2 Statement of the main results
Recall that A is an abelian variety of dimension d defined over the number field k. Furthermore, F/k is cyclic of degree pn where p is an odd prime.
We assume throughout this section that A/k and F/k are such that
p∤|A(k)tor|⋅|At(k)tor|,
p∤∏v∈Sbcv(A,k), where cv(A,k) denotes the Tamagawa number of A at v,
A has good reduction at all p-adic places of k,
no place of bad reduction for A is ramified in F/k, i.e. Sb∩Sr=∅,
the Tate–Shafarevich group Ш(AF) is finite,
Шp(AFH)=0 for all non-trivial subgroups H of G.
Remarks 2.1Our assumptions (a)–(g) recover [12, hypotheses (a)–(h)]. For a fixed abelian variety A/k, the hypotheses (a), (b)
and (c) clearly exclude only finitely many choices of odd prime p, while the additional hypotheses (d), (e) and (f) constitute a mild restriction on the choice of cyclic
field extension F of k of odd, prime-power degree. In order to further illustrate this point, we let S denote any finite set of places of k at which A has good
reduction. We then define a set Σ(S) of rational primes as the union of the set of all prime divisors ℓ of
|A(k)tor|⋅|At(k)tor|⋅∏v∈Sbcv(A,k)⋅∏v∈S|A(κv)|
and the set of all primes ℓ with the property
that A has bad reduction at an ℓ-adic place of k. The set
Σ(S) is then clearly finite and, for any odd prime p∉Σ(S) and any cyclic field extension F of k of p-power degree which is unramified
outside S and with the property that p is unramified in F/ℚ, all of the hypotheses (a)–(f) are satisfied.
The hypothesis (g) is famously conjectured to be true in all cases, and it is straightforward to produce specific
examples for which all of the other hypotheses, including the additional hypothesis (h), are
satisfied (see also Section 5 and Remark 2.13 in this regard). We emphasize that in (h) we allow Шp(AF) to be non-trivial.
An understanding of the G-module structure of the relevant Mordell–Weil groups is key to our approach. We hence begin by applying a result of Yakovlev [22]
in order to obtain such explicit descriptions. This approach is inspired by work of Burns, who first obtained a similar result in [7, Proposition 7.2.6 (i)].
For a non-negative integer m and a ℤp[G]-module M we write M〈m〉 for the direct sum of m
copies of M. Furthermore, we set [m]:={1,…,m}.
Proposition 2.2There exist isomorphisms of Zp[G]-modules of the form
A(F)p≅⊕J≤Gℤp[G/J]〈mJ〉≅At(F)p,
for a set of non-negative integers {mJ:J≤G}.
Proposition 2.2 has the following immediate consequence for the ranks of the relevant Mordell–Weil groups.
Corollary 2.3For any subgroup H of G we have
rk(A(FH))=rk(At(FH))
=∑J>H|G/J|mJ+|G/H|∑J≤HmJ
≤|G/H|rk(A(k)).
Proposition 2.2 combines with Roiter’s lemma (see [13, (31.6)]) to imply the existence of points P(J,j)∈A(F) and P(J,j)t∈At(F) for
J≤G and j∈[mJ] with the property that
(2.1)A(F)p=⊕J≤G⊕j∈[mJ]ℤp[G/J]P(J,j),ℤp[G/J]P(J,j)≅ℤp[G/J],
At(F)p=⊕J≤G⊕j∈[mJ]ℤp[G/J]P(J,j)t,ℤp[G/J]P(J,j)t≅ℤp[G/J].
Furthermore, our choice of points as in (2.1) guarantees that one also has
(2.2)ℚ⊗ℤA(F)=⊕J≤G⊕j∈[mJ]ℚ[G/J]P(J,j),ℚ[G/J]P(J,j)≅ℚ[G/J],
ℚ⊗ℤAt(F)=⊕J≤G⊕j∈[mJ]ℚ[G/J]P(J,j)t,ℚ[G/J]P(J,j)t≅ℚ[G/J].
We now fix sets
𝒫={P(J,j)∈A(F):J≤G,j∈[mJ]},
𝒫t={P(J,j)t∈At(F):J≤G,j∈[mJ]}
such that (2.2) holds. For 0≤t≤n we write Ht for the (unique) subgroup of G of order pn-t and
set
P(t,j):=P(Ht,j),P(t,j)t:=P(Ht,j)t.
We also put mt:=mHt and
eHt:=1|Ht|TrHt=1|Ht|∑g∈Htg.
We write 〈⋅,⋅〉F for the Néron–Tate height pairing A(F)×At(F)→ℝ defined relative to the field F and define a matrix with entries in ℂ[G] by setting
R(𝒫,𝒫t):=(1|Hu|∑τ∈G/Hu〈τ⋅P(u,k),P(t,j)t〉F(τ⋅eHu))(u,k),(t,j),
where (u,k) is the row index with 0≤u≤n, k∈[mu], and
(t,j) is the column index with 0≤t≤n, j∈[mt] (we always order sets of the form {(t,j):0≤t≤n,j∈[mt]} lexicographically).
We note that, since each point P(u,k) belongs to A(FHu), the action of G/Hu on P(u,k) is well-defined.
For any matrix A=(a(u,k),(t,j))(u,k),(t,j) indexed as above we define
At0:=(a(u,k),(t,j))(u,k),(t,j),u,t≥t0
with the convention At0=1 whenever no entries a(u,k),(t,j) with u,t≥t0 exist.
If A is a matrix with coefficients aij in ℂ[G] or ℂp[G], then for any ψ∈G^ we write ψ(A) for the matrix with coefficients
ψ(aij). We also set Rt0(𝒫,𝒫t)=R(𝒫,𝒫t)t0.
Definition 2.4For each character ψ∈G^
we define tψ∈{0,…,n} by the equality
ker(ψ)=Htψ
and call
λψ(𝒫,𝒫t):=det(ψ(Rtψ(𝒫,𝒫t)))
the lower ψ-minor of R(𝒫,𝒫t).
Remark 2.5It is easy to see that the element
∑ψ∈G^λψ(𝒫,𝒫t)eψ∈ℂ[G]
depends upon the choice of points
𝒫 and 𝒫t satisfying (2.2) only modulo ℚ[G]×. Similarly, for any given isomorphism of fields j:ℂ→ℂp, it is clear that the element
∑ψ∈G^j(λψ(𝒫,𝒫t))eψ∈ℂp[G]
depends upon the choice
of points 𝒫 and 𝒫t satisfying (2.1) only modulo ℤp[G]×.
For any order Λ in ℚ[G] that contains ℤ[G] we let C(A,Λ) denote the integrality part
of the equivariant Tamagawa number conjecture (‘eTNC’ for brevity) for the pair (h1(AF)(1),Λ) as formulated by Burns and Flach in [9, Conjecture 4 (iv)].
Similarly, we let C(A,ℚ[G]) denote the rationality part as formulated in [9, Conjecture 4 (iii) or Conjecture 5].
We recall that, under the assumed validity of hypothesis (g), C(A,Λ) takes the form of an equality in the relative K-group K0(Λ,ℝ[G]). For each
embedding j:ℝ→ℂp we denote by Cp,j(A,Λ) the image of this conjectural equality under the
induced map K0(Λ,ℝ[G])→K0(Λp,ℂp[G]). We then say that Cp(A,Λ) is valid if Cp,j(A,Λ) is valid for every
isomorphism j:ℂ→ℂp.
The eTNC is an equality between analytic and algebraic invariants associated with A/k and F/k. In the following we describe and define
the analytic part. We first recall the definition of periods and Galois–Gauss sums of [12, Section 4.4].
We fix Néron models 𝒜t for At over 𝒪k and 𝒜vt for Akvt over 𝒪kv
for each v in Sp and then fix a k-basis {ωb}b∈[d] of the space of invariant differentials H0(At,ΩAt1) which
gives 𝒪kv-bases of H0(𝒜vt,Ω𝒜vt1) for each such v and is also
such that each ωb extends to an element of H0(𝒜t,Ω𝒜t1).
