Integral Iwasawa theory of galois representations for non-ordinary primes

Abstract

In this paper, we study the Iwasawa theory of a motive whose Hodge–Tate weights are 0 or 1 (thence in practice, of a motive associated to an abelian variety) at a non-ordinary prime, over the cyclotomic tower of a number field that is either totally real or CM. In particular, under certain technical assumptions, we construct Sprung-type Coleman maps on the local Iwasawa cohomology groups and use them to define integral p-adic L-functions and (one unconditionally and other conjecturally) cotorsion Selmer groups. This allows us to reformulate Perrin–Riou’s main conjecture in terms of these objects, in the same fashion as Kobayashi’s ±-Iwasawa theory for supersingular elliptic curves. By the aid of the theory of Coleman-adapted Kolyvagin systems we develop here, we deduce parts of Perrin–Riou’s main conjecture from an explicit reciprocity conjecture.

Appendices

Appendix 1: Construction of logarithmic matrices via Wach modules

In [14] and [15], we showed that the theory of Wach modules can be used to study the Iwasawa theory of p-adic representations. The key is to find an explicit basis for the Wach module. In this appendix, we show that the construction of the logarithmic matrix $M_T$ in Sect. 2.2 can be modified to construct an explicit basis for the Wach module $\mathbb {N}(T)$ of T. Here T is as defined in Sect. 2.2, satisfying (H.F.-L.) and (H.S.).

Let $\mathbb {A}_{K}^+=\mathcal {O}_K[[\pi ]]$, which is equipped with the usual semi-linear actions by $\Gamma$ and $\varphi$ (see for example [3]). We write $q=\varphi (\pi )/\pi$.

Definition 3.34

A Wach module with weights in [a; b] is a finitely generated free $\mathbb {A}_{K}^+$-module M such that

• (1) It is equipped with a semi-linear action by $\Gamma$ that is trivial modulo $\pi$;

• (2) There is a semi-linear map $\varphi :M[\pi ^{-1}]\rightarrow M[\varphi (\pi )^{-1}]$ such that $\varphi (\pi ^b M)\subset \pi ^b M$ and that $\pi ^bM/\varphi ^*(\pi ^bM)$ is killed by $q^{b-a}$, where $\varphi ^*(\pi ^bM)$ denotes the $\mathbb {A}_{K}^+$-module generated by the image $\varphi (\pi ^bM)$;

• (3) The actions of $\Gamma$ and $\varphi$ commute.

A Wach module N is equipped with a filtration

$$\begin{aligned} {\text {Fil}}^iN=\{x\in N:\varphi (x)\in q^iN\}. \end{aligned}$$

Let $v_1,\ldots ,v_d$ be an $\mathcal {O}_K$-basis of $\mathbb {D}_K(T)$ such that $v_1,\ldots ,v_{d_0}$ generate ${\text {Fil}}^0\mathbb {D}_K(T)$. Let $C_\varphi$ be the matrix of $\varphi$ with respect to this basis. As in Sect. 2.2,

$$\begin{aligned} C_\varphi =C\left( \begin{array}{c|c} I_{r_0}&{}0\\ \hline 0&{}\frac{1}{p} I_{r-r_0} \end{array} \right) \end{aligned}$$

for some $C\in \mathrm {GL}_d(\mathcal {O}_K)$.

Definition 3.35

For $n\ge 1$, we define

$$\begin{aligned} P_n=C \left( \begin{array}{c|c} I_{r_0}&{}0\\ \hline 0&{}\frac{1}{\varphi ^{n-1}(q)} I_{r-r_0} \end{array} \right) \quad \text {and}\quad M'_n=\left( C_\varphi \right) ^{n}P_{n}^{-1}\cdots P_{1}^{-1}. \end{aligned}$$

Proposition 3.36

The sequence of matrices $\{M'_n\}_{n\ge 1}$ converges entry-wise with respect to the sup-norm topology on $\mathbb {B}_{\mathrm{rig},K}^+$. If $M_T'$ denotes the limit of the sequence, each entry of $M_T'$ are $o(\log (1+X))$. Moreover, $\det (M'_T)$ is, up to a constant in $\mathcal {O}_K^\times$, equal to $\left( \frac{\log (1+\pi )}{\pi }\right) ^g$.

