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 K to be 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)^{∗}).

Now, 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. Thought of in terms of line bundles, this simply maps a line bundle to its degree. 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 April 8, 1998

Now, 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. Now, 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 let Pic^{m}(X) 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, 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 complete local domain which is a 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}; a result of Dickinson says that every A is a subring of such a ring. 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}.

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 A-generator 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 kenrel of multiplication by α on W. Note that W [J] and W^{∗}/JW^{∗} are Cartier dual.

Lemma 3.5. W is A-divisible.

Proof. The easiest way to prove this is to use the fact that μ_{p^{∞}} is an injective ℤ_{p}-module; we give a more explicit proof.

It is clear that it is sufficient to prove the lemma in the case that W^{∗} has rank 1. Also, since the G_{F,Σ}-action is irrelevant here, we can replace μ_{p^{∞}} by ℚ_{p}/ℤ_{p}. Thus we are reduced to showing that

Hom_{ℤ_{p}}(A, ℚ_{p}/ℤ_{p})

is A-divisible.

So let f : A → ℚ_{p}/ℤ_{p} be a homomorphism and fix a non-zero α ∈ A. We wish to define a g : A → ℚ_{p}/ℤ_{p} such that

g(αx) = f(x)

for all x ∈ A. In other words, we wish to extend to all of A the map

g_{0} : αA → ℚ_{p}/ℤ_{p}

given by g_{0}(αx) = f(x). (Note that g_{0} is well-defined since multiplication by α is injective.)

Now, A is finite and flat over ℤ_{p} and thus free of some finite rank m. Note that αA is isomorphic to A as an A-module, and thus also free of rank m over ℤ_{p}. By standard results of algebra we can choose a ℤ_{p}-basis e_{1}, …, e_{m} of A such that p^{n_{1}}e_{1}, …, p^{n_{m}}e_{m} is a ℤ_{p}-basis of αA for some integers n_{1}, …, n_{m}. It is now clear how to extend g_{0} to all of A: we have p^{n_{i}}e_{i} = αa_{i} for some a_{i} ∈ A, and we simply define g(e_{i}) to be any choice of

p^{−n_{i}}f(a_{i}).

This exists since ℚ_{p}/ℤ_{p} is p-divisible. This completes the proof.

Lemma 3.6. For any non-zero w ∈ W, there is an α ∈ A such that αw ≠ 0 and αw ∈ W [𝔪].

Proof. Since W is torsion there is some β ∈ A such that βw = 0. Let a_{1}, …, a_{k} be a set of ideal generators for 𝔪. We first construct an α_{1} ∈ A such that α_{1}w ≠ 0 and α_{1}w ∈ W [a_{1}]. If w ∈ W [a_{1}] then we take α_{1} = 1. If not, then note that a_{1} is nilpotent in the artinian ring A/βA. Thus there is some power of a_{1} which annihilates w. We take α_{1} to be the largest power of a_{1} which does not annihilate w. From here we continue the procedure, replacing w by α_{1}w, and so on.

Corollary 3.7. If W [𝔪]^{G} = 0, then W^{G} = 0.

Proof. This follows immediately from Lemma 3.6 and the A-linearity of the G-action on W.