CHAPTER 13 Lecture 13

1. The Picard Group of a Curve

1.1. The Picard Group. Let X be a smooth, projective, geometrically irreducible curve defined over a field K. For simplicity assume that K is perfect. The geometric irreducibility condition is equivalent to the function field K(X) being linearly disjoint from K̄ over K. In particular, we can identify G = Gal(K̄(X)/K(X)) with G_{K}.

We recall the definition of the Picard group of X, Pic(X). See [Ha, Chapter 2, Section 6] for details. The K̄ points of Pic(X) are easiest to describe: they are in natural bijective correspondence with isomorphism classes of line bundles on \overline{X} = X ×_{Spec K} Spec K̄.

There is another interpretation of Pic(X)(K̄) which is somewhat simpler. Recall that a divisor on X is a formal sum of points of X(K̄); that is, it is an element of the free abelian group ℤ[X(K̄)]. Let K̄(X)^{∗} be the function field of X over K̄. There is a natural map

K̄(X)^{∗} ⟶^{div} ℤ[X(K̄)]

given by

f ↦ ∑_{x∈X(K̄)} ord_{x}(f)x,

where ord_{x}(f) means the order of the zero or pole of f at x. Then

Pic(X)(K̄) ≅ ℤ[X(K̄)]/ div(K̄(X)^{∗}).

It can be shown that div(K̄(X)^{∗}) lies in the kernel of the natural summation map

ℤ[X(K̄)] → ℤ,

so it descends to a map

Pic(X)(K̄) ⟶^{deg} ℤ

called the degree map. This agrees with the usual notion of degree for line bundles. We define

Pic^{0}(X)(K̄)

to be the kernel of the degree map.

It can be shown that Pic^{0}(X) can be given the structure of an abelian variety over K of dimension equal to the genus of X. Often Pic^{0}(X) is called the Jacobian of X. See [Mi-JV].

0Last modified December 17, 1999

If a rational function on \overline{X} has no zeros or pole then it is a constant. Thus there is an exact sequence

0 → K̄^{∗} → K̄(X)^{∗} ⟶^{div} ℤ[X(K̄)] → Pic(X)(K̄) → 0.

We rewrite this as a short exact sequence

0 → K̄(X)^{∗}/K̄^{∗} → ℤ[X(K̄)] → Pic(X)(K̄) → 0;

it is actually an exact sequence of G_{K}-modules (using our identification of G with G_{K}), as is easy to see.

Proposition 1.1. There is an exact sequence

ℤ[X(K̄)]^{G_{K}} → Pic(X)(K) → ker(Br_{K} → H^{2}(G, K̄(X)^{∗})).

(Here Pic(X)(F) is just the F -valued “points” ℤ × Pic^{0}(X)(F) of Pic(X). Note that ℤ[X(K̄)]^{G_{K}} need not equal ℤ[X(K)].)

Proof. Taking the G_{K} = G cohomology of

0 → K̄(X)^{∗}/K̄^{∗} → ℤ[X(K̄)] → Pic(X)(K̄) → 0

yields an exact sequence

ℤ[X(K̄)]^{G_{K}} → Pic(X)(K) → H^{1}(G, K̄(X)^{∗}/K̄^{∗}).

Thus it will be enough to identify H^{1}(G, K̄(X)^{∗}/K̄^{∗}) with the kernel of

H^{2}(G_{K}, K̄^{∗}) → H^{2}(G, K̄(X)^{∗}).

To do this we take G_{K} cohomology of

0 → K̄^{∗} → K̄(X)^{∗} → K̄(X)^{∗}/K̄^{∗} → 0,

yielding

H^{1}(G, K̄(X)^{∗}) → H^{1}(G, K̄(X)^{∗}/K̄^{∗}) → H^{2}(G_{K}, K̄^{∗}) → H^{2}(G, K̄(X)^{∗}).

By Hilbert’s theorem 90 the first term vanishes, which gives the desired identification.

We now restrict to the case where K = F is a number field and relate the above result to the Shafarevich-Tate group.

Proposition 1.2. Along with our running hypotheses, suppose further that X has an F_{v} rational point for all place v of F. Then the natural map

Br_{F} → Br_{F̄(X)}

is injective.