Proof

The proof is the same as that for Proposition 2.5. $\square$

Definition 3.37

For each $\gamma \in \Gamma$, define a matrix $G_\gamma =\left( M_T'\right) ^{-1}\cdot \gamma \left( M_T'\right)$.

We shall show that $G_\gamma$ is a matrix defined over $\mathbb {A}_{K}^+$. Let us first prove the following lemma.

Lemma 3.38

Let $M_{r\times r}(\mathbb {A}_{K}^+)$ be the set of $r\times r$ matrices that are defined over $\mathbb {A}_{K}^+$.

• (a) $P_1\cdot \gamma \left( P_1^{-1}\right) \in I+\pi M_{r\times r}(\mathbb {A}_{K}^+)$;

• (b) If $M\in I+\pi M_{r\times r}(\mathbb {A}_{K}^+)$, then $P_1\cdot \varphi (M)\cdot \gamma (P_1^{-1})\in I+\pi M_{r\times r}(\mathbb {A}_{K}^+)$.

Proof

For (a), we have $P_1\cdot \gamma (P_1^{-1})=C\left( \begin{array}{c|c} I_{r_0}&{}0\\ \hline 0&{}\frac{\gamma \cdot q}{q} I_{r-r_0} \end{array} \right) C^{-1}$ and $\frac{\gamma \cdot q}{q}\in 1+\pi \mathbb {A}_{K}^+$, hence the result.

Let $M=I+\pi N$, then

$$\begin{aligned} P_1\cdot \varphi (M)\cdot \gamma (P_1^{-1})=P_1\gamma \left( P_1^{-1}\right) +\pi \left( qP_1\cdot \varphi (N)\cdot \gamma \left( P_1^{-1}\right) \right) \end{aligned}$$

since $\varphi (\pi )=\pi q$. Both $qP_1$ and $P_1^{-1}$ are defined over $\mathbb {A}_{K}^+$, so (b) follows from (a). $\square$

Proposition 3.39

For all $\gamma$, the matrix $G_\gamma$ is an element of $I+\pi M_{r\times r}(\mathbb {A}_{K}^+)$.

Proof

Since $G_\gamma =\lim _{n\rightarrow \infty }\left( M'_n\right) ^{-1}\cdot \gamma \left( M_n'\right)$, it is enough to show that $\left( M'_n\right) ^{-1}\cdot \gamma \left( M'_n\right)$ is in $I+\pi M_{r\times r}(\mathbb {A}_{K}^+)$ for all n. Let us show this by induction.

We have for all n

$$\begin{aligned} \left( M'_n\right) ^{-1}\cdot \gamma \left( M'_n\right) =P_1\cdots P_{n}\gamma (P_{n}^{-1})\cdots \gamma (P_1^{-1}). \end{aligned}$$

(21)

Hence, the claim for $n=1$ is Lemma 3.38(a).

By definition, $P_n=\varphi ^{n-1}(P_1)$, so we have for $n\ge 2$

$$\begin{aligned} \left( M'_n\right) ^{-1}\cdot \gamma \left( M'_n\right) =P_1\cdot \varphi \left( \left( M'_n\right) ^{-1}\cdot \gamma \left( M'_n\right) \right) \cdot \gamma (P_1^{-1}). \end{aligned}$$

Hence, the inductive step is simply Lemma 3.38(b). $\square$

Lemma 3.40

For all $\gamma$, we have the matrix identity

$$\begin{aligned} P_1\cdot \varphi (G_\gamma )=G_\gamma \cdot \gamma (P_1). \end{aligned}$$

Proof

By (21) and the fact that $P_n=\varphi ^{n-1}(P_1)$, we have

$$\begin{aligned} P_1\cdot \varphi \left( \left( M'_n\right) ^{-1}\cdot \gamma \left( M'_n\right) \right) =P_1\cdots P_{n+1}\gamma (P_{n+1}^{-1}\cdots P_2^{-1}) \end{aligned}$$

and

$$\begin{aligned} \left( \left( M'_n\right) ^{-1}\cdot \gamma \left( M'_n\right) \right) \cdot \gamma (P_1)=P_1\cdots P_n\gamma (P_n^{-1}\cdots P_2^{-1}). \end{aligned}$$

