In this section, we fix notations and recall results from [5] that we shall need.
Cyclotomic extension and Iwasawa algebra
Choose once and for all an odd prime p. Let F be a number field unramified at p. We fix \(\overline{F}\) an algebraic closure of F and denote by \(G_F = {{\,\mathrm{Gal}\,}}(\overline{F}/F)\) the absolute Galois group of F. If v is a prime of F, we denote by \(F_v\) the completion of F at v, \(\mathcal {O}_{F_v}\) its ring of integers and \(G_{F_v}\) the decomposition subgroup of v in \(G_F\). Let \(\mu _{p^n}\) be the group of \(p^n\)th roots of unity for every \(n \geqslant 1\) and \(\mu _{p^\infty } = \cup _{n \geqslant 1} \mu _{p^n}\). We set \(F(\mu _{p^\infty }) = \cup _{n \geqslant 1} F(\mu _{p^n})\) the \(p^\infty \)-cyclotomic extension of F inside \(\overline{F}\). For every \(n \geqslant 1\), we choose a generator \(\epsilon ^{(n)}\) of \(\mu _{p^n}\) with the compatibilities \((\epsilon ^{(n+1)})^p = \epsilon ^{(n)}\), so that \(\varprojlim _n \epsilon ^{(n)}\) is a generator of \(\varprojlim _n \mu _{p^n} \simeq \mathbf {Z}_p(1)\). The cyclotomic character \(\chi : G_F \rightarrow \mathbf {Z}_p^*\) is defined by the relations \(g(\epsilon ^{(n)}) = (\epsilon ^{(n)})^{\chi (g)}\) and it induces a isomorphism \(\chi : {{\,\mathrm{Gal}\,}}(F(\mu _{p^\infty })/F) \simeq \mathbf {Z}_p^*\). In particular, the group \({{\,\mathrm{Gal}\,}}(F(\mu _{p^\infty })/F)\) decomposes as \(\Gamma \times \Delta \) with \(\Gamma \simeq \mathbf {Z}_p\) and \(\Delta \simeq \mathbf {Z}/(p-1)\mathbf {Z}\). For every \(n \geqslant 0\), we denote by \(\Gamma _n\) the unique subgroup of \(\Gamma \) of index \(p^n\). We set \(F_\infty = F(\mu _{p^\infty })^\Delta \) and \(F_n = F_\infty ^{\Gamma _n}\) for every \(n \geqslant 0\).
For \(n \geqslant 1\), we set \(\Lambda _n = \mathbf {Z}_p[{{\,\mathrm{Gal}\,}}(F(\mu _{p^n}/F)]\). Let \(\Lambda = \mathbf {Z}_p[[{{\,\mathrm{Gal}\,}}(F(\mu _{p^\infty })/F)]] = \varprojlim _n \Lambda _n\) be the Iwasawa algebra of \({{\,\mathrm{Gal}\,}}(F(\mu _{p^\infty })/F)\) over \(\mathbf {Z}_p\). The above decomposition of \({{\,\mathrm{Gal}\,}}(F(\mu _{p^\infty })/F)\) implies that \(\Lambda = \mathbf {Z}_p[\Delta ][[\Gamma ]]\). Furthermore, we have an isomorphism \(\mathbf {Z}_p[[\Gamma ]] \simeq \mathbf {Z}_p[[X]]\) induced by \(\gamma \mapsto X+1\) where \(\gamma \) is a topological generator of \(\Gamma \). For \(n \geqslant 1\), let \(\omega _n(X) = (X+1)^p-1\), then this isomorphism induces \(\Lambda _n \simeq \mathbf {Z}_p[\Delta ][X]/(\omega _n)\).
For a Dirichlet character \(\eta \) on \(\Delta \) and a \(\Lambda \)-module R, let \(R^\eta \) be the isotypic component of R, which is given by \(e_\eta \cdot R\) where \(e_\eta = \frac{1}{| \Delta |} \sum _{\delta \in \Delta } \eta ^{-1}(\delta )\delta \). Note that \(R^\eta \) is naturally a \(\mathbf {Z}_p[[\Gamma ]]\)-module. We will say that a \(\Lambda \)-module R has rank r if \(R^\eta \) has rank r over \(\mathbf {Z}_p[[\Gamma ]]\) for all characters \(\eta \) on \(\Delta \).
Given a finitely generated torsion \(\mathbf {Z}_p[[\Gamma ]]\)-module R, there exists a pseudo-isomorphism (i.e. a morphism of \(\mathbf {Z}_p[[\Gamma ]]\)-modules with finite kernel and cokernel)
$$\begin{aligned} R \rightarrow \bigoplus _{i=1}^n \mathbf {Z}_p[[\Gamma ]]/(p^{l_i}) \oplus \bigoplus _{j=1}^m \mathbf {Z}_p[[\Gamma ]]/(f_j^{k_j}) \end{aligned}$$
where \(f_j \in \mathbf {Z}_p[X]\) are distinguished irreducible polynomials (identifying \(\mathbf {Z}_p[[\Gamma ]]\) and \(\mathbf {Z}_p[[X]]\)). Furthermore, the ideals \((p^{l_j})\) and \((f_j^{k_j})\) are uniquely determined by R up to ordering. The characteristic ideal of R is then defined by \(\prod _{i,j} (p^{l_i})\cdot ( f_j^{k_j} ) \subset \mathbf {Z}_p[[\Gamma ]]\). The \(\mu \)-invariant of R is defined by \(\sum _{i=1}^n l_i\) and the \(\lambda \)-invariant of R by \(\sum _{j=1}^m k_j\cdot \deg f_j\).
