From now on assume that ρφ,℘ is residually irreducible and that φ is ℘-isolated. Assume moreover that p is prime to n and the absolute discriminant of K. Taking tensor product by Oφ and quotients by If`, the Kummer map δ : J(`)(K)/pnJ(`)(K) → H1(K, J(`)[pn]) yields a map:
d : J(`)(K)/If` → H1(K, Tap(J(`)))/If`) ' Yk j=1
H1(K, Tφ,nj).
Let
π`: Yk j=1
H1(K, Tφ,nj) → H1(K`, D`) ' H1(K`, Tφ,n) be the projection map to D`. Define κ` := π`(d(PK)).
To describe the behavior of κ` and show that it has the property of Euler systems described in Section 1, it is necessary to study the local structures associated to Tφ,n. Denote as above by Tφ' Oφ,℘2 the Gal(F /F )-stable lattice associated to the representation ρφ,℘and define Vφ := Tφ⊗Kφ ' Kφ,℘2 . Define as above
Tφ,n := Tφ/℘nTφ' (Oφ/℘n)2
so that the multiplication by ℘: Tφ,n → Tφ,n−1 yields a projective system and Tφ= lim←nTφ,n. Define:
Aφ:= Vφ/Tφ' (Kφ,℘/Oφ,℘)2
and denote by Aφ,n' (Oφ/℘n)2 its ℘n-torsion. The natural inclusion Aφ,n ,→ Aφ,n+1
yields an inductive system and Aφ = lim→nAφ,n. Note that the Galois mod-ules Tφ,n and Aφ,n are isomorphic and they correspond to the Galois module associated to the reduction modulo ℘n of the representation ρφ,℘.
For a Gal(F /F )-module M and a prime ideal q of OF, let H1(Kq, M ) denote the direct sum ⊕q0|qH1(Kq0, M ) of the local cohomology groups al all prime ideals q0 of K dividing q.
Define the following finite/singular structures, where M = Aφ,n or Tφ,n: Let q ⊆ OF be a prime ideal such that q - np. The singular part of H1(Kq, M ) is
Hsing1 (Kq, M ) := ⊕q0|qH1(Iq0, M)Gal(Kunrq0 /Kq0),
where the sum is extended over the primes q0 ⊆ OK dividing q, Gq0 ⊆ Gal(K/K) is the choice of a decomposition subgroup at q0 and Iq0 ⊆ Gq0
denotes the inertia subgroup. The finite part is defined by the exactness of the following sequence:
0 → Hfin1 (Kq, M ) → H1(Kq, M )−→ H∂q sing1 (Kq, M ).
The groups Hfin1 (Kq, Aφ,n) and Hsing1 (Kq, Tφ,n) are annihilators of each other under the local Tate pairing h, iq. See [Mi, Chapter I] for the definition of the local Tate pairing and its properties.
Let p | p be a prime ideal of OK and suppose that φ is ordinary at p. Then there is an exact sequence of Ip-modules (where Ip is the inertia subgroup of Gq):
0 → A(p)φ → Aφ→ A(1)φ → 0
such that Ip acts on the free of rank one Oφ,℘-module A(p)φ by the cyclotomic character ² and it acts trivially on the quotient A(1)φ , which is also free of rank one over Oφ,℘. Let
λp : ⊕p0|pH1(Kp0, A(p)φ ) → ⊕p0|pH1(Kp0, Aφ)
be the map of cohomology groups induced by the inclusion A(p)φ ⊆ Aφ, where the sum is over all prime ideals p0 of K dividing p. Define the ordinary part Hord1 (Kp, Aφ) of H1(Kp, Aφ) to be the maximal divisible subgroup of Image(λp). Define
H(p, n) := Hord1 (Kp, Aφ) ∩ H1(Kp, Aφ,n).
For any subgroup H ⊆ H1(Kp, Aφ,n), use the isomorphism Aφ,n ' Tφ,n to define a subgroup H∗ ⊆ H1(Kp, Tφ,n) such that H ' H∗. Then de-fine Hord1 (Kp, Aφ,n) to be the maximal subgroup of H1(Kp, Aφ,n) containing H(p, n) and such that Hord1 (Kp, Aφ,n) and Hord1 (Kp, Aφ,n)∗ are the exact an-nihilators of each other under the local Tate pairing at p.
For any prime ideal q ⊆ OF, let resq : H1(K,Mφ,n) → H1(Kq, Mφ,n) be the restriction map. For primes q - np, denote ∂q(resq(κ)) simply by ∂q(κ), where κ ∈ H1(K, Mφ,n).
Theorem 3.10. Suppose that φ is ℘-isolated, ρφ,℘ is residually irreducible and φ is ordinary at every prime ideal p dividing the residue characteristic p of ℘. Then the class κ` ∈ H1(K, Tφ,n) satisfies the following properties.