In other words,

$$\begin{aligned} P_1\cdot \varphi \left( \left( M'_n\right) ^{-1}\cdot \gamma \left( M'_n\right) \right) =\left( \left( M'_{n+1}\right) ^{-1}\cdot \gamma \left( M'_{n+1}\right) \right) \cdot \gamma (P_1) \end{aligned}$$

Hence the result follows on taking $n\rightarrow \infty$. $\square$

Definition 3.41

We define a free $\mathbb {A}_{K}^+$-module $N_{C_\varphi }$ of rank r, with basis $n_1,\ldots , n_{r}$. With respect to this basis, we equip $N_{C_\varphi }$ with a semi-linear action by $\Gamma$, which is given by the matrix $G_\gamma$ (well-defined by Proposition 3.39) and a semi-linear map $\varphi :N_{C_\varphi }[\pi ^{-1}]\rightarrow N_{C_\varphi }[\varphi (\pi )^{-1}]$, which is given by the matrix $P_1$.

Proposition 3.42

The module $N_{C_\varphi }$ is a Wach module with weights in [0; 1].

Proof

By Proposition 3.39, the action of $\Gamma$ on $N_{C_\varphi }$ is trivial modulo $\pi$.

Since $P_1\in 1/q M_{r\times r}(\mathbb {A}_{K}^+)$, we have

$$\begin{aligned} \varphi \left( \pi N_{C_\varphi }\right) \in \pi N_{C_\varphi }\quad \text {and}\quad q\varphi \left( N_{C_\varphi }\right) \subset \pi ^bN_{C_\varphi }. \end{aligned}$$

Finally, by Lemma 3.40, the actions of $\Gamma$ and $\varphi$ commute, so we are done. $\square$

Theorem 3.43

As Wach modules, $N_{C_\varphi }$ is isomorphic to $\mathbb {N}(T)$. Furthermore,

$$\begin{aligned} \begin{pmatrix}v_1&\cdots&v_{r}\end{pmatrix}M_T'=\begin{pmatrix}n_1&\cdots&n_{r}\end{pmatrix}. \end{aligned}$$

Proof

In order to show that $N_{C_\varphi }\cong \mathbb {N}(T)$, it is enough to show that

$$\begin{aligned} \mathbb {D}_K(T)\cong N_{C_\varphi }\mod \pi \end{aligned}$$

(22)

as filtered $\varphi$-module by [3, Théorm̀e III.4.4].

By definition $P_1\equiv C_\varphi \mod \pi$, so the actions of $\varphi$ agree on the two sides of (22). For the filtration, we have

$$\begin{aligned} {\text {Fil}}^iN_{C_\varphi }= \left\{ \begin{array}{ll} N_{C_\varphi }&{}\quad i\le -1\\ \left( \bigoplus _{1\le j\le r_0}\mathbb {A}_{K}^+\cdot n_j\right) \oplus \left( \bigoplus _{r_0+1\le j\le r}\mathbb {A}_{K}^+\cdot \pi n_j\right) &{}\quad i=0\\ \left( \bigoplus _{1\le j\le r_0}\mathbb {A}_{K}^+\cdot \pi ^in_j\right) \oplus \left( \bigoplus _{r_0+1\le j\le r}\mathbb {A}_{K}^+\cdot \pi ^{i+1}n_j\right) &{}\quad i\ge 1\\ \end{array} \right. \end{aligned}$$

Since ${\text {Fil}}^0\mathbb {D}(T_p(A))$ is generated by $v_1,\ldots ,v_{r_0}$, we see that the filtrations agree on the two sides of (22) as well.