Proof. By global class field theory we have that Br_{F} injects into ⊕_{v} Br_{F_{v}}. The commutative diagram

\xymatrix{Br_{F} \ar[r] \ar@{^{(}->}[d] & Br_{F (X)} \ar[d] \\ _{⨁_{v} Br_{F_{v}}} \ar[r] & _{⨁_{v} Br_{F_{v}(X)}}}

now shows that it will suffice to prove that each map Br_{F_{v}} → Br_{F_{v}(X)} is injective.

For this we will need to use some of the theory of Brauer groups of schemes developed by Asumaya and Grothendieck. We have a diagram

\xymatrix{Spec F_{v}(X) \ar[r] \ar[d] & Spec X_{v} \ar[dl] \\ Spec F_{v}}

where X_{v} = X ×_{Spec F} Spec F_{v} and Spec F_{v}(X) is the generic point of X_{v}. This translates to a diagram

\xymatrix{Br(F_{v}(X)) & Br(X_{v}) \ar[l] \\ Br(F_{v}) \ar[u] \ar[ur]}

of Brauer groups. Since X_{v} is smooth, a theorem of Grothendieck (see [Mi-EC, Chapter 4]) implies that the map

Br(X_{v}) → Br(F_{v}(X))

is injective. Thus it will suffice to show that

Br(F_{v}) → Br(X_{v})

is injective. For this, we use the fact that X_{v} has an F_{v}-rational point. This implies that there is a section to the natural map

X_{v} → Spec F_{v},

and injectivity of Br(F_{v}) → Br(X_{v}) follows immediately.

Corollary 1.3. Let X be a smooth, projective, geometrically irreducible curve over a number field F. Suppose further that X has an F_{v}-rational point for all places v of X. Then any F -rational divisor class is represented by an F -rational divisor; that is, the natural map

ℤ[X(F̄)]^{G_{F}} → Pic(X)(F)

is surjective.

Proof. By Proposition 1.1 it will suffice to prove that the map

Br_{F} → H^{2}(Gal(F̄(X)/F (X)), F̄(X)^{∗})

is injective. By Hilbert’s theorem 90 and the inflation-restriction sequence, this second group injects into Br_{F̄(X)}, so that the map Br_{F} → Br_{F̄(X)} factors through the map we are interested in. Proposition 1.2 thus completes the proof.

1.2. Applications to Shafarevich-Tate Groups. Let E be an elliptic curve defined over a number field F. Let (X, α) be an AG torsor for E and choose a point P_{0} ∈ X(F̄). Then the map

ℤ[X(F̄)] → E(F̄)

defined by sending a point P ∈ X(F̄) to the unique Q ∈ E(F̄) for which

α(Q, P_{0}) = P

descends to an isomorphism of varieties over K

Pic^{0}(X) ≅ E.

(See [Si-AEC, Chapter 10, Theorem 3.8].)

More generally, let us define Pic^{m}(X) to be the inverse image of m ∈ ℤ under the degree map Pic(X) → ℤ. Each of these become isomorphic to Pic^{0}(X) over F̄. In fact, there is a natural action of Pic^{0}(X) on Pic^{m}(X), by addition, which is easily seen to be a G_{F} -equivariant principal homogeneous action. Thus each Pic^{m}(X) is an AG torsor for E. In fact, the natural map

X → Pic^{1}(X)

sending a point of X to its divisor class is an isomorphism of AG torsors for E. We will write X_{m} for Pic^{m}(X).

Now further suppose that X is everywhere locally trivial. Then each X_{m} is as well, simply by “multiplying” the local point on X by m. Thus each X_{m} yields an element ξ_{m} ∈ � (E/F). We claim that ξ_{m} is just mξ_{1}. To prove this one must identify the Baer sum of X_{m} and X_{m′} with X_{m+m′}. Since for once we actually have a candidate variety for X_{m} ⊕ X_{m′} it is possible to show that X_{m} ⊕ X_{m′} ≅ X_{m+m′} by showing that X_{m+m′} satisfies the appropriate universal property; we omit the details. In any event, this fact immediately implies our claim.

We know that � (E/F) is a torsion group, so there is some n > 0 such that nξ_{1} = 0. Thus ξ_{n} = 0, so X_{n} ≅ E as E-torsors. In particular, X_{n} has an F -rational point. Thus X has an F -rational divisor class of degree n. By Corollary 1.3 X has an F -rational divisor. Choose one such and call it D.

