{r}^{D_{r}(Frob_{q}−1)}]^{\frac{1}{n}}} ≡ [β_{r}^{n(Frob_{q}−1)}]^{\frac{1}{n}} · [ξr^{\frac{Dr}{n}}]^{(q−1)}

≡ β_{r}^{(Frob_{q}−1)} · [ξr^{\frac{Dr}{n}}]^{(F rob_{q}−1)}.

Since

k_{r}(Frob_{q}) = β_{r}^{(Frob_{q}−1)} · [ξr^{\frac{Dr}{n}}]^{(F rob_{q}−1)},

we have determined that the singular part of k_{rq} at a prime v dividing q is determined by (the finite part of) k_{r} at v. Specifically,

Proposition.

k_{rq,v,s}(γ_{v}) ≡ k_{r,v}(Frob_{q}) ∈ μ_{n}(F_{q}).

10. Depth. Note that if v ∈ R_{n}, then the four cohomology groups H_{f}^{1}(G_{F_{v}}, μ_{n}), H_{s}^{1}(G_{F_{v}}, Z/nZ), H_{s}^{1}(G_{F_{v}}, μ_{n}), and H_{f}^{1}(G_{F_{v}}, Z/nZ) are all cyclic groups of order n. Moreover the cup-product pairing (to H^{2}(G_{F} , μ_{n}) = Z/nZ) gives perfect dualities

H_{s}^{1}(G_{F_{v}}, μ_{n}) ⊗ H_{f}^{1}(G_{F_{v}}, Z/nZ) → Z/nZ,

and

H_{f}^{1}(G_{F_{v}}, μ_{n}) ⊗ H_{s}^{1}(G_{F_{v}}, Z/nZ) → Z/nZ.

If α is an element in a finite group H, by the depth of α in H let us mean the index of the subgroup generated by α; so α is a generator of H if and only its depth is 1.

Corollary (of the proposition of section 9). The depth of k_{r,v} ∈ H_{f}^{1}(G_{F_{v}}, μ_{n}) is equal to the depth of k_{rq,v,s} ∈ H_{s}^{1}(G_{F_{v}}, μ_{n}).

11. An illustration of how to apply Kolyvagin’s method. Let F_{o} be a real quadratic field, and let m be a positive integer such that F_{o} ⊂ F = Q(m)^{+}. Let ϵ ∈ O_{F_{o}}^{∗} be the fundamental unit, and let c = ±ϵ^{N} be the cyclotomic unit.

Proposition. The prime divisors of the class number of F_{o} divide 2N.

Brief reminder of the outline of the proof. The conjugation involution σ : F_{o} → F_{o} plays an important role in the proof, but we will only hint at this in the telegraphic sketch below. Put n = p a prime number not dividing 2N. Let L/F_{o} be the maximal abelian exponent p extension which is everywhere unramified, and put A = Gal(L/F_{o}). The Galois group Gal(L/Q) is a semi-direct product of A and {1, σ} = Gal(F_{o}/Q) and σ acts (via conjugation) as inversion in the (abelian) group A. We aim to show that A = 0. Let M/F_{o} be the compositum of F_{o}(μ_{p}, c^{1/p}) and L/F_{o}. One shows that these field extensions are linearly independent and that Gal(M/F_{o}) = A × Gal(F_{o}(μ_{p}, c^{1/p})/F_{o}).

Using the Cebotarev Theorem, find a prime number q which, first, splits in F_{o} and letting v and v̄ be the primes of F_{o} lying above q has the following three properties: the elements Frob_{v}, Frobv̄ ∈ Gal(M/F_{o}) project to

1. nonzero elements (each the inverse of the other) in A

2. nonzero elements in Gal(F_{o}(μ_{p}, c^{1/p})/F_{o}))

3. to zero in Gal(F_{o}(μ_{p})/F_{o}).

These three facts give us, respectively, that

1. the primes v and v̄ do not completely split in L.

2. the cycotomic unit c is not locally a p-th power in F_{o,v} nor in F_{o,v̄}.

3. q ∈ R_{p}.

The cyclotomic unit c ∈ F_{o} is equal to N_{F/F_{0}}(κ_{1}), the Norm to F_{o} of the Kolyvagin class κ_{1}. From the corollary to the proposition in section 10, we have that for a q as above, N_{F/F_{0}}(κ_{q}), the Norm to F_{o} of the Kolyvagin class κ_{q}, has singular support precisely at the two primes v and v̄ of F_{o} (the involution σ links its behavior at these two primes). Let

h ∈ Hom(A, Z/pZ) = H_{f}^{1}(G_{F_{o}}, Z/pZ).

Then considering the global invariant of the cup-product of the cohomology classes N_{F/F_{0}}(κ_{q}) and h we have that

h(Frob_{v}) = h(Frobv̄) = 0.

Since this holds for all h ∈ Hom(A, Z/pZ), it follows that the primes v and v̄ split completely in L, contradicting 1. Therefore A = 0.