1. For q - np`, ∂q(κ`) = 0.
2. For p | p, resp(κ`) ∈ Hord1 (Kp, Tφ,n).
3. The equality ∂`(κ`) = Lφ(K) holds in Oφ/℘n up to multiplication by invertible elements.
Proof. For primes not dividing np`, this follows by [Mi, Chapter I, Section 3]. The second part follows form the description of the Kummer map given by [CG, Proposition 4.5]. For the last part, first note that by Proposition 3.5, C` ' Hsing1 (K`, Tφ,n) ' Oφ/℘n. By Equation (17), the image of κ` is then contained in the singular cohomology group. The result follows by Proposition 3.9.
4 Arithmetic applications
4.1 Modular abelian varieties
Let A/F be an abelian variety of GL2-type, that is, [EndQ(A) : Q] = dim(A), where EndQ(A) := End(A) ⊗Z Q. Set E := EndQ(A) and assume that End(A) ' OE, where OE is the ring of integers of E. For any prime ideal
℘ ⊆ OE, denote by A[℘n] the ℘n-torsion in A and by T℘(A) its ℘-adic Tate module. Denote by
ρA,℘ : GF → Aut(T℘(A)) ' GL2(OE,℘)
the associated representation. Finally, denote by n ⊆ OF the arithmetic conductor of A/F .
Definition 4.1. An abelian variety A/F is said to be modular if there exists a newform φ ∈ S2(n), which is an eigenform for the Hecke algebra Tn, such that E = Kφ and ρA,℘ is equivalent to ρφ,℘, where ℘ is a prime ideal of Oφ.
Let K/F be a quadratic totally imaginary extension and assume A/F be modular. Fix a prime ideal ℘ of Oφ of residue characteristic p. The Selmer group Sel℘n(A/K) is defined by the exactness of the following sequence:
0 → Sel℘n(A/K) → H1(K, A[℘n]) → ⊕qH1(Kq, A[℘n])/Im(δq),
where the direct sum is over the set of prime ideals q of OF and Im(δq) is the image of the local Kummer map
δq : A(Kq)/℘nA(Kq) ,→ H1(Kq, A[℘n]).
Note that the image Im(δq) of the Kummer map can be described in terms of the representation Aφ,n as follows:
1. Primes q - np: Im(δq) = Hfin1 (Kq, Aφ,n). This follows from [Mi, Chapter I, Section 3].
2. Primes p dividing p and such that A/K has ordinary reduction at p:
Im(δp) = Hord1 (Kp, Aφ,n). This follows from [CG, Proposition 4.5].
Finally, define the ℘-primary part Sel℘∞(A/K) of the Selmer group to be:
Sel℘∞(A/K) := lim
→nSel℘n(A/K),
where the direct limit is computed by means of the inclusion maps A[℘n] ,→ A[℘n+1].
4.2 Bounding Selmer groups
Let A/K be a modular abelian variety and let φ its associated Hilbert mod-ular form. Define L(A, s) := L(φ, s) and LK(A, s) := LK(φ, s) to be the Hasse-Weil L series of A over F and K. Fix a positive integer n and denote as above by δq : A(Kq)/℘nA(Kq) ,→ H1(Kq, A[℘n]) the local Kummer map.
Recall the conditions on the prime ideal ℘ of residue characteristic p stated in Assumption 1.2. For convenience, these conditions are restated and briefly commented in the following:
Assumption 1.2, Condition 1. ρφ,℘ is residually irreducible. This is equiv-alent to require that the representation of Gal(F /F ) on the ℘-torsion A[℘]
of A is irreducible. If A is an elliptic curve without complex multiplication, thanks to [Se] it is known that A[p] is irreducible for all rational primes p except possibly a finite number of them.
Assumption 1.2, Condition 2. φ is ordinary at all prime ideals p | p.
If the modular abelian variety A is ordinary at p | p, then the associated Hilbert modular form φ is also ordinary at p. For a proof, see for example [Go, Chapter 3, §6.2].
Assumption 1.2, Condition 3. φ is ℘-isolated. Since the dimension of the space of Hilbert modular forms of given weight and level is finite, this condition is verified by all prime ideals ℘ except, possibly, a finite number of them.
Assumption 1.2, Condition 4. Im(δq) = 0 for primes q | n and q - p, where p is the residue characteristic of ℘. This condition is verified by all prime ideals ℘ except, possibly, a finite number of them. This follows from a
theorem of Mattuck which states that, for an abelian variety A of dimension g defined over a finite extension K of Ql, where l is a prime of Q, there is an isomorphism A(K) ' Zg[K:Ql l]× H, where H is a finite group. For references, see for example [CG, Section 4].