From here we will begin to omit some more details. Suppose that n ≥ 3. (The n = 2 case is quite interesting but doesn’t work for our present arguments.) Let O_{X}(D) be the invertible O_{X}-module of rational functions with poles bounded by D. Then by Riemann-Roch H^{0}(X, O_{X}(D)) has dimension n over F, since X has genus 1. Thus we get a map

X → ℙ_{F}^{n−1},

where the ℙ_{F}^{n−1} is the space of linear functionals on H^{0}(X, O_{X}(D)), given by sending x ∈ X to the “evaluation at x” functional. (The range is projective space over F since D is defined over F.) In fact, one can show that this is actually a closed immersion with projectively normal image. In particular, we can realize X as a curve in ℙ_{F}^{n−1} of genus 1 and degree n.

Now let us consider all elements of order n in all Shafarevich-Tate groups of all elliptic curves over F. Our above construction tells us that every such cohomology class is represented by a smooth, projectively normal curve of genus 1 and degree n in ℙ_{F}^{n−1}. There happens to exist a quasi-projective variety C_{n}, called the Chow variety, defined over ℚ, whose F -valued points classify all projectively normal curves in ℙ_{F}^{n−1} of degree n and genus 1. In fact, there are “Chow coordinates” giving an explicit embedding of C_{n} in some large projective space. Thus each element of � (E/F) for some E/F yields a point in C_{n} and thus a point in this large projective space. If we could somehow bound the heights of such points in terms of the coefficients of E (as from the example of Lecture 12, Theorem 1.4 may at least seem a reasonable hope), the fact that there are only a finite number of F -rational points of projective space with bounded height would show that � (E/F) was finite for all elliptic curves E defined over F.

2. Direct Limits of Selmer Groups

2.1. Passage to the Limit. Let B be an abelian variety over a field F and let N be a positive integer. (The reason for the sudden shift in notation will become apparent in the next section.) We consider the N-torsion B[N] as a G_{F} -module with its abelian variety finite/singular structure induced by B. (See Lecture 9, Section 2.2.) Thus we have defined (finite) Selmer groups

H_{f}^{1}(G_{F} , B[N]) = S_{N}(B/F).

Now, choose another positive integer M. Then the inclusion B[N] ↪ B[NM] induces a map

H_{f}^{1}(G_{F} , B[N]) → H_{f}^{1}(G_{F} , B[NM]).

(To check that this map exists one just needs to check that the local maps

H^{1}(G_{v}, B[N]) → H^{1}(G_{v}, B[NM])

send finite parts to finite parts for all v; this follows easily from the definition of the abelian variety structure.) Thus we can form the direct limit (over the multiplicative system of positive integers)

H_{f}^{1}(G_{F} , B_{tors}) =^{def} \underrightarrow{lim}_{N} H_{f}^{1}(G_{F} , B[N]).

In other notation, this is

S_{∞}(B/F) =^{def} \underrightarrow{lim}_{N} S_{N}(B/F).

Recall that there is an exact sequence

0 → B(F)/NB(F) → S_{N}(B/F) → � (B/F)[N] → 0.

Writing B(F)/NB(F) as B(F) ⊗ \frac{1}{N} ℤ/ℤ we have a natural commutative diagram

\xymatrix{0 \ar[r] & B(F) ⊗ \frac{1}{N} ℤ/ℤ \ar[r] \ar[d] & S_{N}(B/F) \ar[r] \ar[d] & � (B/F)[N] \ar[r] \ar[d] & 0 \\ 0 \ar[r] & B(F) ⊗ \frac{1}{NM} ℤ/ℤ \ar[r] & S_{NM}(B/F) \ar[r] & � (B/F)[NM] \ar[r] & 0}

Taking the direct limit yields the following proposition.

Proposition 2.1. There is an exact sequence

0 → B(F) ⊗ ℚ/ℤ → S_{∞}(B/F) → � (B/F) → 0.

This construction yields a covariant function in B/F.

2.2. The Shafarevich-Tate Conjecture. We now state the Shafarevich-Tate conjecture, which we have eluded to many times before.

Conjecture 2.2 (Shafarevich-Tate). � (B/F) is finite for any abelian variety B defined over a number field F.

