INTRODUCTION TO EULER SYSTEMS AND BOUNDS ON TATE-SHAFAREVICH GROUPS

AMNON BESSER

The Selmer and Tate-Šafarevič groups of a Galois representation

K-: A number field

T -: A free ℤ_{p} module of finite rank with a continuous Gal(ℚ̄/ℚ) action, ramified only at the finite number of primes dividing a number N. action.

Let V = T ⊗ ℚ, A defined by the sequence

0 ⟶ T ⟶^{i} V ⟶^{pr} A ⟶ 0.

Assumption: There is a perfect pairing T × A → ℚ/ℤ(1).

Let v be a place of K. The following definitions are due to Bloch and Kato

H_{f}^{1}(K_{v}, V) := Ker H^{1}(K_{v}, V) ⟶^{res} H^{1}(K_{v}^{ur}, V);
H_{f}^{1}(K_{v}, T) := i^{−1}H_{f}^{1}(K_{v}, V);
H_{f}^{1}(K_{v}, A) := Im H_{f}^{1}(K_{v}, V) ↪ H^{1}(K_{v}, V)

⟶^{pr} H^{1}(K_{v}, A),

This is when v ∤ p. Otherwise there is another definition, using the theory of p-adic periods.

\begin{matrix}Sel_{p}(T/K) := Ker H^{1}(K, A) ⟶ \\ ∏_{v} H^{1}(K_{v}, A)/H_{f}^{1}(K_{v}, A), \end{matrix}

\begin{matrix}H_{f}^{1}(K, V) := Ker H^{1}(K, V) ⟶ \\ ∏_{v} H^{1}(K_{v}, V)/H_{f}^{1}(K_{v}, V). \end{matrix}

�_{p}(T/K) = Sel_{p}(T/K)/ Im H_{f}^{1}(K, V)

Examples:

Date: Dec 4, 1996.

2 AMNON BESSER

1. T = ℤ_{p}(1). �_{p}(T/K) is the p-part of the class group of K and is therefore finite.

2. T = T_{p}(E) is the Tate module of a modular elliptic curve E. �_{p}(T/K) is the p part of

�(E/K) := Ker H^{1}(K, E) ⟶ ∏_{v} H^{1}(K_{v}, E)

K is a quadratic imaginary field, some restrictions. Kolyvagin: If ord_{s=1}L(E, s) ≤ 1 then �_{p}(T/K) is finite and 0 for almost every p.

3. Flach - T = Symm^{2}(T_{p}(E))(−1) Then Sel_{p}(T/ℚ) is finite and 0 for almost all p. In all cases certain bounds are known.

4. Results of Nekovář (Inventioness 92): A generalization of 2. We again consider a quadratic imaginary field K. Let f be a new form of weight 2r for Γ_{0}(N). T_{p}(f) is the representation associated to f by Deligne and T = T_{p}(f)(r). One constructs

P (1) ∈ H^{1}(K, T).

Theorem of Nekovář - under certain conditions, if P (1) is not torsion, then �_{p}(T) is finite. Improvement (B) - For almost all primes p, if the reduction of P (1) mod p is not 0, then �_{p}(T) = 0. Nekovář also gives a p-adic analogue of the Gross-Zagier formula that relates the non-torsionness of P (1) to derivatives of a p-adic L-function.

5. Logachev has a general program for considering Euler systems associated with Shimura varieties (still preliminary).

6. Work of Kato - Euler systems comming from K-theory. General program - Method of Euler systems Step 1 - For many positive integers n, usually restricted by some congruences, consider fields K_{n}. K_{1} will be the Hilbert class field of K. K_{n} should be ramified over K_{1} exactly at primes which divide n, and then it is totally ramified. one constructs classes y_{n} ∈ H^{1}(K_{n}, T). This is where we need some input from geometry. Examples: For ℤ_{p}(1) one takes K_{n} = ℚ(μ_{n}) For an elliptic curve E over an imaginary quadratic K one takes K_{n} to be the ring class field of conductor n. y_{n} is the image of a Heenger point. The following relations should be satisfied by the classes y_{n}:

1. The y_{n} are in H_{f} at all primes of good reduction (this is normally no problem away from p)

2. The norm relation: Suppose n = mℓ, ℓ is a prime and ℓ ∤ m. Then we would like a relation of the form

cor_{K_{n},K_{m}}(y_{n}) = T_{ℓ}y_{m}.

Here, T_{ℓ} is some operator on H^{1}(K_{m}, T). In the elliptic case it is actually the ℓ’th Hecke operator. In general, we would like this operator to not be invertible, or more precisely to be divisible by an arbitrary power of p for a suitably chosen ℓ.

3. The congruence relation: see later.

Step 2 - The Kolyvagin derivative

We “inject the ramification of K_{n}/K_{1} into the y_{n}”.

General setting - G group, H ◁ G, finite index. M, A are G-modules with res_{H}(A) trivial. consider m ∈ M^{H} s.t. tr_{G/H} m = 0 and f a 1-cocycle of G/H with values in A.