Motives
Let \(\mathcal {M}\) be a motive defined over F with coefficients in \(\mathbf {Q}\) in the sense of [8]. We denote by \(\mathcal {M}_p\) its p-adic realization and we fix T a \(G_F\)-stable \(\mathbf {Z}_p\)-lattice inside \(\mathcal {M}_p\). Let \(g = \dim _{\mathbf {Q}_p}({{\,\mathrm{Ind}\,}}_F^\mathbf {Q}\mathcal {M}_p)\) and \(g_+ = \dim _{\mathbf {Q}_p}({{\,\mathrm{Ind}\,}}_F^\mathbf {Q}\mathcal {M}_p)^+\) the dimension of the \(+1\)-eigenspace under the action of a fixed complex conjugation on \({{\,\mathrm{Ind}\,}}_F^\mathbf {Q}\mathcal {M}_p\). We set \(g_- = g - g_+\). For every prime v of F dividing p, let \(g_v = \dim _{\mathbf {Q}_p}({{\,\mathrm{Ind}\,}}_{F_v}^{\mathbf {Q}_p} \mathcal {M}_p)\). We have \(g = \sum _{v \mid p} g_v\).
We will assume that, for every prime v of F dividing p,
- (H.-T.):
the Hodge–Tate weights of \(\mathcal {M}_p\), as a \(G_{F_v}\)-representation, are in [0, 1],
- (Cryst.):
the \(G_{F_v}\)-representation \(\mathcal {M}_p\) is crystalline,
- (Tors.):
the Galois cohomology groups \({{\,\mathrm{H}\,}}^0(F_v,T/pT)\) and \({{\,\mathrm{H}\,}}^2(F_v,T/pT)\) are trivial.
We denote by \(T^* = {{\,\mathrm{Hom}\,}}(T,\mathbf {Z}_p(1))\) the Tate dual of T and we set
$$\begin{aligned} M = T \otimes \mathbf {Q}_p/\mathbf {Z}_p, \quad \text {and}, \quad M^* = T^* \otimes \mathbf {Q}_p/\mathbf {Z}_p. \end{aligned}$$
We remark that the dual of \(\mathcal {M}\), which we denote by \(\mathcal {M}^*\), satisfies the hypothesis (Cryst.) and (H.-T.), and \(T^*\), which is a \(G_F\)-stable \(\mathbf {Z}_p\)-lattice inside its p-adic realization \(\mathcal {M}_p^*\), satisfies (Tors.).
We also fix \(\Sigma \) a finite set of primes of F containing the primes dividing p, the archimedean primes and the primes of ramification of \(M^*\). Let \(F_\Sigma \) be the maximal extension of F unramified outside \(\Sigma \), so that \(M^*\) is a \({{\,\mathrm{Gal}\,}}(F_\Sigma /F)\)-module. We remark that \(F(\mu _{p^\infty }) \subseteq F_\Sigma \) since only primes above p and \(\infty \) can be ramified in \(F(\mu _{p^\infty })\). If \(F^\prime \) is an extension of F in \(F(\mu _{p^\infty })\), we will say by abuse that a prime of \(F^\prime \) lies in \(\Sigma \) if it divides a prime of F which is in \(\Sigma \).
Dieudonné modules
If v is a prime of F dividing p, let \(\mathbf {D}_{\mathrm {cris},v}(T)\) be the Dieudonné module associated to T considered as a \(G_{F_v}\)-representation [2, Définition V.1.1]. Then \(\mathbf {D}_{\mathrm {cris},v}(T)\) is a free \(\mathcal {O}_{F_v}\)-module of rank \(\dim _{\mathbf {Q}_p} \mathcal {M}_p\) equipped with a filtration of \(\mathcal {O}_{F_v}\)-modules \(({{\,\mathrm{Fil}\,}}^i \mathbf {D}_{\mathrm {cris},v}(T))_{i \in \mathbf {Z}}\) such that
$$\begin{aligned} {{\,\mathrm{Fil}\,}}^i \mathbf {D}_{\mathrm {cris},v}(T) = \left\{ \begin{array}{l@{\quad }l} 0 &{} \text {for } i \geqslant 1, \\ \mathbf {D}_{\mathrm {cris},v}(T) &{} \text {for } i \leqslant -1. \end{array} \right. \end{aligned}$$
Furthermore, \(\mathbf {D}_{\mathrm {cris},v}(\mathcal {M}_p) :=\mathbf {D}_{\mathrm {cris},v}(T) \otimes \mathbf {Q}_p\) is the usual Fontaine’s filtered \(\phi \)-module associated to \(\mathcal {M}_p\).