Recall that we already know that � (B/F) is torsion (since it is a subgroup of the torsion group H^{1}(G_{F} , B(F̄))). Further, we know that � (B/F)[N] is finite for all N, since S_{N}(B/F) is. We can break the Shafarevich-Tate conjecture up into two somewhat distinct parts.

Conjecture 2.3 (Shafarevich-Tate).

1. The p-primary component � (B/F)_{p} of � (B/F) has trivial p-divisible part. (We already know that it has finite corank since � (B/F)[p] is finite.)

2. � (B/F)[p] = 0 for almost all p.

Let E be an elliptic curve over ℚ. There are two main approaches to the Shafarevich-Tate conjecture. The first, which one might call the classical Fermat descent method, is to fix a prime p and to try to compute � (E/ℚ)_{p}. This sometimes works for small p, but can yield neither (1) nor (2) of Conjecture 2.3. The second method, due to Kolyvagin and Rubin, instead is quite good at showing that � (E/ℚ)_{p} is trivial for almost all p simultaneously.

We now describe a contravariant limit of the Selmer groups, which works well under the assumption that � (B/F) is finite. We use the injection

ℚ/ℤ → ℝ/ℤ

to get a map

B(F) ⊗ ℚ/ℤ → B(F) ⊗ ℝ/ℤ.

Next we push out our exact sequence

0 → B(F) ⊗ ℚ/ℤ → S_{∞}(B/F) → � (B/F) → 0;

that is, we define a group S^{†}(B/F) to be the unique group making

[figure: 0; B(F) ⊗ ℚ/ℤ; S_{∞}(B/F); � (B/F); 0; 0; B(F) ⊗ ℝ/ℤ; S^{†}(B/F); � (B/F); 0]

a commutative diagram with exact rows. Since B(F) is finitely generated, B(F) ⊗ ℝ/ℤ is a compact topological group. Since � (B/F) is assumed to be finite, S^{†}(B/F) has a natural structure of compact topological group. Thus its dual

Σ(B/F) = Hom(S^{†}(B/F), ℝ/ℤ)

is a finitely generated abelian group, and Σ is contravariant in B/F.

3. Conditions on Galois Representations

3.1. Conditions. In this section we will specify the sorts of Galois representations which we will be considering. Fix a prime p > 2 and let A be a local, finite, faithfully flat ℤ_{p}-algebra. (Thus A is free of finite rank over ℤ_{p}.) For example, A could be the ring of integers of a finite extension of ℚ_{p}. We will write 𝔪 for the maximal ideal of A and k for the residue field A/𝔪.

Let F be a number field with our usual conventions and let Σ be a finite set of places of F. We will always assume that Σ contains p and all archimedean places. Let F̄^{Σ} be the maximal extension of F in the fixed algebraic closure F̄ which is unramified outside of Σ and set G_{F,Σ} = Gal(F̄^{Σ}/F).

Let H be a free A-module of rank 2 with an A-linear action of G_{F,Σ}; thus H is an A[G_{F}]-module which is unramified outside of Σ. We will write ρ for the representation

ρ : G_{F,Σ} → Aut_{A}(H) ≅ GL_{2}(A)

and \overline{ρ} for the residual representation

\overline{ρ} : G_{F,Σ} → Aut_{k}(H ⊗_{A} k) ≅ GL_{2}(k).

We are usually only interested in ρ up to conjugation in GL_{2}(A), so that the choice of basis is not important.

We wish to specify the determinant of ρ. Recall that the p-cyclotomic character χ is the map

χ : Gal(F (ζ_{p^{∞}})/F) → ℤ_{p}^{∗}

giving the action of the Galois group on the group of p-power roots of unity. We consider χ as a map to A^{∗} by composing with the structure map ℤ_{p}^{∗} → A^{∗}. We require that the determinant

det ρ : G_{F,Σ} → A^{∗}

of ρ equal χ.