By [3, §II.3],

$$\begin{aligned} \begin{pmatrix}v_1&\cdots&v_{r}\end{pmatrix}M=\begin{pmatrix}n_1&\cdots&n_{r}\end{pmatrix}. \end{aligned}$$

(23)

for some matrix $M\in I+\pi M_{r\times r}(\mathbb {B}_{\mathrm{rig},K}^+)$. For any $\gamma \in \Gamma$,

$$\begin{aligned} \begin{pmatrix}v_1&\cdots&v_{r}\end{pmatrix}\gamma (M)=\begin{pmatrix}n_1&\cdots&n_{r}\end{pmatrix}G_\gamma . \end{aligned}$$

Therefore, $G_\gamma =M\cdot \gamma (M^{-1})=M_T'\cdot \gamma \left( M_T'\right) ^{-1}$. But $M_T'\in I+\pi M_{r\times r}(\mathbb {B}_{\mathrm{rig},K}^+)$ also. Hence,

$$\begin{aligned} M\cdot \left( M_T'\right) ^{-1}\in \left( I+\pi M_{r\times r}(\mathbb {B}_{\mathrm{rig},K}^+)\right) ^\Gamma . \end{aligned}$$

This implies that $M=M_T'$ as required. $\square$

We now use the theory of Wach modules to prove an integrality result that is used in Proposition 2.9. Recall from [14, §3.1] and [15, §3.1] that for any $x\in \mathbb {N}(T)^{\psi =0}$, we have $(1-\varphi )x\in (\varphi ^*\mathbb {N}(T))^{\psi =0}\subset \mathbb {B}_{\mathrm{rig},K}^+\otimes \mathbb {D}_K(T)$. Furthermore, we have an $\mathcal {O}_K\otimes \Lambda$-basis for $(\varphi ^*\mathbb {N}(T))^{\psi =0}$ of the form $(1+\pi )\varphi (n_1),\ldots ,(1+\pi )\varphi (n_r)$.

Lemma 3.44

Let $x\in \mathbb {N}(T)^{\psi =1}$, then $(1\otimes \varphi ^{-n-1})\circ (1-\varphi )x$ is congruent to an element of $(\mathbb {A}_{K}^+)^{\psi =0}\otimes \mathbb {D}_K(T)$ modulo $\varphi ^{n+1}(\pi )\mathbb {B}_{\mathrm{rig},K}^+\otimes \mathbb {D}_K(T)$.

Proof

By [14, Lemma 3.3], there exists $x_1,\ldots ,x_d\in (\mathbb {A}_{K}^+)^{\psi =0}$ such that

$$\begin{aligned} (1-\varphi )x=\sum _{i=1}^rx_i(1+\pi )\varphi (n_i)=\begin{pmatrix} v_1&\hdots&v_r \end{pmatrix}\cdot C_\varphi \cdot (1+\pi )\varphi (M)\cdot \begin{pmatrix} x_1\\ \vdots \\ x_d \end{pmatrix} . \end{aligned}$$

Note that we have abused notation to write $v_i\cdot (\star )$ for $(\star )\otimes v_i\in \mathbb {B}_{\mathrm{rig},K}^+\otimes \mathbb {D}_K(T)$. Thus, on applying $(1\otimes \varphi ^{-n-1})$, we have

$$\begin{aligned} (1\otimes \varphi ^{-n-1})\circ (1-\varphi )x=\begin{pmatrix} v_1&\hdots&v_r \end{pmatrix}\cdot C_\varphi ^{-n}\cdot \varphi (M)\cdot \begin{pmatrix} x_1\\ \vdots \\ x_d \end{pmatrix} . \end{aligned}$$

Therefore, it is enough to show that $C_\varphi ^{-n}\cdot \varphi (M)$ is congruent to some element in $\mathbb {A}_{K}^+$ modulo $\varphi ^{n+1}(\pi )\mathbb {B}_{\mathrm{rig},K}^+$.

If we apply $\varphi$ to the Eq. (23), we have the relation

$$\begin{aligned} M=C_\varphi \cdot \varphi (M)\cdot P^{-1}. \end{aligned}$$

Since $M\equiv I\mod \pi$, we have $M\equiv C_\varphi \cdot P^{-1}$ modulo $\pi$. On iterating, we have

$$\begin{aligned} M\equiv C_\varphi ^{n}\cdot \varphi ^{n-1}(P^{-1})\cdots P^{-1}\mod \varphi ^n(\pi ), \end{aligned}$$

which implies that

$$\begin{aligned} \varphi (M)\equiv C_\varphi ^{n}\cdot \varphi ^{n}(P^{-1})\cdots \varphi (P^{-1})\mod \varphi ^{n+1}(\pi ). \end{aligned}$$

Recall that $P^{-1}$ is defined over $\mathbb {A}_{K}^+$, hence we are done. $\square$

Appendix 2: Linear algebra: Proof of Proposition 3.3

Our goal in this appendix is to provide a proof of Proposition 3.3.