Lemma:

∑_{σ∈G/H} σ(m) ⊗ f(σ) ∈ (M ⊗ A)^{G}

Proof:

τ ∑_{σ∈G/H} σ(m) ⊗ f(σ) = ∑_{σ∈G/H}(τσ(m)) ⊗ τf(σ) =

∑_{σ∈G/H}(τσ(m)) ⊗ (f(τσ) − f(τ)) =

∑_{σ∈G/H} σ(m) ⊗ f(σ) − (tr_{G/H} m) ⊗ f(τ)

In H^{1} - start with m ∈ H^{1}(H, M) s.t. cor_{G/H} m = 0. Take f as before. Then ∑_{σ∈G/H} σ(m)⊗f(σ) ∈ H^{1}(H, M ⊗A)^{G}. It “essentially” comes from H^{1}(G, M ⊗A).

Call the resulting element m ∪ k f.

What this is good for? - Consider a prime ℓ such that:

• ℓ ∤ Np;

• ℓ is inert in K;

• If λ is the unique prime of K over ℓ, then Frob(λ) = id on A_{p}(1).

Many such by Chebotarev.

A_{p} defines an étale smooth sheaf A_{p} on spec(O_{K}[N^{−1}]). Let S = spec(O_{K})

Consider the long exact sequence:

→ H^{1}(S, A_{p}) → H^{1}(S − {λ}, A_{p})
→ H_{{λ}}^{2}(S, A_{p}) → H^{2}(S, A_{p})

Our assumptions imply that we have elements in H_{{λ}}^{2}(S, A_{p}) ≅ A_{p}(1). H^{2}(S, A_{p}) is expected to be very small, so up to killing it, we should find elements in H^{1}(S − {λ}, A_{p}) mapping on them. This is what the Kolyvagin derivative does

Let ℓ as above. Suppose that p|T_{ℓ}. Then the norm relation implies

cor_{K_{n},K_{m}}(y_{n}) (mod p) = 0,

so with an appropriate f we can form y_{n} (mod p) ∪ k f ∈ H^{1}(K_{m}, T/p ⊗ A) (In all the known cases Δ = Gal(K_{n}/K_{m}) is cyclic and when we insist its order be divisible by p we can take f : Δ → ℤ/p to be the reduction).

Proposition: y_{n} (mod p) ∪ k f is unramified at primes not dividing Nℓ. It’s “essentially unramified” at primes dividing N. It is ramified at λ.

The following recipe is given for computing the ramification: Since both y_{n}, y_{m} are unramified, we have y_{n} = inf y′_{n}, y_{m} = inf y′_{m} with y′_{n} and y′_{m} in T_{Frob(λ)}. The norm relation is translated into T_{ℓ}y′_{m} = |Δ|y′_{n}. Consider u = (T_{ℓ}/p)y′_{m} − (|Δ|/p)y′_{n} ∈ (T_{Frob(λ)})_{p}. From 0 → T → T → T/p → 0 we get T_{Frob(λ)})_{p} ≅ T/p and the computation is

res_{(K_{m})_{λ}}(y_{n} (mod p) ∪ k f) = u ∪ [f]

where [f] is the cohomology class of f.

We want an expression which depends only on y′_{m}. Here enters the congruence relation: There should be some operator F_{ℓ} such that y′_{n} = F_{ℓ}y′_{m}. It used to be part of the definition of an Euler system. Some ideas of Rubin, Kato and Perrin-Riou can help simplify things:

Generalized norm relation

cor_{K_{n},K_{m}}(y_{n}) = \begin{cases}T_{ℓ}y_{m}. & ℓ ∤ m \\ y_{m} & ℓ|m \end{cases}

implies having (y′_{np^{k}}) ∈ lim_{←} T_{λ_{k}} s.t.

(∗) T_{ℓ}y′_{mp^{k}} = |Δ|y′_{np^{k}}

The inverse limit is often p-torsion free so we can recover the y′_{n} from the y′_{m} once we can solve (*) independently of k.

E.g. in the elliptic case we have Frob(λ) = Frob(ℓ)^{2} = ϕ^{2}. ϕ satisfies ϕ^{2} −T_{ℓ}ϕ+ℓ = 0. We use coinvarints of ϕ^{2} so ϕ^{2} = 1. We get T_{ℓ}ϕ = ℓ+1 = |Δ|, so F_{ℓ} = Frob(ℓ)^{−1}.

We use the above to make classes P (ℓ) ∈ H^{1}(K, A_{p})

P (ℓ) = cor_{K_{1},K}(y_{ℓ} (mod p) ∪ k f)

More generally one has classes P (n) for n a product of distinct ℓ’s Basic property, which can be arranged for sufficiently many primes: with n, m, ℓ and λ as above,

P (m)_{λ} ≠ 0 ⟹ P (n)_{λ,sin} ≠ 0.

P (n)_{λ,sin} is the image of P (n) under the composition of res_{K_{λ}} and projection on H^{1}(K_{λ}, A_{p})/H_{f}^{1}(K_{λ}, A_{p}).

Step 4 - Local Tate duality

There is a perfect pairing ⟨, ⟩ : H^{1}(K_{v}, A_{p}) × H^{1}(K_{v}, A_{p}) → ℤ/p.