For each v in Sℂ we fix a ℤ-basis {γv,a}a∈[2d]
of H1(σv(At)(ℂ),ℤ). For each v in Sℝ we let c denote complex conjugation and
fix a ℤ-basis {γv,a+}a∈[d] of H1(σv(At)(ℂ),ℤ)c=1. For each v in Sℝ, resp. Sℂ, we then define periods by setting
Ωv(A/k):=|det(∫γv,a+ωb)a,b|,resp. Ωv(A/k):=|det(∫γv,aωb,c(∫γv,aωb))a,b|,
where in the first matrix (a,b) runs over [d]×[d] and in the second matrix (a,b) runs over [2d]×[d].
In our special case all characters are one-dimensional and, moreover, |G| is odd. Therefore the definitions of [12]
simplify and we set
Ω(A/k):=∏v∈S∞Ωv(A/k),
w∞(k):=i|Sℂ|.
For each place v in Sr we write I¯v⊆G for the inertia group of v and Frv for the natural Frobenius in G/I¯v.
We define the non-ramified characteristicuv by
uv(ψ):={-ψ(Frv-1),ψ|I¯v=1,1,ψ|I¯v≠1,
and
u(ψ):=∏v∈Sruv(ψ).
For each character ψ∈G^ we then define the modified Galois–Gauss sum by setting
τ*(ℚ,indkℚ(ψ)):=u(ψ)τ(ℚ,indkℚ(ψ))∈(ℚc)×,
where each individual Galois–Gauss sum τ(ℚ,⋅) is as defined by Martinet in [19]. For each character ψ∈G^
we set
ℒψ*=ℒA,F/k,ψ*:=LSr*(A,ψˇ,1)τ*(ℚ,indkℚ(ψ))dΩ(A/k)w∞(k)d∈ℂ×,
where here for each finite set Σ of places of k we write LΣ*(A,ψ,1) for the leading term in the Taylor expansion at
s=1 of the Σ-truncated ψ-twisted Hasse–Weil-L-function of A. Without any further mention we will always
assume that the functions LΣ(A,ψ,s) have analytic continuation to s=1 (as conjectured in [9, Conjecture 4 (i)]) and recall that they are then expected
to have a zero of order rψ:=dimℂ(eψ(ℂ⊗ℤA(F))) (this is the rank conjecture [9, Conjecture 4 (ii)]).
We finally define
ℒ*=ℒA,F/k*:=∑ψ∈G^ℒA,F/k,ψ*eψ∈ℂ[G]×
and note that the element ℒ* defined in [12, Theorem 5.1] specialises precisely to our definition.
Theorem 2.6Conjecture C(A,Q[G]) is valid if and only if
ℒψ*λψ(𝒫,𝒫t)-1∈ℚ(ψ)
for all ψ∈G^ and furthermore, for any γ∈Gal(Q(ψ)/Q),
ℒψγ*λψγ(𝒫,𝒫t)-1=γ(ℒψ*λψ(𝒫,𝒫t)-1)
for any, or equivalently every, choice of points P and Pt such that (2.2) holds.
Remarks 2.7(i) From the definitions of u(ψ), w∞(k) and the definition of local Euler factors it is immediately clear that in the statement
of Theorem 2.6 we can replace ℒψ* by
ℒ~ψ*:=L*(A,ψˇ,1)τ(ℚ,indkℚ(ψ))dΩ(A/k).
(ii) The explicit conditions on elements of the form ℒψ*λψ(𝒫,𝒫t)-1 given in
Theorem 2.6 generalise and refine the predictions given by Fearnley and Kisilevsky in [16, 17]. For details see
[2, Example 5.2]. In particular, we note that the numerical computations performed by Fearnley and Kisilevsky can be interpreted via
Theorem 2.6 as supporting evidence for conjecture C(A,ℚ[G]).
We fix a generator σ of G and define Σ to be the diagonal matrix indexed by pairs (t,j) and (s,i) with σpt-1 at the diagonal entry
associated to (t,j) and zeros elsewhere.
For any matrix A=(a(u,k),(t,j))(u,k),(t,j) indexed by tuples (u,k) and (t,j) as above we define
At0:=(a(u,k),(t,j))(u,k),(t,j),u,t≤t0,
once again with the convention At0=1 whenever no entries a(u,k),(t,j) with u,t≤t0 exist. We recall that for each character ψ∈G^
we defined tψ such that ker(ψ)=Htψ. We define the upper ψ-minor of Σ by
δψ:=det(ψ(Σtψ-1)).
It is easy to see that for another choice of generator of G, say τ, one has
∑ψ∈G^δψ(σ)δψ(τ)eψ∈ℤp[G]×.
Under our current hypotheses on the data (A,F/k,p) and the additional hypothesis that Шp(AF)=0, and for any intermediate field L of F/k, we shall say that
BSDp(L) holds if, for any choice of ℤ-bases {Qi} and {Rj} of A(L) and At(L) respectively and of isomorphism j:ℂ→ℂp, one has that
j(L*(A/L,1)⋅(|dL|)ddet(〈Qi,Rj〉L)⋅∏v∈S∞LΩv(A/L))∈ℤp×.
Here dL denotes the discriminant of the field L and each period Ωv(A/L)
is as defined above but relative to the field L rather than k. It will become apparent in the proof of Theorem 2.8 below that the validity of BSDp(L) is
equivalent to the validity of the p-part of the eTNC for the pair (h1(AL)(1),ℤ). We recall that hypotheses (a), (b) and (h) justify the fact that no orders of
torsion subgroups of Mordell–Weil groups, Tamagawa numbers or orders of Tate–Shafarevich groups occur in this formulation, and furthermore note that,
by explicitly computing integrals, the periods Ωv(A/L)
can be related to those obtained by integrating measures as occurring in the classical formulation of the
Birch and Swinnerton-Dyer conjecture – see, for example, Gross [18, p. 224].
For the remainder of this section, we assume that C(A,ℚ[G]) is valid. It is then easy to see that, for any order Λ in ℚ[G] that contains ℤ[G], the
validity of Cp,j(A,Λ) is independent of the choice of isomorphism j:ℂ→ℂp, and so we fix such a j for the remainder of this section.
In fact, all relevant elements of ℂ[G] appearing in the statements of our results will actually belong to ℚ[G]
(as a consequence of an easy application of Theorem 2.6) and so we will consider them simultaneously as elements of ℚp[G]⊂ℂp[G]
in the natural way without any explicit mention of j.
Let ℳ denote the maximal ℤ-order in
ℚ[G]. For any ψ∈G^, let 𝒪ψ
be the valuation ring of ℚp(ψ). Let 𝔭ψ
be the (unique) prime ideal of 𝒪ψ above p.
We write v𝔭ψ for the normalised valuation defined by 𝔭ψ.
Theorem 2.8Let P and Pt be any choice of points such that (2.1) holds. We assume that Шp(AF)=0. Then the following are equivalent.
BSDp(L) is valid for all intermediate fields L of F/k.
For each ψ∈G^ one has
v𝔭ψ(ℒψ*λψ(𝒫,𝒫t))=bψ where bψ:=∑s=0tψ-1psms.
One has
∑ψ∈G^ℒψ*λψ(𝒫,𝒫t)δψeψ∈ℳp×.
To describe the full range of implications of the validity of Cp(A,ℤ[G]) requires yet more work and some further notations.
For each finite extension L/k and natural number n we write Sel(pn)(AL) for the Selmer group associated
to the isogeny [pn]. We define the p-primary Selmer group by
Selp(AL):=lim→Sel(pn)(AL).
We recall that one then obtains a canonical short exact sequence
0→ℚp/ℤp⊗ℤA(F)→Selp(AF)→Шp(AF)→0
of ℤp[G]-modules, from which upon taking Pontryagin duals one derives a canonical short exact sequence
(2.3)0→Шp(AF)∨→Selp(AF)∨→A(F)p*→0.
We will throughout use this canonical short exact sequence to fix identifications of the torsion subgroup (Selp(AF)∨)tor with Шp(AF)∨ and
of (Selp(AF)∨)tf with A(F)p*.
In [12] a suitable integral model RΓf(k,Tp,F(A)) of the finite support cohomology of Bloch and
Kato for the base change through F/k of the p-adic Tate module of At is defined and then used in order to define
an equivariant regulator which is essential to the explicit reformulation of Cp(A,ℤ[G])
(see [12, Theorem 5.1]). We will recall this reformulation in Section 3.
By [12, Lemma 4.1], RΓf(k,Tp,F(A)) is under our current hypotheses a perfect complex of
ℤp[G]-modules which is acyclic outside degrees 1 and 2 and whose cohomology groups in degrees 1 and 2 canonically
identify with At(F)p and Selp(AF)∨ respectively.