We will assume that
- (Fil.):
\(\sum _{v \mid p} \dim _{\mathbf {Q}_p} {{\,\mathrm{Fil}\,}}^0 \mathbf {D}_{\mathrm {cris},v}(T) \otimes \mathbf {Q}_p= g_-\),
- (Slopes):
the slopes of \(\phi \) are in \(]-1,0[\).
We may choose \(\{u_1,\ldots ,u_{g_v}\}\) a \(\mathbf {Z}_p\)-basis of \(\mathbf {D}_{\mathrm {cris},v}(T)\) such that \(\{u_1,\ldots ,u_{d_v}\}\) is a basis for \({{\,\mathrm{Fil}\,}}^0\mathbf {D}_{\mathrm {cris},v}(T)\) for some \(d_v\). We call such a basis a Hodge-compatible basis and fix one for the rest of the paper. Then, from our hypotheses, the matrix of the crystalline Frobenius \(\phi \) with respect to this basis is of the form
$$\begin{aligned} C_{\phi ,v} = C_v \left( \begin{array}{c|c} I_{d_v} &{} 0 \\ \hline 0 &{} \frac{1}{p} I_{g_v - d_v} \end{array}\right) \end{aligned}$$
(1)
where \(C_v \in {{\,\mathrm{GL}\,}}_{g_v}(\mathbf {Z}_p)\) and \(I_n\) is the identity matrix of size n.
Let \(\mathbf {D}_{\mathrm {cris},v}(T^*)\) be the Dieudonné module associated to \(T^*\). There is a natural pairing
$$\begin{aligned} \mathbf {D}_{\mathrm {cris},v}(T) \times \mathbf {D}_{\mathrm {cris},v}(T^*) \rightarrow \mathbf {D}_{\mathrm {cris},v}(\mathbf {Z}_p(1)) \simeq \mathbf {Z}_p, \end{aligned}$$
(2)
with respect to which \({{\,\mathrm{Fil}\,}}^i \mathbf {D}_{\mathrm {cris},v}(T^*)\) is the orthogonal complement of \({{\,\mathrm{Fil}\,}}^{-i}\mathbf {D}_{\mathrm {cris},v}(T)\) and \(\phi ^{-1}\) is the dual of \(p\phi \). In particular, \(\mathbf {D}_{\mathrm {cris},v}(T^*)\) also satisfies the hypotheses (Fil.) and (Slopes).
Example 1.1
Let A be an abelian variety defined over F with good supersingular reduction at every prime dividing p. Let \(T_p(A)=\varprojlim _n A[p^n]\) be the p-adic Tate module of A and let \(V_p(A)=T_p(A)\otimes _{\mathbf {Z}_p}\mathbf {Q}_p\). Then \(V_p(A)\) is a \(G_F\)-representation and \(T_p(A)\) a \(G_F\)-stable \(\mathbf {Z}_p\)-lattice of \(V_p(A)\) which satisfy all the hypotheses (Crys.), (H.-T.), (Tors.), (Fil.) and (Slopes).
Decomposition of Perrin–Riou’s big logarithm map
Let v be a prime of F dividing p. For \(i \geqslant 0\), the projective limit of the Galois cohomology groups \({{\,\mathrm{H}\,}}^i(F_v(\mu _{p^n}),T)\) relative to the corestriction maps is denoted by \({{\,\mathrm{H}\,}}_{\mathrm {Iw}}^i(F_v,T)\). Recall that \({\text {H}}_{\mathrm {Iw}}^1(F_v,T)\) is a \(\Lambda \)-module of rank \(g_v\) [26, Proposition A.2.3 ii)].
We set \(\mathcal {H}= \mathbf {Q}_p[\Delta ] \otimes _{\mathbf {Q}_p}\mathcal {H}(\Gamma )\) where \(\mathcal {H}(\Gamma )\) is the set of elements \(f(\gamma - 1)\) with \(\gamma \in \Gamma \) and \(f(X) \in \mathbf {Q}_p[[X]]\) is convergent on the p-adic open unit disk. Perrin–Riou’s big logarithm map is a \(\Lambda \)-homomorphism [25]
$$\begin{aligned} \mathcal {L}_{T,v} : {\text {H}}_{\mathrm {Iw}}^1(F_v,T) \rightarrow \mathcal {H}\otimes _{\mathbf {Z}_p} \mathbf {D}_{\mathrm {cris},v}(T) \end{aligned}$$
which interpolates Kato’s dual exponential maps [13, II Sect. 1.2]
$$\begin{aligned} \exp _n^* : {{\,\mathrm{H}\,}}^1(F_v(\mu _{p^n}),T) \rightarrow F_v(\mu _{p^n}) \otimes _{\mathbf {Z}_p} \mathbf {D}_{\mathrm {cris},v}(T). \end{aligned}$$
As in [5], we may define for \(n \geqslant 1\),
$$\begin{aligned} C_{v,n} = \left( \begin{array}{c|c} I_{d_v} &{} 0 \\ \hline 0 &{} \Phi _{p^n}(1+X) I_{g_v - d_v} \end{array}\right) C_v^{-1} \quad \text {and} \quad M_{v,n} = (C_{\phi ,v})^{n+1} C_{v,n} \cdots C_1, \end{aligned}$$
(3)
where \(\Phi _{p^n}\) is the \(p^n\)th cyclotomic polynomial. By Proposition 2.5 in op. cit., the sequence \((M_{v,n})_{n \geqslant 1}\) converges to some \(g_v \times g_v\) logarithmic matrix over \(\mathcal {H}\), which we denote by \(M_v\). This allows to decompose \(\mathcal {L}_{T,v}\) into
$$\begin{aligned} \mathcal {L}_{T,v} = (u_1, \ldots , u_{g_v}) \cdot M_v \cdot \begin{pmatrix} {{\,\mathrm{Col}\,}}_{T,v,1}\\ \vdots \\ {{\,\mathrm{Col}\,}}_{T,v,g_v} \end{pmatrix} \end{aligned}$$
(4)
where \({{\,\mathrm{Col}\,}}_{T,v,i}, i \in \{1, \ldots ,g_v\}\) are \(\Lambda \)-homomorphisms from \({\text {H}}_{\mathrm {Iw}}^1(F_v,T)\) to \(\Lambda \). More details on the decomposition (4) are given in Sect. 3.2.