This requirement has some interesting consequences. First, let λ be a place of F not in Σ. Then we can choose a Frobenius element Fr_{λ} for λ in G_{F,Σ}. We will call

T_{λ} = tr ρ(Fr_{λ}) ∈ A

the λ^{th} Hecke operator for reasons that will become clear later. −T_{λ} is the coefficent of x in the characteristic polynomial

x^{2} − T_{λ}x + det ρ(Fr_{λ})

of Fr_{λ}. By the p-cyclotomic determinant condition

det ρ(Fr_{λ}) = χ(Fr_{λ}) = N_{F/ℚ}(λ),

since Fr_{λ} is just N_{F/ℚ}(λ) powers on residue fields and thus on roots of unity. Thus we have an Eichler-Shimura type relation

Fr_{λ}^{2} −T_{λ} Fr_{λ} +N_{F/ℚ}(λ) = 0.

Next, assume that F has at least one real embedding. Then let c be any choice of complex conjugation for this embedding; equivalently c is the non-trivial element of a decomposition group of a real valuation. Since c has order 2, \overline{ρ}(c) has order 2 in GL_{2}(k). However, χ(c) = −1 since complex conjugation acts as inversion on roots of unity, so det \overline{ρ}(c) = −1. It now follows from an easy computation that \overline{ρ}(c) is not a scalar matrix, so \overline{ρ} is called an odd representation.

3.2. An Example : Tate Modules of Elliptic Curves. Let E be an elliptic curve over F. Let Σ consist of p, the archimedean places and those places at which E/F has bad reduction. Take A = ℤ_{p} and set

H = \underleftarrow{lim}_{ν} E[p^{ν}],

the maps of the inverse system being

E[p^{ν+1}] ⟶^{p} E[p^{ν}].

Then by [Si-AEC, Chapter 3, Proposition 7.1] H is free of rank 2 over ℤ_{p}. H has a natural G_{F} action which by [Si-AEC, Chapter 7, Proposition 4.1] is unramified outside of Σ. Further the Weil pairing shows that the determinant

G_{F,Σ} → ∧_{A}^{2}H

is just the p-cyclotomic character χ. (See [Si-AEC, Chapter 3, Proposition 8.3].) So H satisfies all of our conditions. The Hecke operators T_{λ} for λ ∉ Σ are computed in [Si-AEC, Chapter 5, Proposition 2.3]; they are given by

T_{λ} = 1 + N_{F/ℚ}(λ) − |E(k_{λ})| ∈ ℤ ⊆ ℤ_{p},

where k_{λ} is the residue field of F_{λ}. (Note that this is independent of the choice of p.)

3.3. Principal Polarizations. We continue to let A and H be as before. In this section we consider pairings

(·, ·) : H ⊗_{ℤ_{p}} H → ℤ_{p}(1).

We say that (·, ·) is G_{F,Σ}-equivariant if

(gm, gn) = g(m, n)

for all m, n ∈ H and g ∈ G_{F,Σ}. We say that (·, ·) is A-Hermetian if

(am, n) = (m, an)

for all m, n ∈ H and a ∈ A. Finally, we say that (·, ·) is skew-symmetric if

(m, n) = −(n, m)

for all m, n ∈ H.

Lemma 3.1. Let

(·, ·)_{ψ} : H ⊗_{ℤ_{p}} H → ℤ_{p}(1)

be a G_{F,Σ}-equivariant, A-Hermetian, skew-symmetric self-pairing on H. Let

ψ : H → Hom_{ℤ_{p}}(H, ℤ_{p}(1))

be the induced map, given by sending m ∈ H to the map sending n ∈ H to (m, n)_{ψ}. Then if ψ is an isomorphism of ℤ_{p}-modules, it is also an isomorphism of A[G_{F,Σ}]- modules.

Proof. Note that ℤ_{p} acts on the ℤ_{p}(1) (or equivalently on the H) in Hom_{ℤ_{p}}(H, ℤ_{p}(1)), while A acts only on the H and G_{F,Σ} acts on both factors in the usual adjoint way. The lemma now follows immediately from some easy computations.

Lemma 3.2. An A-Hermetian, skew-symmetric pairing

(·, ·) : H ⊗_{ℤ_{p}} H → ℤ_{p}(1)

is automatically G_{F,Σ}-equivariant.

Proof. A pairing as above is by definition the same as a ℤ_{p}-homomorphism

∧_{A}^{2}H → ℤ_{p}(1),

where ∧_{A}^{2}H is the second exterior power of H. Furthermore, it follows immediately from the definitions that G_{F,Σ}-equivariance of the pairing is the same as G_{F,Σ}- equivariance of the map on the exterior power. But here G_{F,Σ}-equivariance is automatic: by the p-cyclotomic determinant condition, G_{F,Σ} acts on ∧_{A}^{2}H via the cyclotomic character, which is exactly how G_{F,Σ} acts on ℤ_{p}(1). ℤ_{p}-linearity now completes the proof.