We recall that given any complex E with just two non-zero cohomology modules Hm(E) and Hn(E), n>m, the complex
τ≤nτ≥mE represents an element in the Yoneda ext-group Extℤp[G]n-m+1(Hn(E),Hm(E)). Here τ is the truncation of
complexes preserving cohomology in the indicated degrees. In this way, RΓf(k,Tp,F(A)) uniquely determines a class δA,K,p in Extℤp[G]2(Selp(AF)∨,At(F)p). The element δA,K,p is furthermore
perfect, meaning that it can be represented as a Yoneda 2-extension by a four term exact sequence in which each of the two middle modules is perfect when considered
as an object of D(ℤp[G]).
We will use Proposition 2.2
to fix an explicit 2-syzygy of the form
(2.4)0→M→ιF0→F1→A(F)p*→0,
in which we set
M:=⊕(t,j)ℤp[G/Ht]
and both F0 and F1 are finitely generated free ℤp[G]-modules
and then use the exact sequence (2.4) to compute Extℤp[G]2(A(F)p*,At(F)p) via the
canonical isomorphism
Extℤp[G]2(A(F)p*,At(F)p)≃Homℤp[G](M,At(F)p)/ι*(Homℤp[G](F0,At(F)p)).
If we now assume that Шp(AF) vanishes, we may identify Selp(AF)∨ and A(F)p*, so that δA,F,p
uniquely determines an element of the above quotient. We will prove (see Lemmas 4.2 and 4.3 below)
that we may choose a representative Φ∈Homℤp[G](M,At(F)p) of δA,F,p with the following properties:
for every j∈[mn], Φ restricts to send an element x(n,j) of the (n,j)-th direct summand ℤp[G] to x(n,j)P(n,j)t.
For a fixed choice of points 𝒫 and 𝒫t such that the conditions in (2.1) hold and of a representative Φ∈Homℤp[G](M,At(F)p) as above,
we fix a canonical ℤp[G/Ht]-basis element e(t,j)
of each direct summand ℤp[G/Ht] of M and fix any elements Φ(t,j),(s,i) of ℤp[G] with the property that
(2.5)Φ(e(s,i))=∑(t,j)Φ(t,j),(s,i)P(t,j)t.
We thus obtain an invertible matrix (Φ(t,j),(s,i))(t,j),(s,i) with entries in ℤp[G], which by abuse of notation
we shall also denote by Φ. The matrix Φ is of the form
(2.6)((Φ(t,j),(s,i))t,s<n0⋮00…0Imn)
with Imn denoting the identity mn×mn matrix.
Recall the definition of tψ in Definition 2.4.
We define the lower ψ-minor of Φ by setting
εψ(Φ):=det(ψ(Φtψ)).
We note firstly that, since the chosen points P(t,j)t satisfy (2.1), each element εψ(Φ)
(and, indeed, even the matrix ψ(Φtψ)) is independent of our particular
choice of Φ(t,j),(s,i)∈ℤp[G] with the property that (2.5) holds.
Theorem 2.9Let P and Pt be any choice of points such that (2.1) holds. Assume that Шp(AF)=0.
Let Φ∈HomZp[G](M,At(F)p) be any representative of δA,F,p such that (P1) and (P2) hold.
Then Cp(A,Z[G]) is valid if and only if
(2.7)∑ψ∈G^ℒψ*λψ(𝒫,𝒫t)⋅εψ(Φ)⋅δψeψ∈ℤp[G]×.
Remark 2.10Theorem 2.9 can be reformulated in terms of explicit congruences.
Via Theorem 2.9, we now obtain completely explicit predictions concerning congruences in the augmentation filtration
of the integral group ring ℤp[G] for leading terms at s=1 of the relevant Hasse–Weil-L-functions of A normalised by
our twisted regulators. We recall that such predictions constitute a refinement and generalisation of the congruences for modular
symbols that are conjectured by Mazur and Tate in [20].
In order to state such conjectural congruences, we require the following notation: if the inequality rk(A(FJ))≤|G/J|rk(A(k)) of Corollary 2.3 is strict for some subgroup J of G, we may and will denote by H=Ht0
the smallest non-trivial subgroup of G with the property that mH≠0. Hence t0 is the maximal index with the
properties mt0≠0 and t0<n. We then define
ℒ:={∑ψ∈G^ℒψ*det(ψ(R(𝒫,𝒫t)))eψ,if rk(A(FJ))=|G/J|rk(A(k)) for every J,∑ψ|H≠1ℒψ*λψ(𝒫,𝒫t)eψ,otherwise.
We also let IG,p denote the kernel of the augmentation map ℤp[G]→ℤp.
Corollary 2.11Let P and Pt be any choice of points such that (2.1) holds. Assume that Шp(AF)=0.
Let Φ∈HomZp[G](M,At(F)p) be any representative of δA,F,p such that (P1) and (P2) hold.
If Cp(A,Z[G]) is valid, then the following hold:
ℒ belongs to the ideal IG,ph of ℤp[G], where h:=∑t<nmt.
One has ϵ:=det(𝟏G(Φ))∈ℤp×.
One has
v:=(-1)d⋅|Sr|LSr*(A/k,1)⋅(|dk|)dΩ(A)⋅det(𝟏G(R(𝒫,𝒫t)))∈ℤp×.
One has
ℒ≡vϵ⋅∏t<n(σpt-1)mt(modIG,ph+1).
The theory of organising matrices developed by Burns and the second named author in [10] allows one to derive the
containment ℒ∈IG,ph of Corollary 2.11 (i) from the assumed validity of conjecture Cp(A,ℤ[G]) in
situations in which Шp(AF) is non-trivial. In this greater level of generality, it furthermore leads to explicit statements concerning annihilation of Tate–Shafarevich groups and (generalised) strong main conjectures of the kind that Mazur and Tate
ask for in [20, Remark after Conjecture 3]. Namely, we obtain the following result.
Theorem 2.12Let P,Pt be any choice of points such that (2.1) holds. If Cp(A,Z[G]) is valid, then the following hold:
ℒ belongs to the ideal IG,ph of ℤp[G], where h:=∑t<nmt.
ℒ annihilates the ℤp[G]-module Шp(AFt).
There exists a (finitely generated) free ℤp[G]-submodule
Π
of Selp(AF)∨ of (maximal) rank mn
with the property that ℒ generates the Fitting ideal of Selp(AF)∨/Π.
Remark 2.13It will become clear in the course of the proof that, provided that there exist sets of points 𝒫 and 𝒫t
such that (2.1) holds from which one may construct the element ℒ, Theorem 2.12 remains valid even if hypothesis (h) fails to hold. This
fact is relevant because, as we will see in Section 5, it allows us to obtain numerical supporting evidence for Cp(A,ℤ[G]) (via verifying the
explicit assertions of Theorem 2.12) in a wider range of situations.
Let #:ℤp[G]→ℤp[G] denote the involution induced by g↦g-1. Recalling that the Cassels–Tate pairing
induces a canonical isomorphism between Шp(AF)∨ and Шp(AFt), we immediately obtain the following corollary:
Corollary 2.14Under the assumptions of Theorem 2.12 one has that the element L# of IG,ph annihilates
the Zp[G]-module Шp(AF).
4 The proofs
4.1 The proof of Proposition 2.2
In this subsection we will prove Proposition 2.2. The existence of global points P(t,j) and P(t,j)t
such that (2.1) holds is then an immediate consequence of Roiter’s lemma (see [13, (31.6)]).
To ease notation we set
H1:=H1(CA,Ff,∙)=At(F)p
and
H2:=H2(CA,Ff,∙)=Selp(AF)∨.
We recall that, for any intermediate field L of F/k, we may and will use the
relevant canonical short exact sequence of the form (2.3) to identify (Selp(AL)∨)tor with Шp(AL)∨ and
(Selp(AL)∨)tf with A(L)p*.
Under the assumed validity of hypotheses (a)–(e), the result of [11, Proposition 3.1] directly combines with hypothesis (h) to imply that, for every non-trivial subgroup J of G, the Tate cohomology group
H^-1(J,Htf2) vanishes and the module (H2)J is torsionfree.
By the definition of Tate cohomology,
we have that the finite group H^-1(J,H2) identifies with a submodule of (H2)J and therefore vanishes too.