Signed Coleman maps
Let \(I_v\) be a subset of \(\{1,\ldots ,g_v\}\). We set
$$\begin{aligned} {{\,\mathrm{Col}\,}}_{T,I_v} : {\text {H}}_{\mathrm {Iw}}^1(F_v,T) \rightarrow \bigoplus _{i = 1}^{|I_v|} \Lambda , \quad \mathbf {z} \mapsto ({{\,\mathrm{Col}\,}}_{T,v,i}(\mathbf {z}))_{i \in I_v}. \end{aligned}$$
These maps are called signed Coleman maps. We recall results about them that we shall need.
Lemma 1.2
([5, Proposition 2.20, Lemma 3.22])
- 1.
For any character \(\eta \) on \(\Delta \), the \(\eta \)-isotypic component of the image of the signed Coleman map \({{\,\mathrm{Im}\,}}{{\,\mathrm{Col}\,}}_{T,I_v}^\eta \) is a \(\mathbf {Z}_p[[\Gamma ]]\)-module of rank \(|I_v|\) contained in a free \(\mathbf {Z}_p[[\Gamma ]]\)-module with finite index.
- 2.
The \(\Lambda \)-module \({{\,\mathrm{Ker}\,}}{{\,\mathrm{Col}\,}}_{T,I_v}\) is free of rank \(g_v - |I_v|\).
Let
$$\begin{aligned} \langle \cdot , \cdot \rangle _n : {{\,\mathrm{H}\,}}^1(F_v(\mu _{p^n}),T) \times {{\,\mathrm{H}\,}}^1(F_v(\mu _{p^n}),T^*) \rightarrow {{\,\mathrm{H}\,}}^2(F_v(\mu _{p^n}),\mathbf {Z}_p(1)) \simeq \mathbf {Z}_p, \end{aligned}$$
be Tate’s local pairing. If \(x = (x_n)_n\) and \(y = (y_n)_n\) are elements of \({\text {H}}_{\mathrm {Iw}}^1(F_v,T)\) and \({\text {H}}_{\mathrm {Iw}}^1(F_v,T^*)\) then the elements
$$\begin{aligned} \sum _{\sigma \in {{\,\mathrm{Gal}\,}}(F_v(\mu _{p^n})/F_v)}\langle x_n, \sigma (y_n) \rangle \sigma \in \mathbf {Z}_p[{{\,\mathrm{Gal}\,}}(F_v(\mu _{p^n})/F_v)], \end{aligned}$$
are compatible under the natural projection maps
$$\begin{aligned} \mathbf {Z}_p[{{\,\mathrm{Gal}\,}}(F_v(\mu _{p^{n+1}})/F_v)] \rightarrow \mathbf {Z}_p[{{\,\mathrm{Gal}\,}}(F_v(\mu _{p^n})/F_v)], \end{aligned}$$
thus, they define an element in \(\Lambda \). This defines Perrin–Riou’s pairing
$$\begin{aligned} {\text {H}}_{\mathrm {Iw}}^1(F_v,T) \times {\text {H}}_{\mathrm {Iw}}^1(F_v,T^*) \rightarrow \Lambda . \end{aligned}$$
(5)
Since all our hypotheses (Crys.), (H.-T.), (Tors.), (Fil.) and (Slopes) are satisfied by \(\mathcal {M}^*\) and \(T^*\), we carry out all of the constructions of paragraph 1.4 for \(T^*\) with respect to the dual basis of our fixed basis \(\{u_1,\ldots ,u_{g_v}\}\) for the pairing (2) and similarly define signed Coleman maps for \(T^*\).
Then, we have the following relation.
Lemma 1.3
([21, Lemma 3.2]) Let \(I_v\) be a subset of \(\{1,\ldots ,g_v\}\) and \(I_v^c\) its complement. Then \({{\,\mathrm{Ker}\,}}{{\,\mathrm{Col}\,}}_{T,I_v}\) is the orthogonal complement of \({{\,\mathrm{Ker}\,}}{{\,\mathrm{Col}\,}}_{T^*,I_v^c}\) relative to Perrin–Riou’s pairing (5).