Lemma 3.45

Let W be a free $\mathbb {Z}_p$-module of rank $\mathfrak {d}$ and let $W^\prime$ be a free, rank $\mathfrak {d}-1$ direct summand of W. Then the collection $\{W^\prime +\mathbb {Z}_p\cdot v: v\in W\}$ of submodules of W is totally ordered (with respect to inclusion).

Proof

This follows from the fact that the quotient $W/W^\prime$ is a free $\mathbb {Z}_p$-module of rank one. $\square$

Lemma 3.46

Let W be as in the previous lemma. Let $\mathfrak {D}$ be a finite collection of rank $\mathfrak {d}-1$ direct summands of W and let $W_0=\cup _{\mathfrak {D}}\,W^\prime$ be their union. For any $k\in \mathbb {Z}^+$ we have,

$$\begin{aligned} p^kW\cup W_0\ne W. \end{aligned}$$

Proof

Choose any element $w=w_0 \in W-W_0$ (such an element clearly exists). If $w_0\not \in p^kW$, we are done, otherwise write $w_0=p^k w_1$. Observe that $w_1\not \in W_0$ (if not, $w_0$ would be an element of $W_0$ as well). Now if $w_1\not \in p^kW$, we are done again. Otherwise we may continue with this process, which eventually has to terminate. $\square$

Lemma 3.47

For $\left( \begin{array}{cc}a &{} b\\ c&{} d \end{array} \right) \in \mathrm {GL}_2(\mathbb {Z}_p)$, the set $\left\{ \frac{ax+by}{cx+dy}: x,y \in \mathbb {Z}_p^\times \right\}$ has infinite cardinality.

Proof

Since $\left( \begin{array}{cc}a &{} b\\ c&{} d \end{array} \right) \in \mathrm {GL}_2(\mathbb {Z}_p)$, either $c\ne 0$ or $d\ne 0$; say the first holds true. Observe that

$$\begin{aligned} \displaystyle {\frac{ax+by}{cx+dy}=\frac{a}{c}-\frac{(ad-bc)/{c}}{cx+dy}}. \end{aligned}$$

Since $ad-bc\ne 0$ and $cx+dy$ takes on infinitely many values as $x,y\in \mathbb {Z}_p^\times$ vary, the proof follows. $\square$

Lemma 3.48

Let $W, \mathfrak {D}$ and $W_0$ be as in Lemma 3.46. Let $W_1,W_2 \in \mathfrak {D}$ and suppose $v_1,v_2 \in W-W_0$ verify

$$\begin{aligned} W_1\oplus \mathbb {Z}_p\cdot v_1=W=W_2\oplus \mathbb {Z}_p\cdot v_2. \end{aligned}$$

Then, one can choose $\alpha ,\beta \in \mathbb {Z}_p$ so that

• (a) $v=\alpha v_1+ \beta v_2 \in W-W_0$,

• (b) $W_1\oplus \mathbb {Z}_p\cdot v=W_2\oplus \mathbb {Z}_p\cdot v=W$.

Proof

Fix a basis $\mathfrak {B}_1$ of $W_1$ and $\mathfrak {B}_2$ of $W_2$. Let $x_1$ be the $v_2$-coordinate of $v_1$ with respect to the basis $\mathfrak {B}_2\cup \{v_2\}$ and $x_2$ be the $v_1$-coordinate of $v_2$ with respect to the basis $\mathfrak {B}_1 \cup \{v_1\}$. We may assume without loss of generality that $v_p(x_i)>0$ for $i=1,2$, as otherwise, say in case $v_p(x_1)=0$, it would follow that $span \left( \mathfrak {B}_2,v_1\right) =span \left( \mathfrak {B}_2,x_1\cdot v_2\right) =W$ and thus the choice $\alpha =1$ and $\beta =0$ (thus $v=v_1$) would work. Let $X=\left( \begin{array}{cc}x_1 &{} 1\\ 1&{} x_2 \end{array} \right)$ and let $Y=\left( \begin{array}{cc}a &{} b\\ c&{} d \end{array} \right) \in \mathrm {GL}_2(\mathbb {Z}_p)$ be such that $YX=1$ (such Y exists since $\det (X)\in \mathbb {Z}_p^\times$ thanks to our hypothesis on $v_p(x_i)$).