Remark. We have gotten so far, and yet we have only used the κ_{r}’s for r = q a prime number. There are more κ_{r}’s just waiting to give us even more precise information…

12. A list of existent “Euler Systems”, and sample applications that can be gotten from each of them.

The reason for the quotation-marks around Euler System is that in the exposition in this class, I have adopted only one example of the present collection of variations of the notion of “Euler System”. And there is some considerable variation in the structure of of the others. The concept is still in flux, which is one of the exciting things about it; we don’t know what shape a unified theory of “Euler Systems” will eventually have (or if there will be such a unified theory). But whatever form it will take, it has amazing applications. The variant Euler Systems are listed roughly in order of how difficult it is to understand them, except possibly for Kato’s and Flach’s systems. For Kato’s, it is necessary to know algebraic K-theory; for Flach’s it is good to know it. Any single choice of the topics listed below would make a reasonable subject to be studied in a (full semester) seminar; e.g. 1.b.

Let p be an odd prime number, O a discrete valuation ring finite over Z_{p}, K/Q a finite field extension, and χ : G_{K} → O^{∗} a nontrivial character of finite order prime to p with χ(P) ≠ 1 for all primes P of K (equivalently: no prime P above p splits completely in the “splitting field” of χ over K, i.e., the finite field extension F/K be such that G_{F} = ker(χ) ⊂ G_{K}). The character χ can be thought of as a character on Gal(F/K). By the χ-part of a finite abelian p-group A with Gal(F/K)-action, we mean the O-submodule, A^{χ}, of A ⊗_{Z_{p}} O, defined by:

A^{χ} := {a ∈ A ⊗_{Z_{p}} O | g · a = χ(g)a; (g ∈ Gal(F/K)) }.

1. The Cyclotomic Euler System (K = Q).

Applications:

a. Set K = Q. Let p, O, χ, and F be as above, but suppose that χ is an even character. The order of the χ-part of the (p-primary component of the) ideal class group of F is equal to the order of the χ-part of the (p-primary component of the) group O_{F}^{∗} /C_{F} (where C_{F} is the group of cyclotomc units of F.

b. Iwasawa’s “Main Conjecture” (over Q).

References: [6], [9].

2. The Elliptic Unit Euler System.

Applications:

a. Let p, O, χ, and F be as above, and let K be a quadratic imaginary field. Let C_{F} denote the elliptic units of the field F. The order of the χ-part of the (p-primary component of the) ideal class group of F is equal to the order of the χ-part of the (pprimary component of the) group O_{F}^{∗} /C_{F} (where C_{F} is the group of cyclotomc units of F.

b. The “Main Conjecture” (over K).

References: [4], [9].

3. The Heegner Point Euler System. (If we can schedule it conveniently, Mak will be giving some more introductory lectures on this topic.)

Applications (using results of Gross-Zagier; Bump, Friedberg, and Hoffstein; and Murty and Murty):

a. An elliptic curve E defined over Q has Mordell-Weil rank equal to the order of vanishing at s = 1 of its L-function, L(E, s), provided that order of vanishing is ≤ 1. In such a case, the Shafarevich-Tate group of E over Q is finite.

References: [1], [2], [3], [5], [9].

4. Kato’s Euler System.

Applications Let K = Q.

a. An elliptic curve E defined over Q has Mordell-Weil rank equal to the order of vanishing at s = 1 of its L-function, L(E, s), provided that order of vanishing is ≤ 1. In such a case, the Shafarevich-Tate group of E over Q is finite.

b. Analogous result to the above for the χ-parts of the Mordell-Weil group and the Shafarevich-Tate group of E over F provided that the order of vanishing at s = 1 of the L-function, L(E, χ, s) is ≤ 1.

References: [9], [10].

5. Flach’s System.

I have some notes to a course I gave on this topic about five years ago and if you wish to have a copy I can get one to you. For the nature of this Euler System and its applications, I refer to Tom Weston’s lectures, which begin today.

Bibliography.

[1]. Gross, B.: Kolyvagin’s work on modular elliptic curves. Pages 235-256 in: L-functions and arithmetic Durham (1989) London Math. Soc. Lecture Notes 153 Cambridge Univ. Press (1991)

[2]. Kolyvagin, V.A.: Euler Systems, pp. 435-483 in The Grothendieck Festschrift, vol. II Birkhaüser (1991)

[3]. Mazur, B.: On the passage from local to global in number theory (specifically: pp. 28-33 in this) Bull. A.M.S. 29 (1993)

[4]. Rubin, K.: The main conjectures of Iwasawa Theory for imaginary quadratic fields, Inv. Math. 75 (1984) 409-423

[5]. Rubin, K.: The work of Kolyvagin on the arithmetic of elliptic curves, pp. 128-136 in Arithmetic of Complex Manifolds, LNM 1399 Springer-Verlag (1989)

[6]. Rubin, K.: The Main Conjecture. (published as an appendix: pp.397-420 in Serge Lang’s “Cyclotomic Fields I and II” (Combined Second Edition) Springer-Verlag (1990)

[7]. Rubin, K.: Kolyvagin’s system of Gauss sums, Arithmetic Algebraic Geometry (van der Geer, Oort and Steenbrink, eds.) Progress in Math 89 Birkhaüser (1991)

[8]. Rubin, K.: Euler systems and exact formulas in number theory, Jber.d Dt. Math.-Verein 98 (1996) 30-39.

[9]. Rubin, K.: “Euler systems” to appear (Princeton Univ. Press) and can be gotten from the WEB

[10]. Scholl, A.: An introduction to Kato’s Euler System. In the Proceedings of the 1996 Durham Conference on modular forms and Galois representations.