Remark 1.4
In [21], there is an additional hypothesis that \(g_+ = g_-\) and F is abelian over \(\mathbf {Q}\) with degree prime to p. However, the proof of Lemma 3.2 in op. cit. applies in the setting considered in the present article in verbatim.
Signed Selmer groups
Let \(\underline{I} = (I_v)_{v \mid p}\) be a tuple of sets indexed by the primes v of F dividing p and where each \(I_v\) is a subset of \(\{1, \ldots ,g_v\}\).
Tate’s local pairing
$$\begin{aligned} {{\,\mathrm{H}\,}}^1(F_v(\mu _{p^n}),M^*) \times {{\,\mathrm{H}\,}}^1(F_v(\mu _{p^n}),T) \rightarrow {{\,\mathrm{H}\,}}^2(F_v(\mu _{p^n}),\mu _{p^\infty })\simeq \mathbf {Q}_p/\mathbf {Z}_p\end{aligned}$$
passes to the limit relative to restriction and corestriction and defines a pairing
$$\begin{aligned} {{\,\mathrm{H}\,}}^1(F_v(\mu _{p^\infty }),M^*) \times {\text {H}}_{\mathrm {Iw}}^1(F_v,T) \rightarrow \mathbf {Q}_p/\mathbf {Z}_p. \end{aligned}$$
(6)
Definition 1.5
We define \({{\,\mathrm{H}\,}}^1_{I_v}(F_v(\mu _{p^\infty }),M^*) \subseteq {{\,\mathrm{H}\,}}^1(F_v(\mu _{p^\infty }),M^*)\) as the orthogonal complement of \({{\,\mathrm{Ker}\,}}{{\,\mathrm{Col}\,}}_{T,I_v}\) under the pairing (6).
The assumption \({{\,\mathrm{H}\,}}^2(F_v,T/pT) = 0\)(Tors.) implies by Tate’s duality that \({{\,\mathrm{H}\,}}^0(F_v,M^*)=0\), thus \({{\,\mathrm{H}\,}}^0(F_{v,\infty },M^*) = 0\) since \({{\,\mathrm{Gal}\,}}(F_{v,\infty }/F_v) \simeq \mathbf {Z}_p\) is a pro-p-group. In particular, by the inflation-restriction exact sequence, we have
$$\begin{aligned} {{\,\mathrm{H}\,}}^1(F_{v,\infty },M^*) \simeq {{\,\mathrm{H}\,}}^1(F_v(\mu _{p^\infty }),M^*)^\Delta \end{aligned}$$
since the order of \(\Delta \) is \(p-1\) and \({{\,\mathrm{H}\,}}^0(F_v(\mu _{p^\infty }),M^*)\) is finite of order a power of p, and for \(n \geqslant 0\), we have
$$\begin{aligned} {{\,\mathrm{H}\,}}^1(F_{v,n},M^*) \simeq {{\,\mathrm{H}\,}}^1(F_{v,\infty },M^*)^{\Gamma _n}. \end{aligned}$$
We set
$$\begin{aligned} {{\,\mathrm{H}\,}}^1_{I_v}(F_{v,\infty },M^*) = {{\,\mathrm{H}\,}}^1_{I_v}(F_v(\mu _{p^\infty }),M^*)^\Delta \quad \text {and} \quad {{\,\mathrm{H}\,}}^1_{I_v}(F_{v,n},M^*) = {{\,\mathrm{H}\,}}^1_{I_v}(F_{v,\infty },M^*)^{\Gamma _n}. \end{aligned}$$
We also have signed Coleman maps for \(T^*\). For \(n \geqslant 0\), let \(({{\,\mathrm{Ker}\,}}{{\,\mathrm{Col}\,}}_{T^*,I_v^c})_n\) be the image of \({{\,\mathrm{Ker}\,}}{{\,\mathrm{Col}\,}}_{T^*,I_v^c}\) under the natural map \({\text {H}}_{\mathrm {Iw}}^1(F_v,T^*) \rightarrow {{\,\mathrm{H}\,}}^1(F_{v,n},T^*)\). Again by (Tors.), we have the exact sequence
$$\begin{aligned} 0 \rightarrow {{\,\mathrm{H}\,}}^1(F_{v,n},T^*) \xrightarrow {i_n} {{\,\mathrm{H}\,}}^1(F_{v,n},\mathcal {M}_p^*) \xrightarrow {\pi _n} {{\,\mathrm{H}\,}}^1(F_{v,n},M^*)\rightarrow 0. \end{aligned}$$
(7)
The image of \(({{\,\mathrm{Ker}\,}}{{\,\mathrm{Col}\,}}_{T^*,I_v^c})_n\) under \(i_n\) generates a \(\mathbf {Q}_p\)-vector space in \({{\,\mathrm{H}\,}}^1(F_{v,n},\mathcal {M}_p^*)\), and we denote by \(\overline{({{\,\mathrm{Ker}\,}}{{\,\mathrm{Col}\,}}_{T^*,I_v^c})_n}\) the image of this \(\mathbf {Q}_p\)-vector space in \({{\,\mathrm{H}\,}}^1(F_{v,n},M^*)\) under \(\pi _n\).