Consider $W_0\, \cap \, span \left( v_1,v_2\right)$. Since $v_1 \not \in W_0$, it follows that this intersection is a finite union of $\mathbb {Z}_p$-lines, say spanned by $\{\alpha _iv_1+\beta _iv_2\}_{i=1}^d$ (with $\alpha _i,\beta _i \in \mathbb {Z}_p$). Let $\mathfrak {X}=\{\alpha _i/\beta _i : \beta _i\ne 0\}$, note that it is a finite subset of $\mathbb {Q}_p$. Use Lemma 3.47 to choose $x,y \in \mathbb {Z}_p^\times$ such that $\frac{ax+by}{cx+dy} \not \in \mathfrak {X}$. Set $\alpha =ax+by$ and $\beta =cx+dy$. Note that we have by definitions

$$\begin{aligned} Y\left[ \begin{array}{c}x\\ y \end{array}\right] =\left[ \begin{array}{c}\alpha \\ \beta \end{array}\right] , \end{aligned}$$

or equivalently that

$$\begin{aligned} \left( \begin{array}{cc}x_1 &{} 1\\ 1&{} x_2 \end{array} \right) \left[ \begin{array}{c}\alpha \\ \beta \end{array}\right] =X\left[ \begin{array}{c}\alpha \\ \beta \end{array}\right] =\left[ \begin{array}{c}x\\ y \end{array}\right] . \end{aligned}$$

(24)

Observe that $v:=\alpha v_1 +\beta v_2 \not \in W_0$ (as $\alpha /\beta \not \in \mathfrak {X}$), so v satisfies (a). Furthermore,

$$\begin{aligned} v=\alpha v_1 +\beta v_2 \equiv (\alpha x_1 + \beta )\cdot v_2=x\cdot v_2 \mod W_2 \end{aligned}$$

and

$$\begin{aligned} v \equiv (\alpha + \beta x_2)\cdot v_1=y\cdot v_1 \mod W_1 \end{aligned}$$

We therefore conclude (using the fact $x,y \in \mathbb {Z}_p^\times$) that

$$\begin{aligned} span \left( W_1,v\right) =span \left( W_1,y\cdot v_1\right) =span \left( W_1,v_1\right) =W, \end{aligned}$$

and hence

$$\begin{aligned} span \left( W_2,v\right) =span \left( W_2,x\cdot v_2\right) =span \left( W_1,v_2\right) =W, \end{aligned}$$

which shows that v verifies (b) as well. $\square$

Lemma 3.49

Let W be as in the previous lemma and let $\{w_1,\dots ,w_\mathfrak {d}\}$ be a given basis of W. For any non-negative integer k, one can find elements $\{w_{\mathfrak {d}+1},\dots , w_{\mathfrak {d}+k}\}\subset W$ so that for any $I \subset \{1,\dots ,\mathfrak {d}+k\}$ of size $\mathfrak {d}$, the set $\{w_j\}_{j\in I}$ spans W.

Proof

We prove the lemma by induction on k. When $k=0$, the assertion is clear. Suppose that for $k\ge 1$ we have found a set $\{w_{\mathfrak {d}+1},\dots , w_{\mathfrak {d}+k-1}\}$. Let $\mathfrak {S}$ denote the collection of subsets of $1,\dots ,\mathfrak {d}+k-1$ of size $\mathfrak {d}-1$ and let $\mathfrak {D}=\{span \left( \{w_i\}_{i\in S}\right) : S\in \mathfrak {S}\}$ be a set of free, rank $\mathfrak {d}-1$ direct summands of W. Set $W_0=\cup _{\mathfrak {D}}\,W^\prime$, observe that $W_0$ is a proper subset of W. For any $w \in W- W_0$ and $S\in \mathfrak {S}$, the submodule $span \left( \{w\}\cup \{w_i\}_{i \in S}\right) \subset W$ is of finite index. Fix $S \in \mathfrak {S}$ and define $W_S:=span \left( w_i: i \in S\right) .$