Lemma 3.3 (Schur’s Lemma). Let Π be a profinite group and ρ : Π → GL_{n}(R) a continuous homomorphism, where R is a complete local ring with residue field k. Suppose that the residual representation \overline{ρ} is absolutely irreducible. Then the only matrices which commute with the image of ρ are the scalar matrices.

Proposition 3.4. Let

(·, ·)_{ψ} : H ⊗_{ℤ_{p}} H → ℤ_{p}(1)

be a G_{F,Σ}-equivariant, A-hermetian, skew-symmetric pairing on H. Let

ψ_{0} : ∧_{A}^{2}H → ℤ_{p}(1)

be the induced map on the exterior square. Then (·, ·)_{ψ} is perfect if and only if Hom_{ℤ_{p}}(∧_{A}^{2}H, ℤ_{p}(1)) is free of rank 1, generated by ψ_{0}.

Proof. First suppose that (·, ·)_{ψ} is perfect. Let

ψ : H ⟶^{≅} Hom_{ℤ_{p}}(H, ℤ_{p}(1))

be the map induced by (·, ·)_{ψ} and let

f : H → Hom_{ℤ_{p}}(H, ℤ_{p}(1))

be any other G_{F,Σ}-equivariant homomorphism induced from some map

f_{0} : ∧_{A}^{2}H → ℤ_{p}(1).

Then

ψ^{−1} ∘ f : H → H

is a G_{F,Σ}-equivariant endomorphism of H. In other words, the matrix of ψ^{−1} ∘ f commutes with the image of G_{F,Σ} in Aut_{A}(H). Thus, by Lemma 3.3, ψ^{−1} ∘ f is a scalar. So f = aψ for some a ∈ A, so f_{0} = aψ_{0}. Thus ψ_{0} is an Agenerator of Hom_{ℤ_{p}}(∧_{A}^{2}H, ℤ_{p}(1)). (The above argument in fact shows that every G_{F,Σ}-equivariant, A-Hermetian self-pairing from H to ℤ_{p}(1) is skew-symmetric.)

It remains to show that Hom_{ℤ_{p}}(∧_{A}^{2}H, ℤ_{p}(1)) is free of rank 1 over A. This, however, is clear from the fact that ∧_{A}^{2}H is free of rank 1 over A, so that Hom_{ℤ_{p}}(∧_{A}^{2}H, ℤ_{p}(1)) has the same rank over ℤ_{p} as A does.

For the converse, we suppose that ψ_{0} is an A-generator of the free rank 1 Amodule Hom_{ℤ_{p}}(∧_{A}^{2}H, ℤ_{p}(1)). The G_{F,Σ}-action plays no role here, so we choose an A-basis x, y of H and a ℤ_{p}-generator ζ of ℤ_{p}(1). Now

∧_{A}^{2}H = (x ∧ y)A

and

ℤ_{p}(1) = ζℤ_{p}.

ψ_{0} : (x ∧ y)A → ζℤ_{p} can thus be simply written as

(x ∧ y)A ↦ ζ tr(a),

where

tr : A → ℤ_{p}

is a generator of the free rank 1 A-module Hom(A, ℤ_{p}). (tr is called a Gorenstein trace.)

We now recover (·, ·)_{ψ} by (x, x)_{ψ} = 0, (y, y)_{ψ} = 0 and (x, y)_{ψ} = ζ tr(1). The map

ψ : xA ⊕ yA → Hom_{ℤ_{p}}(yA, ζℤ_{p}) ⊕ Hom_{ℤ_{p}}(xA, ζℤ_{p})

can now be computed as

(xa, yb) ↦ (a · tr, b · tr),

and this is an isomorphism since tr is an A-basis for Hom_{ℤ_{p}}(A, ℤ_{p}). This completes the proof.

Definition 7. A G_{F,Σ}-equivariant, A-Hermetian, skew-symmetric perfect pairing

H ⊗_{ℤ_{p}} H → ℤ_{p}(1)

will be called a principal polarization or simply a polarization.