Lemma 1.6
For any \(n \geqslant 0\), \(\overline{({{\,\mathrm{Ker}\,}}{{\,\mathrm{Col}\,}}_{T^*,I_v^c})_n} \) is the orthogonal complement of \(({{\,\mathrm{Ker}\,}}{{\,\mathrm{Col}\,}}_{T,I_v})_n\) under Tate’s local pairing
$$\begin{aligned} {{\,\mathrm{H}\,}}^1(F_{v,n},M^*) \times {{\,\mathrm{H}\,}}^1(F_{v,n},T) \rightarrow {{\,\mathrm{H}\,}}^2(F_{v,n},\mu _{p^\infty })\simeq \mathbf {Q}_p/\mathbf {Z}_p. \end{aligned}$$
Moreover, \(\overline{({{\,\mathrm{Ker}\,}}{{\,\mathrm{Col}\,}}_{T^*,I_v^c})_n} = {{\,\mathrm{H}\,}}^1_{I_v}(F_{v,n},M^*)\). In particular, \({{\,\mathrm{H}\,}}^1_{I_v}(F_{v,n},M^*)\) is a divisible group.
Proof
By Lemma 1.3 and bilinearity of Tate’s pairing, the orthogonal complement of \(({{\,\mathrm{Ker}\,}}{{\,\mathrm{Col}\,}}_{T,I_v})_n\) under Tate’s pairing contains \(\overline{({{\,\mathrm{Ker}\,}}{{\,\mathrm{Col}\,}}_{T^*,I_v^c})_n}\). The reverse inclusion follows from the exactness of the sequence (7). As already remarked, by (Tors.), one has \({{\,\mathrm{H}\,}}^1(F_{v,n},M^*) = {{\,\mathrm{H}\,}}^1(F_v(\mu _{p^\infty }),M^*)^{\Gamma _n\times \Delta }\) and by duality \({\text {H}}_{\mathrm {Iw}}^1(F_v,T)_{\Gamma _n\times \Delta } = {{\,\mathrm{H}\,}}^1(F_{v,n},T)\). It follows that \(\overline{({{\,\mathrm{Ker}\,}}{{\,\mathrm{Col}\,}}_{T^*,I_v^c})_n} = {{\,\mathrm{H}\,}}^1_{I_v}(F_{v,n},M^*)\). \(\square \)
Let w be a prime of F not dividing p and let K be a finite extension of \(F_w\). Define
$$\begin{aligned} {\text {H}}_{\mathrm {unr}}^1(K,\mathcal {M}^*_p) = {{\,\mathrm{Ker}\,}}({{\,\mathrm{H}\,}}^1(K,\mathcal {M}^*_p) \rightarrow {{\,\mathrm{H}\,}}^1(K_\mathrm {unr},\mathcal {M}^*_p)) \end{aligned}$$
where \(K_\mathrm {unr}\) the maximal unramified extension of K. Let \({\text {H}}_{\mathrm {unr}}^1(K,M^*)\) be the image of \({\text {H}}_{\mathrm {unr}}^1(K,\mathcal {M}^*_p)\) under the natural map
$$\begin{aligned} {{\,\mathrm{H}\,}}^1(K,\mathcal {M}^*_p) \rightarrow {{\,\mathrm{H}\,}}^1(K,M^*) \end{aligned}$$
and \({\text {H}}_{\mathrm {unr}}^1(K,T^*)\) the inverse image of \({\text {H}}_{\mathrm {unr}}^1(K,\mathcal {M}^*_p)\) under
$$\begin{aligned} {{\,\mathrm{H}\,}}^1(K,T^*) \rightarrow {{\,\mathrm{H}\,}}^1(K,\mathcal {M}_p^*). \end{aligned}$$
We remark that \({\text {H}}_{\mathrm {unr}}^1(K,M^*)\) is divisible by definition and recall that it is the orthogonal complement of \({\text {H}}_{\mathrm {unr}}^1(K,T)\) under Tate’s local pairing (see [3, Proposition 3.8]). If \(K^\prime \) is an infinite algebraic extension of \(F_w\), we define the subgroup
$$\begin{aligned} {\text {H}}_{\mathrm {unr}}^1(K^\prime ,M^*)=\varinjlim _{K} {\text {H}}_{\mathrm {unr}}^1(K,M^*) \subset {{\,\mathrm{H}\,}}^1(K^\prime ,M^*) \end{aligned}$$
where the limit runs through the finite extensions K of \(F_w\) contained in \(K^\prime \) and is taken with respect to the restriction maps.