Let:

H_{f}^{1}(K_{v}, A_{p}) be the preimage in H^{1}(K_{v}, A_{p}) of H_{f}^{1}(K_{v}, A).

H_{sin}^{1}(K_{v}, A_{p}) := H^{1}(K_{v}, A_{p})/H_{f}^{1}(K_{v}, A_{p})

(Singular cohomology, this definition is due to Mazur).

For v ∤ Np it turns out that H_{f}^{1} = H_{ur}^{1} := Ker H^{1}(K_{v}, •) → H^{1}(K_{v}^{ur}, •). Also, H_{f}^{1} is its own annihilator under ⟨, ⟩.

\begin{matrix}Sel(K, A_{p}) := Ker H^{1}(K, A_{p}) ⟶ \\ ∏_{v} H^{1}(K_{v}, A_{p})/H_{f}^{1}(K_{v}, A_{p}). \end{matrix}

The assumption under which we prove the theorem is P (1) ≠ 0.

Step 5 - How to show that Sel(K, A_{p}) is trivial (more generally one needs to show that it’s generated by known classes).

Let L = K(A_{p}).

Proposition: H^{i}(Gal(L/K), A_{p}) = 0 for all i, almost all p.

This is usually a consequence of a “big image of Gal(L/K) in Aut(A_{p})” (Serre type results for elliptic curves).

0 → H^{1}(Gal(L/K), A_{p}) → H^{1}(K, A_{p}) →
H^{1}(L, A_{p})^{Gal(L/K)} → H^{2}(Gal(L/K), A_{p})

Shows that

H^{1}(K, A_{p}) ≅ H^{1}(L, A_{p})^{Gal(L/K)}
≅ Hom_{Gal(L/K)}(Gal(ℚ̄/L), A_{p})

We consider one method that uses 1 dimensionality (Other methods need more close understanding of the localization). For that we need to consider the algebra 𝔽 of all Gal automorphisms of A_{p}. We also split everything according to characters of Gal(K/ℚ) and we may extend A_{p} for that if we need to. For any S ∈ H^{1}(K, A_{p}) a f.d 𝔽-module corresponds a field L_{S}/L cut out by S, and we have:

[ , ]_{S} : S × Gal(L_{S}/L) → A_{p}.

6 AMNON BESSER

s.t.

x ∈ S and [x, Gal(L_{S}/L)]_{S} = 0 ⟹ x = 0.
Gal(L_{S}/L) ↪ Hom_{𝔽}(S, A_{p}).

Step 6 Idea of the proof is: If x ∈ Sel(K, A_{p}) consider some P (ℓ) for a good ℓ. We have (global reciprocity)

∑_{v}⟨x_{v}, P (ℓ)_{v}⟩ = 0.

If λ is the unique prime of K dividing ℓ, x_{v} ∈ H_{f}^{1},P (ℓ)_{v} ∈ H_{f}^{1} for all v ≠ λ. So

⟨x_{λ}, P (ℓ)_{λ}⟩ = 0.

Want to deduce x_{λ} = 0, and from that for enough λ that x = 0. The pairing ⟨ , ⟩ induces nondegenerate pairings:

⟨ , ⟩ : H_{f}^{1}(K_{λ}, A_{p}) × H_{sin}^{1}(K_{λ}, A_{p}) → ℤ/p(sth).

It is Gal(K/ℚ) equivariant so we may decompose according to the characters of Gal(K/ℚ). because the conditions imposed on ℓ:

H_{f}^{1}(K_{λ}, A_{p}) ≅ H^{1}(K_{λ}^{ur}/K_{λ}, A_{p}) ≅ A_{p}

The rightmost isomorphism is just evaluation at Frob(λ), so for x ∈ Sel(K, A_{p}), [x, Frob(λ)] = 0 iff x_{λ} = 0. The 1-dimensionality requirement is that A_{p} decomposes into 1-dim 𝔽-eigenspaces. Then one has duality between the corresponding eigenspaces in H_{f} and H_{sin} (throwing away a finite number of p’s). Corollary: If x ∈ Sel and P (n) are in the same eigenspace, then P (n)_{λ,sin} ≠ 0 and ⟨x_{λ}, P (n)_{λ}⟩ = 0 imply x_{λ} = 0. A careful application of Chebotarev shows that we can generate Gal(L_{S}/L) with Frob(λ) for “good” ℓ. Step 7 - End of proof Assume that P (1) is in some eigenspace. We consider x in another eigenspace of Sel(K, A_{p}). (we can’t show vanishing of the eigenspace of P (1) of course). We assume we have enough good ℓ’s such that P (ℓ) is in the same eigenspace as x. Let S = span(x, P (1)). Consider ℓ s.t. P (1)_{λ} ≠ 0. Their Frob’s will still span Gal(L_{S}/L).

P (1)_{λ} ≠ 0 ⟹ P (ℓ)_{λ,sin} ≠ 0 ⟹ x_{λ} = 0

INTRODUCTION TO EULER SYSTEMS AND BOUNDS ON TATE-SHAFAREVICH GROUPS 7

So x = 0