In fact, it follows from Proposition 3.4 that a principal polarization exists if and only if A is Gorenstein. See [Ti, Section 1].

3.4. Moment Representations. We let A and H be as before. For any ν > 0, Sym_{A}^{ν} H is a free A-module of rank ν +1; however, in order to get the theory to work properly we need to twist the symmetric powers. Precisely, if ν = 2m + 1 is odd we define

W^{ν} = Sym_{A}^{ν} H ⊗_{ℤ_{p}} ℤ_{p}(−m)

and if ν = 2m + 2 is even we define

W^{ν} = Sym_{A}^{ν} H ⊗_{ℤ_{p}} ℤ_{p}(−m).

(Note that W^{1} and W^{2} are still just Sym_{A}^{1} H and Sym_{A}^{2} H respectively.) So W^{ν} is free of rank ν + 1 over A.

We will write

W_{ν} = (W^{ν})^{∗} = Hom(W^{ν}, μ_{p^{∞}}) = Hom(Sym_{A}^{ν} H, μ_{p^{∞}}(m))

for the Cartier dual of W^{ν}. We will also write ρ_{ν} for the representation

G_{F,Σ} → Aut_{A/J}(W_{ν})

induced by H and ρ_{ν}^{J} for the restriction of ρ^{ν} to W_{ν}[J].

If H has a principal polarization, it is not difficult to show that

W_{2} ≅ End_{A}^{0}(H) ⊗_{ℤ_{p}} ℚ_{p}/ℤ_{p},

where End_{A}^{0}(H) is the module of A-linear endomorphisms of H of trace 0. For relations of this fact to the deformation theory of Galois representations see [Maz, Chapters 4 and 5].

3.5. Properties of Representations. In this section we summarize some properties of representations which will be useful to us. Let A be a ring as before. Let W^{∗} be a free A-module of finite rank n with an A-linear action of G = G_{F,Σ}. Let W = Hom(W^{∗}, μ_{p^{∞}}) be its Cartier dual. W is made into an A-module by the action of A on W^{∗}. For any ideal J of A of finite index we define

W [J] = ∩_{α∈J}W [α],

where W [α] is the kernel of multiplication by α on W. Note that W [J] and W^{∗}/JW^{∗} are Cartier dual.

Lemma 3.5. Let α ∈ A be a non-zero divisor. Then α divides some power of l in A and αA has finite index in A.

Proof. Since A is finite over ℤ_{l} and α is a non-zero divisor, α necessarily satisfies some monic linear equation of the form

α^{n} + a_{n−1}α^{n−1} + ⋯ + a_{1}α + a_{0} = 0

with a_{i} ∈ ℤ_{l} and a_{0} ≠ 0. Thus α divides a_{0}, which proves the first statement. Since a_{0}A ⊆ αA and a_{0}A visibly has finite index in A (as A is free of finite rank over ℤ_{l}), the second statement is now clear as well.

Lemma 3.6. Let W be an A-module and let t be a non-zero element of W annihilated by some power of l. Then there exists α ∈ A such that αt ≠ 0 and αt ∈ W [𝔪].

Proof. Since t is l-power torsion, there is a largest power l^{n} of l such that t_{0} = l^{n}t ≠ 0; thus t_{0} ∈ W [l]. Note that W [l] is a module over the artinian ring A/lA. Let α_{1}, …, α_{m} be generators of 𝔪 in A; they also generate the maximal ideal of A/lA and are therefore nilpotent in this ring. It follows that some power of α_{1} annihilates t_{0}; let n_{1} be the smallest such power, and set t_{1} = α_{1}^{n_{1}−1}t_{0}. Continuing in this way, we obtain a non-zero element t_{m} = αt where α = l^{n}α_{1}^{n_{1}−1} ⋯ α_{m}^{n_{m}−1}. This t_{m} is killed by every generator of 𝔪, so it is clearly in W [𝔪].

Lemma 3.7. Let W be a ℤ_{l}-torsion A-module. If W [𝔪] = 0, then W = 0.

Proof. Suppose that W ≠ 0 and let t be a non-zero element of W. By Lemma 3.6 there is some α ∈ A such that αt is non-zero and annihilated by 𝔪. Thus W [𝔪] ≠ 0, which yields the desired contradiction.