Let \(F^\prime \) be one of \(F(\mu _{p^\infty })\), \(F_\infty \) or \(F_n\) for some \(n\geqslant 0\). We set
$$\begin{aligned} \mathcal {P}_{\Sigma ,\underline{I}}(M^*/F^\prime ) = \prod _{w \in \Sigma , w \not \mid p} \frac{{{\,\mathrm{H}\,}}^1(F_w^\prime ,M^*)}{{\text {H}}_{\mathrm {unr}}^1(F_w^\prime ,M^*)} \times \prod _{w \mid p}\frac{{{\,\mathrm{H}\,}}^1(F_w^\prime ,M^*)}{{{\,\mathrm{H}\,}}^1_{I_v}(F_w^\prime ,M^*)}. \end{aligned}$$
Definition 1.7
Let \(F^\prime \) be \(F(\mu _{p^\infty })\), \(F_\infty \), or \(F_n\) for some \(n \geqslant 0\). The \(\underline{I}\)-Selmer group of \(M^*\) over \(F^\prime \) is defined by
$$\begin{aligned} {{\,\mathrm{Sel}\,}}_{\underline{I}}(M^*/F^\prime ) = {{\,\mathrm{Ker}\,}}({{\,\mathrm{H}\,}}^1(F_\Sigma /F^\prime ,M^*) \rightarrow \mathcal {P}_{\Sigma ,\underline{I}}(M^*/F^\prime )) \end{aligned}$$
where the map is the composition of localization at each \(w\in \Sigma \) followed by the projection in the appropriate quotient.
Let \(\mathcal {I}_p\) be the set of tuples \(\underline{I} = (I_v)_{v \mid p}\) indexed by the primes v of F dividing p and where each \(I_v\) is a subset of \(\{1,\ldots ,g_v\}\) and such that \(\sum _{v \mid p} |I_v| = g_-\). From observations about the expected \(\Lambda \)-corank of the Selmer group of a supersingular abelian variety, Büyükboduk and Lei have made the following conjecture [5, Remark 3.27].
Conjecture 1.8
For any \(\underline{I} \in \mathcal {I}_p\) and any even Dirichlet character \(\eta \) on \(\Delta \), \({{\,\mathrm{Sel}\,}}_{\underline{I}}(M^*/F(\mu _{p^\infty }))^\eta \) is a cotorsion \(\mathbf {Z}_p[[\Gamma ]]\)-module (i.e. its Pontryagin dual is a torsion \(\mathbf {Z}_p[[\Gamma ]]\)-module).
Remark 1.9
When \(F=\mathbf {Q}\) and \(\mathcal {M}\) is the Tate module of a supersingular elliptic curve with \(a_p=0\), for a good choice of basis of the Dieudonné module, the signed Selmer groups with \(\underline{I} \in \mathcal {I}_p\) coincide with Kobayashi plus and minus Selmer groups [19] (see [5, Appendix 4]). Conjecture 1.8 is known in that case op. cit.. Furthermore, Sprung [6] as well as Lei, Loeffler and Zerbes [20] have proved that this conjecture holds in cases of p-supersingular elliptic curves with \(a_p\ne 0\) and p-non-ordinary eigenforms, respectively.
Remark 1.10
The definition of the signed Selmer groups does not depend on the choice of \(\Sigma \). If \(\eta \) is the trivial character on \(\Delta \), then \({{\,\mathrm{Sel}\,}}_{\underline{I}}(M^*/F(\mu _{p^\infty }))^\eta \simeq {{\,\mathrm{Sel}\,}}_{\underline{I}}(M^*/F_\infty )\). It follows from the definition that, for any \(\underline{I}\), the Pontryagin dual of \({{\,\mathrm{Sel}\,}}_{\underline{I}}(M^*/F_\infty )\) is a finitely generated \(\mathbf {Z}_p[[\Gamma ]]\)-module since \({{\,\mathrm{H}\,}}^1(F_\Sigma /F_\infty ,M^*)\) is [9, Proposition 3]. In the remainder of this article, we study these Selmer groups.
In the next section, we shall need twisted signed Selmer groups. Let us explain now what they are. For \(s \in \mathbf {Z}\), we set \(M^*_s = M^* \otimes \chi ^s_{| \Gamma }\) where \(\chi _{|\Gamma } : \Gamma \simeq \mathbf {Z}_p\). As a \({{\,\mathrm{Gal}\,}}(\overline{F}/F_\infty )\)-module, \(M_s^* = M^*\), thus \({{\,\mathrm{H}\,}}^1(F_\infty ,M^*_s) = {{\,\mathrm{H}\,}}^1(F_\infty ,M^*)\otimes \chi ^s_{| \Gamma }\) and for a prime v of F, \({{\,\mathrm{H}\,}}^1(F_{v,\infty },M^*_s) = {{\,\mathrm{H}\,}}^1(F_{v,\infty },M^*)\otimes \chi ^s_{| \Gamma }\) and \({{\,\mathrm{H}\,}}^0(F_{v,\infty },M^*_s)=0\). At primes dividing p, we set
$$\begin{aligned} {{\,\mathrm{H}\,}}^1_{I_v}(F_{v,\infty },M^*_s) = {{\,\mathrm{H}\,}}^1_{I_v}(F_{v,\infty },M^*) \otimes \chi ^s_{| \Gamma }. \end{aligned}$$
Therefore, for \(F^\prime \) being \(F_\infty \) or \(F_n\) for some \(n \geqslant 0\), we can define twisted \(\underline{I}\)-Selmer groups \({{\,\mathrm{Sel}\,}}_{\underline{I}}(M^*_s/F^\prime )\) as above with local condition at p induced by \({{\,\mathrm{H}\,}}_{I_v}^1(F_{v,\infty },M^*_s)\). We remark that \({{\,\mathrm{Sel}\,}}_{\underline{I}}(M^*_s/F_\infty ) \simeq {{\,\mathrm{Sel}\,}}_{\underline{I}}(M^*/F_\infty ) \otimes \chi ^s_{| \Gamma }\) as \(\mathbf {Z}_p[[\Gamma ]]\)-modules.