Furthermore, since the complex CA,Ff,∙ is perfect and acyclic outside degrees 1 and 2, for each subgroup J of G the group H^1(J,H1) is
isomorphic to H^-1(J,H2). In addition, since G is cyclic, the Tate cohomology of each J is periodic of order 2 and so H^-1(J,H1) also vanishes.
We next note that, since G is a p-group, hypothesis (a) implies that At(F)p=H1 is torsion-free.
We apply the main result [22, Theorem 2.4] of Yakovlev to see that both At(F)p=H1 and A(F)p*=Htf2 are ℤp[G]-permutation
modules, that is, that there exist isomorphisms of the form
At(F)p≃⊕J≤Gℤp[G/J]〈rJ〉,A(F)p*≃⊕J≤Gℤp[G/J]〈sJ〉
for some sets of non-negative integers {rJ} and {sJ}.
But the Néron–Tate height pairing induces an isomorphism of ℂp[G]-modules between ℂp⊗ℤpAt(F)p and
ℂp⊗ℤpA(F)p* and so by rank considerations we find that rJ=sJ=:mJ for every J.
Finally, it is easy to see that the ℤp-linear dual of a permutation module is again a permutation module of the same
form. Therefore the canonical isomorphism A(F)p**≃A(F)p shows that one also has that
A(F)p≃⊕J≤Gℤp[G/J]〈mJ〉.
4.2 The proof of Theorem 2.9
Recall that in addition to our running hypothesis (a)–(h) we also assume that Шp(AF)=0. In particular,
we identify Selp(AF)∨ with A(F)p* via the canonical map in (2.3).
We fix an isomorphism of fields j:ℂ→ℂp. From formula (3.4) and the discussion in Section 3.1 it is clear that it will be enough to show that
(4.1)-χG,p(CA,Ff,∙,(λA,FNT,j)-1)=δG,p(∑ψ∈G^j(λψ(𝒫,𝒫t))ϵψ(Φ)δψeψ)
(we recall that, since we have assumed the validity of C(A,ℚ[G]), one actually has that the validity of Cp,j(A,ℤ[G]) is equivalent to the
validity of Cp(A,ℤ[G])).
We begin by defining, for every pair (s,i), an element P(s,i)*∈A(F)p* by setting, for every pair (t,j) and element τ of G,
(4.2)P(s,i)*(τP(t,j))={1,if s=t,i=j and τ∈Hs,0,otherwise.
Lemma 4.1We have
A(F)p*=⊕(s,i)ℤp[G/Hs]P(s,i)*
with each summand Zp[G/Hs]P(s,i)* isomorphic to Zp[G/Hs].
Proof.
If γ∈G, then
(4.3)(γP(s,i)*)(τP(t,j))=P(s,i)*(γ-1τP(t,j))=1
⇔s=t,i=j and γ≡τ(modHs).
Hence we have
γP(s,i)*=P(s,i)*
for γ∈Hs. Moreover, it easily follows that the
maps γP(s,i)* with γ∈G/Hs form a ℤp-basis of A(F)p* (actually the ℤp-dual basis of
τP(t,j) with τ∈G/Ht).
∎
We now proceed to fix an explicit 2-syzygy of the form (2.4). For this purpose, we first recall
that Ht=〈σpt〉. For each pair (t,j) corresponding to the subgroup Ht of G
and j∈[mt] we hence have a 2-extension
0→ℤp[G/Ht]→ιtℤp[G]→σpt-1ℤp[G]→πt,jℤp[G/Ht]P(t,j)*→0.
In this sequence we let ιt denote the (well-defined) map which sends the image of
an element x∈ℤp[G] under the natural surjection ℤp[G]→ℤp[G/Ht] to the element TrHtx of ℤp[G],
while πt,j sends the element 1 of ℤp[G] to the element P(t,j)*∈A(F)p* defined in (4.2).
Lemma 4.1 then implies that, summing over all pairs (t,j) we obtain a 2-extension
0⟶M⟶ιF0⟶ΘF1⟶πA(F)p*⟶0.
with
F0=F1=X:=⊕(t,j)ℤp[G],
M:=⊕(t,j)ℤp[G/Ht].
We now recall that we have a canonical isomorphism
Extℤp[G]2(A(F)p*,At(F)p)≃Homℤp[G](M,At(F)p)/ι*(Homℤp[G](F0,At(F)p))
under which an element ϕ of Homℤp[G](M,At(F)p) corresponds to the element ϵ(ϕ) of the group Extℤp[G]2(A(F)p*,At(F)p) which has the bottom row of the commutative diagram with exact rows
as a representative. In this diagram X(ϕ) is defined as the push-out of ι and ϕ.
We now proceed to prove that, when considering perfect elements
ϵ(ϕ)∈Extℤp[G]2(A(F)p*,At(F)p),
one may without loss of generality restrict attention to a special class of elements
ϕ∈Homℤp[G](M,At(F)p).
Lemma 4.2For all subgroups J of G one has
Extℤp[G]2(ℤp[G],ℤp[G/J])=0,
Extℤp[G]2(ℤp[G/J],ℤp[G])=0.
Proof.
Claim (i) is clear. Concerning claim (ii), we first note that since ℤp[G/J] is
ℤp-torsion-free, there is an isomorphism of the form
Extℤp[G]2(ℤp[G/J],ℤp[G])≅H2(G,Homℤp(ℤp[G/J],ℤp[G])).
But the G-module Homℤp(ℤp[G/J],ℤp[G]) is cohomologically-trivial and therefore
Extℤp[G]2(ℤp[G/J],ℤp[G])=0,
as required.
∎
Lemma 4.2 now implies that we can without loss of generality restrict attention to those
elements ϕ of Homℤp[G](M,At(F)p) which satisfy (P2) and, in addition,
by the argument of [3, Lemma 4.3], which are furthermore injective.
Lemma 4.3Suppose ϕ∈HomZp[G](M,At(F)p) has all of the properties described in the previous paragraph. Then the
element ϵ(ϕ) of ExtZp[G]2(A(F)p*,At(F)p) is perfect if
and only if ϕ is an isomorphism.
Proof.
The fact that ϕ restricts to send an element x(n,j) of the (n,j)-th direct summand ℤp[G] to x(n,j)P(n,j)t immediately implies that cok(ϕ)=cok(ϕ′) where
ϕ′:⊕(t,j),t<nℤp[G/Ht]→⊕(t,j),t<nℤp[G/Ht]P(t,j)t
is the map obtained by restriction of ϕ. Since ϕ is injective, the commutative diagram (4.4) implies that the 2-extension ϵ(ϕ)
is perfect if and only cok(ϕ)=cok(ϕ′) is cohomologically trivial. Note that Hn-1 clearly acts trivially on cok(ϕ′). So, if cok(ϕ′) is cohomologically trivial, then
cok(ϕ′)/pcok(ϕ′)=H^0(Hn-1,cok(ϕ′))=0.
It then follows that cok(ϕ′) must itself vanish, as required.
∎
We henceforth fix Φ∈Homℤp[G](M,At(F)p) representing the element
δA,F,p∈Extℤp[G]2(Selp(AF)∨,At(F)p)
which is specified by the complex CA,Ff,∙. Recall that by our current assumption Шp(AF)=0 we identify
Selp(AF)∨ and A(F)p*. By Lemmas 4.2 and 4.3 we may and will assume that Φ is an
isomorphism and furthermore that the matrix defined in (2.5) is of the form (2.6).
Having justified our choice of homomorphism Φ, we now proceed to compute the term
-χG,p(CA,Ff,∙,(λA,FNT,j)-1)
that occurs
in (4.1) via a generalisation of the computations done in [3, Section 4]. For brevity, given any ℤp[G]-module N, resp.
ℤp[G]-homomorphism h, we set
Nℂp:=ℂp⊗ℤpN,resp. hℂp:=ℂp⊗ℤph.
For any choice of respective splittings
s1:Xℂp→Mℂp⊕im(Θ)ℂp and s2:Xℂp→ker(π)ℂp⊕A(F)ℂp*
of the short exact sequences induced by scalar extension of
0→M→𝜄X→Θim(Θ)→0 and 0→ker(π)→X→𝜋A(F)p*→0
respectively,
we write 〈λA,FNT,j∘Φℂp,Θ,s1,s2〉 for the composite ℂp[G]-automorphism of Xℂp given by
Xℂp→s1Mℂp⊕im(Θ)ℂp
→(Φℂp,id)At(F)ℂp⊕im(Θ)ℂp
→(λA,FNT,j,id)A(F)ℂp*⊕im(Θ)ℂp=A(F)ℂp*⊕ker(π)ℂp→s2-1Xℂp.