Remark 4.2. Suppose that Assumptions 1.1 is verified and that the special value LK(φ, 1) is not zero. Suppose also that there are infinitely many prime ideals ℘ verifying Assumption 1.2. Then there are infinitely many prime ideals ℘ verifying Assumption 1.2 and such that ℘ - LK(φ).
The following theorem corresponds to Theorem 1.4 of the Introduction and proves the existence of Euler systems relative to A/K and ℘n, where ℘ satisfies Assumption 1.2.
Theorem 4.3. Suppose that K/F satisfies Assumption 1.1. If ℘ satisfies Assumption 1.2, then for every positive integer n the set
{κ` : ` is a ℘n-admissible prime}
is an Euler system relative to A/K and ℘n. More precisely:
1. The set of ℘n-admissible primes controls the Selmer group Sel℘n(A/K).
2. Primes q 6= `: resq(κ`) ∈ Im(δq).
3. The equality ∂`(κ`) = Lφ(K) holds in Hsing1 (K`, A[℘n]) ' Oφ/℘n up to multiplication by invertible elements in Oφ/℘n.
Proof. By the generalization of [BD2, Theorem 3.2], the set of ℘n-admissible primes controls the Selmer group Sel℘n(A/K). The equality of ∂`(κ`) and LK(φ) up to invertible elements follows from Theorem 3.10, statement 3. The description of resq(κ`) for primes q - n and primes p | p comes from Theorem 3.10, statements 1 and 2, combined with Condition 2 of Assumption 1.2 enjoyed by ℘. Since, by Condition 4 in Assumption 1.2, the image of the Kummer map at primes dividing n but not p is trivial, the result follows.
The following theorem explains how to use the Euler system of Theorem 4.3 to control the Selmer group of A/K.
Theorem 4.4. Suppose that K/F satisfies Assumption 1.1 and the central critical value LK(A, 1) of LK(A, s) is not zero. If ℘ satisfies Assumption 1.2 and ℘ - LK(φ), then Sel℘n(A/K) = 0 for every n. It follows that the
℘-primary part Sel℘∞(A/K) of the Selmer group is trivial too.
Proof. Assume that Sel℘n(A/K) 6= 0 and fix s ∈ Sel℘n(A/K) a non-zero element. As in [BD2, Theorem 3.2], choose, among infinitely many, a ℘n -admissible prime ` so that res`(s) 6= 0 and consider the class κ` built in Section 3.6. By the choice of ℘ and Theorem 4.3, it follows that resq(κ`) is orthogonal to resq(s) with respect to the local Tate pairing h, iq for all prime ideals q 6= `. By the global reciprocity law of class field theory, it follows that h∂`(κ`), res`(s)i` = 0. (18) Since ` is ℘n-admissible, it follows as recalled above that
Hsing1 (K`, Tφ,n) ' Oφ/℘n
and, by the same argument as in [BD2, Lemma 2.6], its orthogonal com-plement Hfin1 (K`, Aφ,n) is isomorphic to Oφ/℘n too. Since ∂`(κ`) 6= 0 by Theorem 4.3 and res`(s) 6= 0, Equation (18) contradicts the non degeneracy of the local Tate pairing. It follows that the Selmer group Sel℘n(A/K) is trivial. The last statement follows from the definition of Sel℘∞(A/K).
Corollary 4.5. Suppose that K/F satisfies Assumption 1.1 and the central critical value LK(A, 1) of LK(A, s) is not zero. Suppose also that there exists at least one prime ideal ℘ verifying Assumption 1.2 and such that ℘ - LK(φ).
Then the rank of A(K) is zero.
Proof. Let ℘ be as in the statement. Then, by Theorem 4.4, the Selmer group Sel℘(A/K) is trivial. Since the Kummer map
A(K)/℘A(K) ,→ Sel℘(A/K) is injective, the rank of A(K) is zero.
Under stronger hypotheses on A such that the existence of infinitely many prime ideals ℘ verifying Assumption 1.2, it is possible to prove the following more compact result. The first statement corresponds to Corollary 1.7 of the Introduction.
Corollary 4.6. Suppose that Assumption 1.1 is verified. Suppose also that there exists infinitely many prime ideals ℘ satisfying Assumption 1.2.
1. If LK(A, 1) 6= 0, then the rank of A(K) is zero.
2. If L(A, 1) 6= 0, then the rank of A(F ) is zero.
Proof. To prove the first statement about the rank of A(K), choose, among infinitely many, a prime ideal ℘ satisfying Assumption 1.2 and such that
℘ - LK(φ). This is possible by Remark 4.2. Then apply Corollary 4.5.
To prove the second statement on the rank of A(F ), choose, by Remark 3.8, a quadratic totally imaginary extension K/F verifying Assumption 1.1 and such that the special value LK(A, 1) is not zero. Then apply the first part of this Corollary and use that A(F ) ⊆ A(K).
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