Similarly, we can define signed Selmer groups for M using the signed Coleman maps \({{\,\mathrm{Col}\,}}_{T^*,I_v}\), as well as twisted signed Selmer groups for M as above. We remark that if \(\underline{I}\) is an element of \(\mathcal {I}_p\), then \(\underline{I}^c = (I^c_v)_{v \mid p}\) satisfies \(\sum _{v \mid p} I_v^c = g - g_- = g_+ = \dim _{\mathbf {Q}_p} ({{\,\mathrm{Ind}\,}}_F^\mathbf {Q}\mathcal {M}_p^*)^-\). In particular, Conjecture 1.8 is expected to hold for the signed Selmer groups of M.
Bloch–Kato’s Selmer groups
Let \(n \geqslant 0\) and w be a prime of \(F_n\) dividing p. We recall that Bloch and Kato [3] defined the \(\mathbf {Q}_p\)-subspace of \({{\,\mathrm{H}\,}}^1(F_{n,w},\mathcal {M}_p^*)\)
$$\begin{aligned} {{\,\mathrm{H}\,}}^1_f(F_{n,w},\mathcal {M}_p^*) = {{\,\mathrm{Ker}\,}}({{\,\mathrm{H}\,}}^1(F_{n,w},\mathcal {M}_p^*) \rightarrow {{\,\mathrm{H}\,}}^1(F_{n,w},\mathbf {B}_{\mathrm {cris}}\otimes \mathcal {M}_p^*)) \end{aligned}$$
where \(\mathbf {B}_{\mathrm {cris}}\) is Fontaine’s ring of crystalline periods [7]. Let \({{\,\mathrm{H}\,}}^1_f(F_{n,w},M^*)\) be the image of \({{\,\mathrm{H}\,}}^1_f(F_{n,w},\mathcal {M}_p^*)\) under the natural map
$$\begin{aligned} {{\,\mathrm{H}\,}}^1(F_{n,w},\mathcal {M}_p^*) \rightarrow {{\,\mathrm{H}\,}}^1(F_{n,w},M^*). \end{aligned}$$
We set
$$\begin{aligned} \mathcal {P}_{\Sigma ,f}(M^*/F_n) = \prod _{w \in \Sigma , w \not \mid p} \frac{{{\,\mathrm{H}\,}}^1(F_{n,w},M^*)}{{\text {H}}_{\mathrm {unr}}^1(F_{n,w},M^*)} \times \prod _{w \mid p} \frac{{{\,\mathrm{H}\,}}^1(F_{n,w},M^*)}{{{\,\mathrm{H}\,}}^1_f(F_{n,w},M^*)}. \end{aligned}$$
Then, the Bloch–Kato’s Selmer group of \(M^*\) over \(F_n\) is defined by
$$\begin{aligned} {{\,\mathrm{Sel}\,}}_{\mathrm {BK}}(M^*/F_n) = {{\,\mathrm{Ker}\,}}({{\,\mathrm{H}\,}}^1(F_\Sigma /F_n,M^*) \rightarrow \mathcal {P}_{\Sigma ,f}(M^*/F_n)) \end{aligned}$$
and we set \({{\,\mathrm{Sel}\,}}_{\mathrm {BK}}(M^*/F_\infty ) = \varinjlim _n {{\,\mathrm{Sel}\,}}_{\mathrm {BK}}(M^*/F_n)\).
Recall that the definition of the signed Coleman maps and thus of the signed Selmer groups depends on a choice of Hodge-compatible basis of \(\oplus _{v \mid p} \mathbf {D}_{\mathrm {cris},v}(T)\).
Lemma 1.11
([4, Lemma 8.1]) There exists a Hodge-compatible basis of \(\oplus _{v \mid p} \mathbf {D}_{\mathrm {cris},v}(T)\) such that for any \(\underline{I} \in \mathcal {I}_p\)
$$\begin{aligned} {{\,\mathrm{H}\,}}^1_f(F_v,M^*) = {{\,\mathrm{H}\,}}^1_{I_v}(F_v,M^*). \end{aligned}$$
In particular, for such a basis,
$$\begin{aligned} {{\,\mathrm{Sel}\,}}_{\mathrm {BK}}(M^*/F) = {{\,\mathrm{Sel}\,}}_{\underline{I}}(M^*/F). \end{aligned}$$
The basis of the lemma is a strongly admissible basis in the sense of [5, Definition 3.2].