We also write X∙ for the perfect complex of ℤp[G] modules X⟶ΘX with the first term placed in degree 1 and the modules
H1(X∙) and H2(X∙) identified with M and A(F)p* respectively via the top row of diagram (4.4).
By unwinding the definition of refined Euler characteristics and using their basic functoriality properties one then finds that,
independently of the choice of splittings s1 and s2, one has
-χG,p(CA,Ff,∙,(λA,FNT,j)-1)=-χG,p(X∙,Φℂp-1∘(λA,FNT,j)-1)
=δG,p(detℂp[G](〈λA,FNT,j∘Φℂp,Θ,s1,s2〉)).
The proof of equality (4.1), and hence of Theorem 2.9, will thus be achieved by the following explicit computation.
Proposition 4.4There exist splittings s1 and s2 as above with the property that
detℂp[G](〈λA,FNT,j∘Φℂp,Θ,s1,s2〉)=∑ψ∈G^j(λψ(𝒫,𝒫t))ϵψ(Φ)δψeψ.
Proof.
Let {w(s,i):s=0,…,n,i∈[ms]} be the standard basis of X. For each (s,i) we write
Ws=W(s,i) for the kernel of the canonical map
ℂp[G]→ℂp[G/Hs],
so that Ws=(σps-1)ℂp[G]=(1-eHs)ℂp[G]. We then have a commutative diagram
with furthermore ⊕(s,i)W(s,i) equal to im(Θ)ℂp=ker(π)ℂp.
We now fix the required splittings s1 and s2 by summing over all pairs (s,i) the splittings of the short exact sequences in (4.5) given by
(4.6)ℂp[G]→ℂp[G/Hs]⊕Ws,1↦(1|Hs|,σps-1),
(4.7)ℂp[G]→ℂp[G/Hs]P(s,i)*⊕Ws,1↦(P(s,i)*,1-eHs),
respectively. Note that for the inverse map in (4.7) we have
(P(s,i)*,0)↦eHs and (0,σps-1)↦σps-1.
After these preparations we proceed to compute the matrix ΛNT(Φ) which represents 〈λA,FNT,j∘Φℂp,Θ,s1,s2〉
with respect to the fixed ℂp[G]-basis {w(s,i)} of Xℂp. From (4.6) and (2.5) it follows easily that the composite
of s1 and (Φℂp,id) maps w(s,i) to
(1|Hs|∑(t,j)Φ(t,j),(s,i)P(t,j)t,(…,σps-1,…))
in
At(F)ℂp⊕im(Θ)ℂp=(⊕(t,j)ℂp[G/Ht]P(t,j)t)⊕(⊕(t,j)Wt)
with the only non-zero component in ⊕(t,j)Wt at the
(s,i)-spot. By Lemma 4.5 below this is further mapped by (λA,FNT,j,id) to
(1|Hs|∑(t,j)Φ(t,j),(s,i)∑(u,k)(∑τ∈G/Huj(〈τP(u,k),P(t,j)t〉F)τeHu)P(u,k)*,(…,σps-1,…)).
Rearranging the summation and applying the map s2-1 as described in (4.7) we obtain
∑(u,k)(1|Hs|∑(t,j)Φ(t,j),(s,i)∑τ∈G/Huj(〈τP(u,k),P(t,j)t〉F)τeHu)w(u,k)+(σps-1)w(s,i).
We now fix a character ψ∈G^. We have that
ψ(1|Hs|∑(t,j)Φ(t,j),(s,i)∑τ∈G/Huj(〈τP(u,k),P(t,j)t〉F)τeHu)={1|Hs|∑(t,j)Φ(t,j),(s,i)∑τ∈G/Huj(〈τP(u,k),P(t,j)t)〉Fψ(τ),u≥tψ,0,u<tψ,
while ψ(σps-1) is equal to 0 if and only if s≥tψ.
We immediately obtain that
det(ψ(ΛNT(Φ)))=j(λψ(𝒫,𝒫t))⋅εψ(Φ)⋅δψ,
as required.
∎
We finally provide the relevant lemma used in the course of the above proof.
Lemma 4.5We have
λA,FNT,j(P(t,j)t)=∑(u,k)(∑τ∈G/Huj(〈τP(u,k),P(t,j)t〉F)τeHu)P(u,k)*.
Proof.
Recall that λA,FNT,j is induced by 〈⋅,⋅〉F:A(F)×At(F)→ℂ. For Pt∈At(F)
we explicitly have
λA,FNT,j(Pt)=j(〈⋅,Pt〉F).
Let f∈A(F)p* denote the map defined by the right-hand side
of the equation in Lemma (4.5). From (4.3) we immediately see that
eHuP(u,k)*=P(u,k)*. For each pair (v,l) and γ∈G/Hv we hence have that
f(γP(v,l))=∑(u,k)∑τ∈G/Huj(〈τP(u,k),P(t,j)t〉F)(τP(u,k)*)(γP(v,l))
=j(〈γP(v,l),P(t,j)t〉F)
=(λA,FNT,j(P(t,j)t))(γP(v,l)).∎
4.3 The proof of Theorem 2.6
We set MA,F=h1(AF)(1) and let us recall that the approach used in [9] in order to formulate the conjecture C(A,ℚ[G]) relies upon the theory of
categories of virtual objects. Although focused on the study of p-parts of the relevant equivariant Tamagawa numbers for prime numbers p, the exact same techniques
involved in the proof of [12, Proposition 4.2] allow one to translate this more technical language into the one of refined Euler characteristics employed throughout
this article. For this reason, we will avoid any explicit mention of categories of virtual objects throughout this proof. Furthermore, since we only need to consider
the relevant realisations and cohomology spaces associated to MA,F as modules over the semisimple algebras ℚ[G] or ℝ[G], we may directly reformulate
C(A,ℚ[G]) in terms of the determinants of certain endomorphisms of free ℝ[G]-modules, which is what we do in the sequel.
The leading term L*(MA,F,0) at s=0 of the ℚ[G]-equivariant motivic L-function of MA,F is given by
∑χ∈Ir(G)eχL*(A,χˇ,1).
We let once again
λA,FNT:ℝ⊗ℤAt(F)→ℝ⊗ℤA(F)*
denote the canonical isomorphism induced by the Néron–Tate height pairing and
αA,F:ℝ⊗ℚ⊕v∈S∞FH0(Fv,Hv(MA,F))→ℝ⊗ℚHdR(MA,F)/F0
denote the canonical period isomorphism described by Deligne in [14] (see also [9, Section 3]
for a general description of the modules involved and [12, Section 4.3] for more details in the relevant special case).
We now note that the ℚ[G]-modules
X:=ℚ⊗ℤAt(F) and Y:=ℚ⊗ℤA(F)*,
resp.
Z:=⊕v∈S∞FH0(Fv,Hv(MA,F)) and W:=HdR(MA,F)/F0,
are isomorphic (as a consequence, for instance, of [1, p. 110]) and hence, since ℚ[G] is semisimple, there exist ℚ[G]-modules M and N
with the property that both X⊕M≅Y⊕M and Z⊕N≅W⊕N are free ℚ[G]-modules. In the sequel we (choose bases and so) fix identifications of
X⊕M, Y⊕M, Z⊕N and W⊕N with direct sums of copies of ℚ[G] and hence regard λA,FNT⊕idℝ⊗ℚM and
αA,F⊕idℝ⊗ℚN as elements of K1(ℝ[G]).
The validity of conjecture C(A,ℚ[G]) is then equivalent
to the containment
L*(MA,F,0)/(detℝ[G](αA,F⊕idℝ⊗ℚN)detℝ[G](λA,FNT⊕idℝ⊗ℚM))∈ℚ[G]×.
But it is clear from the proof of Lemma 4.5 that, independently of our choice of fixed identifications,
(4.8)detℝ[G](λA,FNT⊕idℝ⊗ℚM)/∑χ∈Ir(G)eχλχ(𝒫,𝒫t)∈ℚ[G]×,
and it is straightforward to deduce from the proof of [12, Lemma 4.5] that, independently of our choice of fixed identifications,
(4.9)detℝ[G](αA,F⊕idℝ⊗ℚN)/∑χ∈Ir(G)eχw∞(k)d⋅Ω(A/k)τ*(ℚ,indkℚ(χ))d∈ℚ[G]×.
Equalities (4.8) and (4.9), combined with the fact that the Euler factors involved in the truncation of each of the leading terms
LSr*(A,ψˇ,1) live by definition in ℚ[G]×, therefore imply
that the validity of C(A,ℚ[G]) is equivalent to the containment
∑ψ∈G^ℒψ*λψ(𝒫,𝒫t)eψ∈ℚ[G]×.
By [2, Lemma 2.9] this containment is equivalent to the explicit condition described in Theorem 2.6, as required.
4.4 The proof of Theorem 2.8
We assume now that C(A,ℚ[G]) is valid and proceed to prove the explicit interpretation of Cp(A,ℳ) claimed in Theorem 2.8.
We begin by noting that, for any fixed isomorphism of fields j:ℂ→ℂp, the respective maps jG,* restrict to give the vertical arrows in a natural
commutative diagram with exact rows of the form
We note that the exactness of the rows follows from [9, Lemma 11]. We now proceed to prove several useful results.
Lemma 4.6Conjecture Cp(A,M) holds if and only if
jG,*(TΩ(h1(AF)(1),ℤ[G]))∈K0(ℤp[G],ℚp[G])tor.
Proof.
The equality
TΩ(h1(AF)(1),ℳ)=μ(TΩ(h1(AF)(1),ℤ[G]))
proved in [9, Theorem 4.1]
combines with the commutativity of the right-hand square of diagram (4.10) to imply that
jG,*(TΩ(h1(AF)(1),ℳ))=μp(jG,*(TΩ(h1(AF)(1),ℤ[G]))).
The exactness of the bottom row of diagram (4.10) thus completes the proof.
∎
Lemma 4.7Conjecture Cp(A,M) holds if and only if
∑ψ∈G^ℒψ*j(λψ(𝒫,𝒫t))ϵψ(Φ)δψeψ∈ℳp×.
Proof.
Lemma 4.6 combines with equalities (3.3) and (4.1) to imply that Cp(A,ℳ) holds if and only if
μp(δG,p(∑ψ∈G^ℒψ*j(λψ(𝒫,𝒫t))ϵψ(Φ)δψeψ))=0.
We next note that the respective maps δG,p induce vertical (bijective) arrows in a commutative diagram of the form
This completes the proof of the lemma.
∎
Lemma 4.8We have εψ(Φ)∈Zp[ψ]×.
Proof.
The map
Φ⊗ℳp:M⊗ℤp[G]ℳp→At(F)p⊗ℤp[G]ℳp
is an isomorphism of
ℳp-modules. Since ℳp contains the ℚp[G]-rational idempotents,
Φ⊗ℤp[ψ]:M⊗ℤp[G]ℤp[ψ]→At(F)p⊗ℤp[G]ℤp[ψ]
is an isomorphism of ℤp[ψ]-modules. It is now easy to see that Φ⊗ℤp[ψ] is represented by ψ(Φtψ).
∎
We now proceed to give the proof of Theorem 2.8.
The equivalence of (i) and (iv) follows directly upon combining Lemmas 4.7 and 4.8.
Furthermore it is straightforward to compute the valuation of each element δψ. One has
v𝔭ψ(δψ)=bψ
with
bψ defined as in Theorem 2.8, and hence (iii) and (iv) are clearly equivalent.
In order to prove the equivalence of (i) and (ii), we will use (a special case of) a general fact which we now describe. If H is any subgroup
of G, we write
ρHG:K0(ℤp[G],ℚp[G])→K0(ℤp[H],ℚp[H])
for the natural
restriction map and
q0H:K0(ℤp[H],ℚp[H])→K0(ℤp,ℚp)
for the
natural map induced by sending an element [P,ϕ,Q] of K0(ℤp[H],ℚp[H]) to the element [PH,ϕH,QH] of K0(ℤp,ℚp).
By [6, Theorem 4.1] one then has that
(4.11)K0(ℤp[G],ℚp[G])tor=⋂H≤Gker(q0H∘ρHG).
The functoriality properties of the element TΩ(h1(AF)(1),ℤ[G]) with respect to the maps ρHG and q0H proved in
[9, Proposition 4.1] then imply that, for any subgroup H of G,
(q0H∘ρHG)(jG,*(TΩ(h1(AF)(1),ℤ[G])))=j0,*(TΩ(h1(AFH)(1),ℤ)),
and so Lemma 4.6 combines with (4.11) to imply that Cp(A,ℳ) holds if and only if, for every intermediate field L
of F/k, the element jGL/L,*(TΩ(h1(AL)(1),ℤ)) vanishes, that is, if and only if the p-part of the eTNC holds for the pair
(h1(AL)(1),ℤ). Noting that it is easy to check that the set of data (A/L,L/L,p) satisfies all the hypotheses of Theorem 2.9
for any such field L (see for instance [11, Lemma 3.4] for a proof of a more general assertion), all that
is left to do in order to prove the equivalence of (i) and (ii) is to apply Theorem 2.9. Indeed,
any choice of ℤ-bases {Qi} and {Rj} of A(L) and At(L) respectively satisfy condition (2.1) for the set of data (A/L,L/L,p),
while an explicit computation proves that
τ*(ℚ,indLℚ(𝟏GL/L))w∞(L)=|dL|.
4.5 The proof of Corollary 2.11
For brevity we set
λψ:=λψ(𝒫,𝒫t),ϵψ:=ϵψ(Φ),u:=∑ψ∈G^ℒψ*λψϵψδψeψ.
By Theorem 2.9 the validity of Cp(A,ℤ[G]) is equivalent to the containment u∈ℤp[G]×, which we assume holds throughout the proof.
We also let ε:ℤp[G]→ℤp denote the augmentation map.
We begin by noting that claim (ii) is just the ψ=𝟏G special case of Lemma 4.8, and proceed now to deduce claim (iii) from it.
One clearly has that
ℒ𝟏G*/(λ𝟏Gϵ𝟏Gδ𝟏G)=ε(u)∈ℤp× with δ𝟏G equal by
definition to 1, while a straightforward computation shows that
τ*(ℚ,indkℚ(𝟏G))w∞(k)=(-1)|Sr||dk|.
Claim (ii) therefore indeed implies that
(4.12)v=ℒ𝟏G*/λ𝟏G=ε(u)⋅ϵ𝟏G=ε(u)⋅ϵ
belongs to ℤp×, as required.
In order to prove the remaining claims, we first note that, if rk(A(FJ))=|G/J|rk(A(k)) for every subgroup J of G, then h=0 by Proposition 2.2 while Φ can be chosen to be the identity matrix by property (P2) and each element δψ is simply equal to 1 by convention. In any such case, claim (i) therefore reduces to the trivial statement ℒ=u∈ℤp[G] while claim (iv) simply reads
u≡v(modIG,p) and follows directly from (4.12). We therefore may and will henceforth assume that
the inequality rk(A(FJ))≤|G/J|rk(A(k)) of Corollary 2.3 is strict for some subgroup J of G. We recall that H=Ht0 denotes
the smallest non-trivial subgroup of G with the property that mH≠0.
In order to prove claim (i), we note first that for each ψ∈G^ we have
ψ|H≠1⇔ker(ψ)⊆Ht0 and ker(ψ)≠Ht0⇔tψ>t0.
From the definitions of ϵψ and δψ we immediately deduce that, for each character ψ∈G^ such that ψ|H≠1,
ϵψ=1,δψ=δ:=∏j=0t0(σpj-1)mj.
Since δeψ=0 for each ψ such that ψ|H=1, we deduce that ℒ=δu∈δℤp[G]⊆IG,ph, as required.
Finally, claim (iv) follows from (4.12) because u is clearly congruent to ε(u)=v/ϵ modulo IG,p and therefore ℒ=δu is congruent to δvϵ modulo IG,ph+1, as required.
4.6 The proof of Theorem 2.12
We begin by defining a (free) ℤp[G]-submodule
P:=⊕j∈[mn]ℤp[G]P(n,j)*
of A(F)p* and then fix, as we may, an injective lift
κ:P→Selp(AF)∨
of the inclusion
P⊆A(F)p* through the canonical projection of (2.3).
We also fix, as we may, a representative of the perfect complex CA,Ff,∙ of the form C1→C2 in which both C1 and C2 are finitely generated,
cohomologically-trivial ℤp[G]-modules.
We then obtain a commutative diagram with exact
rows and columns of the form
in which we have set
N:=⊕t<n,j∈[mt]ℤp[G/Ht]
and we simply define the arrow im(κ)→C2, as we may since im(κ) is a free ℤp[G]-module, by the commutativity of the upper right square.
The ℤp[G]-modules D1 and D2 are finitely generated and cohomologically-trivial,
and hence the central arrow of the bottom row of this diagram defines an object D∙ of Dp(ℤp[G]) which is acyclic outside of
degrees 1 and 2 and has identifications of H1(D∙) with N and of H2(D∙) with
cok(κ). We analogously define an object B∙ of Dp(ℤp[G]) represented by
the perfect complex of ℤp[G]-modules
⊕jℤp[G]P(n,j)t→0im(κ).
Following [10, Section 2.1.4] we next define an idempotent
eN:=∑ψ∈ΥNeψ
in ℚp[G] by letting ΥN be the subset
of G^ comprising characters ψ with the property that
eψ(ℂp⊗ℤpN)=0.
For any object
C∙ of Dp(ℤp[G]) we then obtain an object
eNC∙:=eNℤp[G]⊗ℤp[G]𝕃C∙
of Dp(eNℤp[G]). In particular, the exact triangle represented by diagram
(4.13) induces an exact triangle in Dp(eNℤp[G]) of the form
(4.14)eNB∙→eNCA,Ff,∙→eND∙→eNB∙[1].
But eND∙⊗eNℤp[G]eNℂp[G] is
acyclic and an immediate application of the additivity criterion of [5, Corollary 6.6] to triangle (4.14) implies that one has
(4.15)-χeNℤp[G],eNℂp[G](eND∙,0)
=-χeNℤp[G],eNℂp[G](eNCA,Ff,∙,eN(λA,FNT,j)-1)
+χeNℤp[G],eNℂp[G](eNB∙,λ′),
where λ′ denotes the canonical isomorphism
eN(ℂp⊗ℤpim(κ))=eN(ℂp⊗ℤpSelp(AF)∨)
→eN(λA,FNT,j)-1eN(ℂp⊗ℤpAt(F)p)=eN(ℂp⊗ℤp⊕jℤp[G]P(n,j)t).
If we now write
φ:⊕jℤp[G]P(n,j)t→⊕jℤp[G]P(n,j)*
for the canonical isomorphism that maps an element P(n,j)t to the element
P(n,j)*, then one finds that
(4.16)χeNℤp[G],eNℂp[G](eNB∙,λ′)
=δeNℤp[G],eNℂp[G](deteNℂp[G](λ′∘eN(ℂp⊗ℤp(κ∘φ))))
=-δeNℤp[G],eNℂp[G](∑ψ∈ΥNj(λψ(𝒫,𝒫t))eψ),
where the last equality follows from Lemma 4.5.
The assumed validity of Cp(A,ℤ[G]) therefore combines via (3.4) with equalities (4.15) and (4.16) to imply that, in the
terminology of [10, Section 2.3.2],
the element
j*(∑ψ∈ΥNℒψ*λψ(𝒫,𝒫t)-1eψ)=j*(ℒ)
of eNℂp[G] is a characteristic element for eND∙. The result [10, Lemma 2.6] therefore implies that there exists a characteristic element ℒ′ for
D∙ in ℂp[G] with the property that
eNℒ′=j*(ℒ).
Since D∙ is clearly an admissible complex
of ℤp[G]-modules (in the terminology of [10, Section 2.1.1]), the results of [10, Corollary 3.3] therefore imply
that the element j*(ℒ) belongs to the ideal IG,ph~
of ℤp[G], with h~:=dimℚp(ℚp⊗ℤpcok(κ)G), and furthermore
generates Fittℤp[G](cok(κ)). To proceed with the proof, we first note that
h~=dimℚp(ℚp⊗ℤpcok(κ)G)=dimℚp(ℚp⊗ℤp⊕t<nℤp[G/Ht]G〈mt〉)=∑t<nmt=h.
We have hence proved that j*(ℒ) belongs to IG,ph, as stated in claim (i) of Theorem 2.12. Furthermore, Π:=im(κ) is clearly a finitely generated, free ℤp[G]-submodule of Selp(AF)∨ of maximal rank mn, and so the fact that the element j*(ℒ)
generates Fittℤp[G](cok(κ)) proves claim (iii) of Theorem 2.12. To complete the proof, it is enough to note that, since im(κ) is torsion-free,
the canonical composite homomorphism
Шp(AF)∨→∼(Selp(AF)∨)tor⊆Selp(AF)∨→cok(κ)
is injective and hence that one has that
Fittℤp[G](cok(κ))⊆Annℤp[G](Шp(AF)∨).
Recalling finally that the Cassels–Tate pairing induces a canonical isomorphism between Шp(AF)∨ and Шp(AFt) completes the proof of claim (ii) and thus of Theorem 2.12.
5 Examples
In this section we gather some evidence, mostly numerical, in support of conjecture Cp(A,ℤ[G]).
Our aim is to verify statements that would not follow in an straightforward manner from the validity of the Birch and Swinnerton-Dyer conjecture for all intermediate fields
of F/k. Because of the equivalence of statements (i)–(ii) in Theorem 2.8 we therefore choose not to focus on presenting evidence for conjecture
Cp(A,ℳ) (although we also used our MAGMA programs to produce numerical evidence
for Cp(A,ℳ) by verifying statement (iii) of Theorem 2.8).
Throughout this section A will always denote an elliptic curve.
5.1 Verifications of conjecture Cp(A,ℤ[G])
For the verification of Cp(A,ℤ[G]) using Theorem 2.9 it is necessary to have explicit knowledge
of a map Φ that represents the extension class δA,F,p. Whenever A(F)p is not projective as a ℤp[G]-module,
we are currently not able to numerically compute Φ, so we only deal with examples in which A(F)p is projective. To the best of our knowledge there are currently three instances of theoretical evidence (in situations in which our fixed cyclic extension F/k is not trivial):
In [4], it is shown that for each elliptic curve A/ℚ with the property that L(A/ℚ,1)≠0 there are infinitely many
primes p and for each such prime p infinitely many (cyclic) p-extensions F/ℚ such that Cp(A,ℤ[Gal(F/ℚ)]) holds.
All of these examples satisfy our hypotheses and are such that A(F)p vanishes.
In [12, Theorem 1.1], conjecture Cp(A,ℤ[Gal(F/ℚ)]) is proved for certain elliptic curves A/ℚ,
where F denotes the Hilbert p-classfield of an imaginary quadratic field k. This result combines with the functoriality properties of the eTNC to
imply the validity of conjecture Cp(A,ℤ[Gal(F/k)]). In these examples one has that A(F)p is a free ℤp[Gal(F/k)]-module of rank one.
In [12, Corollary 6.2], certain S3-extensions F/K are considered. Let k and L denote the quadratic and cubic
subfield of F/K respectively. Under certain additional assumptions it is then shown that the validity of the Birch and Swinnerton-Dyer conjecture for A over the fields k,K and L implies
the validity of Cp(A,ℤ[Gal(F/K)]). Again by functoriality arguments, the validity of
Cp(A,ℤ[Gal(F/k)]) follows. We note that the assumptions are such that one again has that A(F)p is a free ℤp[Gal(F/k)]-module of rank one.
In the rest of this section we are concerned with numerical evidence. In [2, Section 6] there is a list of examples
of elliptic curves A/ℚ and dihedral extensions F/ℚ of order 2p for which conjecture Cp(A,ℤ[Gal(F/ℚ)]) is numerically verified.
Here the quadratic subfield k is real and A(F)p vanishes. Again by functoriality arguments we obtain examples where conjecture Cp(A,ℤp[Gal(F/k)]) is numerically
verified. There are two further analogous numerical verifications in dihedral examples in [12, Section 6.3], one of degree 10 and one of
degree 14, both of them with the property that A(F)p is a free ℤp[Gal(F/k)]-module of rank one.
In the following we fix an odd prime p and let q denote a prime such that q≡1(modp). Let F denote
the unique subfield of ℚ(ζq)/ℚ of degree p and take k to be ℚ.
For p∈{3,5,7} and q<50 we went through the list of semistable elliptic curves of rank one and conductor N<200 and
checked numerically whether L(A/ℚ,χ,1)=0 and L′(A/ℚ,χ,1)≠0 for a non-trivial character χ of G, and in addition,
whether our hypotheses are satisfied.
This resulted in a list of 50 examples (27 for p=3, 20 for p=5 and 3 for p=7). In each of these examples we could find
a point R such that A(F)p=ℤp[G]R and numerically verify conjecture Cp(A,ℤ[G]).
We now describe in detail an example with [F:ℚ]=7.
Let A be the elliptic curve
A:y2+xy+y=x3+x2-2x.
This is the curve 79a1 in Cremona’s notation. It is known that A(ℚ) is free of rank one generated by
P1=(0,0) and that Ш(Aℚ)=0. Moreover it satisfies the hypotheses used throughout the paper.
We take p=7 and let F be the unique subfield of ℚ(ζ29) of degree 7. Explicitly,
F is the splitting field of
f(x)=x7+x6-12x5-7x4+28x3+14x2-9x+1
and we let α denote a root of f.
Using the MAGMA command Points it is easy to find a point R of infinite order in A(F)∖A(ℚ),
R=(117(31α6+23α5-373α4-135α3+814α2+372α-86),
117(-35α6-83α5+380α4+771α3-811α2-1321α+232)).
By Proposition 2.2 we know that A(F)p is a permutation module, hence A(F)p≃ℤp[G]. Furthermore,
[11, Proposition 3.1] now implies that Шp(AF)=0.
We set Q1:=TrF/ℚ(R)=(34,-38) and easily verify that Q1=-4P1.
We checked numerically that ℤp[G]R=A(F)p.
Computing numerical approximations to the leading terms using Dokchitser’s MAGMA implementation of
[15] we obtain the following vector for (ℒχ*/λχ(𝒫,𝒫t))χ∈G^:
(-0.077586206896551724152,
-0.49999999999999999992+2.1906431337674115362i,
-0.49999999999999999996+0.62698016883135191886i,
-0.49999999999999999998-0.24078730940376432202i,
-0.49999999999999999992-2.1906431337674115362i,
-0.49999999999999999996-0.62698016883135191886i,
-0.49999999999999999998+0.24078730940376432202i).
This is very close to
(-9116,ζ73+ζ72+ζ7,-ζ75-ζ74-ζ7-1,-ζ75-ζ73-ζ7-1,
-ζ73-ζ72-ζ7-1,ζ75+ζ74+ζ7,ζ75+ζ73+ζ7).
It is now easy to verify the rationality conjecture C(A,ℚ[G]) by the criterion of Theorem 2.6.
Moreover, the valuations of -9/116 and ζ73+ζ72+ζ7 at 𝔭χ
are 0, so that by Theorem 2.8 we deduce the validity of Cp(A,ℳ).
Finally, one easily checks that
-9116≡ζ73+ζ72+ζ7(mod(1-ζ7)),
so that
the element in (2.7) is actually a unit in ℤp[G], thus (numerically) proving Cp(A,ℤ[G]).
5.2 Evidence in support of conjecture Cp(A,ℤ[G])
In this subsection we collect evidence for statements that we have shown to follow from the validity of Cp(A,ℤ[G]) and focus on situations in which A(F)p is not
ℤp[G]-projective. In particular, we aim to verify claim (i) of Theorem 2.12.
Since we can neither compute the module Шp(AF) nor a map Φ as required, we are not able to verify any other claim of either Corollary 2.11 or Theorem 2.12.
Again we want to focus on evidence which goes beyond implications of the Birch and Swinnerton-Dyer conjecture for A over all intermediate fields of F/k.
We assume the notation of Theorem 2.12, so in particular set h=∑t<nmt. If k=ℚ and mn=0, then
the element ℒ is essentially the Mazur–Tate modular element (see [20]) and the validity of the Birch and Swinnerton-Dyer conjecture would imply that
it belongs to IG,p in the type of situations under consideration. Hence, if mn=0, we only searched for examples where h>1.
If F/k is cyclic of order p, then IG,ph=(σ-1)hℳp for all h≥1. Letting u and δ denote
the elements defined in the proof of Corollary 2.11, we hence note that, if Cp(A,ℳ) is valid, then u∈ℳ× and the proof of Corollary 2.11 clearly shows that ℒ=δu is contained in δℳp=IG,p.
We therefore further restricted our search for interesting examples to cases where [F:k]=pn with n≥2.
Restricted by the complexity of the computations and the above considerations
we are therefore lead to consider the following types of examples:
A(F)p≃ℤpm0⊕ℤp[G/H1]m1⊕ℤp[G]m2, [F:ℚ]=32, (m0,m1,m2)=(1,1,0),
A(F)p≃ℤpm0⊕ℤp[G/H1]m1⊕ℤp[G]m2, [F:ℚ]=32,
(m0,m1,m2)=(m0,0,0), m0≥2.
We note that, whenever Шp(AF) is trivial, the validity of Cp(A,ℤ[G])
implies via Corollary 2.11 that h is the exact order of vanishing, i.e., that
ℒ∈IG,ph∖IG,ph+1.
However, this need not be true if Шp(AF) is non-trivial. In such cases, by Theorem 2.12 (iii), Cp(A,ℤ[G]) does predict
that ℒ generates the Fitting ideal of Selp(AF)∨ since mn=0 immediately implies Π=0.
Let q denote a prime such that q≡1(mod32). We let F denote
the unique subfield of ℚ(ζq)/ℚ of degree 9 and take k to be ℚ.
We checked two examples of type (i), namely those given by the pairs
(A,q)∈{(681c1,19),(1070a1,19)}.
In both cases we were able
to find a points P0 and P1 such that A(F)p=ℤpP0⊕ℤp[G/H]P1, where H denotes the subgroup of order 3. Each time we numerically
found that Шp(AF)=0 (predicted by the Birch and Swinnerton-Dyer conjecture for A over F) and verified that h is the precise order of vanishing, as predicted
by Corollary 2.11.
Concerning examples of type (ii) went through the list of semistable elliptic curves of rank two and conductor N<750 and
produced by numerically checking L-values and derivatives a list of twelve examples satisfying the necessary hypotheses. In each of these examples we had h=m0=2 and
could numerically verify the containment ℒ∈IG,p2. Whenever Шp(AF) was trivial, we also checked that ℒ∉IG,p3.
We finally present one example in detail.
Let A be the elliptic curve
A:y2+y=x3+x2-2x.
This is the curve 389a1 in Cremona’s notation. It is known that A(ℚ) is free of rank two generated by
P1=(0,0) and P2=(-1,1) and that Ш(Aℚ)=0. Moreover it satisfies the hypotheses required to apply Theorem 2.12 (see Remark
2.13).
Computing numerical approximations to the leading terms we find that the order of vanishing at each non-trivial character is 0.
The rank part of the Birch and Swinnerton-Dyer conjecture therefore predicts that rk(A(F))=2.
We checked that 〈P1,P2〉ℤ is 3-saturated in A(F) and therefore (conjecturally) conclude that
A(F)p=〈P1,P2〉ℤp≃ℤp2.
The Birch and Swinnerton-Dyer conjecture predicts that |Шp(AF)|=81. We therefore cannot test for the precise
order of vanishing.
Computing leading terms, periods and regulators we find the following numerical approximations to
(ℒχ*/λχ(𝒫,𝒫t))χ∈G^:
(-1.243243,1.500000+2.598076i,1.500000-2.598076i,
0.358440+2.032818i,0.286988-0.104455i,-3.645429+3.058878i,
0.358440-2.032818i,0.286988+0.104455i,-3.645429-3.058878i).
The actual computation was done with a precision of 30 decimal digits.
This is very close to
(-4637,3ζ3+3,-3ζ3,
2ζ93-ζ92+2ζ9,-ζ94-2ζ93+2ζ92-2,ζ95+2ζ94+2ζ93+ζ92,
-2ζ95+ζ94-2ζ93-2ζ92+ζ9-2,-ζ95-2ζ94+2ζ93-2ζ9,2ζ95-2ζ93-ζ9-2).
It is now easy to verify the rationality conjecture C(A,ℚ[G]) by the criterion of Theorem 2.6.
We finally find that
ℒ=-σ+2σ2-σ3+2σ5-2σ6-2σ7+2σ8
and easily check that ℒ∈IG,p2.
und
Daniel Macias Castillo