We first prove that there is an element $v_S\in W-W_0$ such that

$$\begin{aligned} W_S+\mathbb {Z}_p\cdot v_S=W. \end{aligned}$$

(25)

Indeed, pick any $w\in W-W_0$. If $W_S+\mathbb {Z}_p\cdot w=W$, we are done. Otherwise we may use Lemma 3.46 to choose $w_1\in W-(W_S+\mathbb {Z}_p\cdot v \cup W_0)$, for which we have

$$\begin{aligned} W_S+\mathbb {Z}_p\cdot w_1\supsetneq W_S+\mathbb {Z}_p\cdot w. \end{aligned}$$

This process has to terminate and when it does, we have found the desired $v_S$ verifying (25).

Using Lemma 3.48 iteratively, one obtains an element $v \in W-W_0$ such that

$$\begin{aligned} W_S+\mathbb {Z}_p\cdot v= W \end{aligned}$$

for every $S \in \mathfrak {S}$. We set $w_{\mathfrak {d}+k}:=v$. $\square$

Proof of Proposition 3.3

Let $\mathfrak {B}=\{v_1,\ldots ,v_{g_-},w_{g_-+1},\dots ,w_{g}\}$ be any $\mathbb {Z}_p$-basis of $\mathbb {D}_p(T)$ such that $\{v_1,\ldots ,v_{g_-}\}$ forms a basis of ${\text {Fil}}^0\mathbb {D}_p(T)$. Form the dual basis

$$\begin{aligned} \mathfrak {B}^\prime =\{v_1^\prime ,\ldots ,v_{g_-}^\prime ,w_{g_-+1}^\prime ,\ldots ,w_{g}^\prime \}\subset \mathbb {D}_p(T^*(1)). \end{aligned}$$

Consider the free $\mathbb {Z}_p$-module $W:=\mathbb {D}_p(T^*(1))/{\text {Fil}}^0\mathbb {D}_p(T^*(1))$ of rank $g_-$ and for an element $v\in \mathbb {D}_p(T^*(1))$, let $\bar{v}$ denote its image in W. It is easy to see that $\{\bar{v}_1^\prime ,\ldots ,\bar{v}_{g_{-}}^\prime \}$ forms a basis of W. Use Lemma 3.49 (with $\mathfrak {d}=g_-$ and $k=g_+$) to obtain a set $\{\bar{v}_1^\prime ,\ldots ,\bar{v}_{g}^\prime \}$ such that for any $\underline{I}\in \mathfrak {I}_p$,

$$\begin{aligned} span \left( \bar{v}_i^\prime : i\in \underline{I}\right) =W. \end{aligned}$$

One can lift the set $\{\bar{v}_1^\prime ,\ldots ,\bar{v}_{g}^\prime \}$ to a basis $\mathfrak {B}_{ad }^\prime =\{{v}_1^\prime ,\ldots ,{v}_{g}^\prime \}$ of $\mathbb {D}_p(T^*(1))$ and the basis $\mathfrak {B}_{ad }$ dual to $\mathfrak {B}_{ad }^\prime$ gives us an admissible basis of $\mathbb {D}_p(T)$, which completes the first part of the proof.

The proof that a strongly admissible basis exists is similar and we only provide a sketch of its proof after inverting p. The technical details to conclude an integral version of this result are identical to the arguments above we have assembled in the course of deducing the first part concerning admissibility. To ease notation, let $\mathcal {V}=\mathbb {D}_p(T^*(1))\otimes \mathbb {Q}_p$ and $\mathcal {W}={\text {Fil}}^0\mathbb {D}_p(T^*(1))\otimes \mathbb {Q}_p$. Set also $\mathcal {T}=(1-\varphi )^{-1}(p\varphi -1)$ and $\mathcal {W}^\prime =\mathcal {T}^{-1}(\mathcal {W})\otimes \mathbb {Q}_p$. (Note that $\mathcal {T}$ is invertible thanks to our running assumptions.) Set $r=\dim \mathcal {W}= \mathcal {W}^\prime$ and $r+s=\dim \mathcal {V}$. We choose a basis $\{v^\prime _i\}$ inductively as follows:

• Choose any $v_1 \notin \mathcal {W} \cup \mathcal {W}^\prime$.

• For $k\le s-1$, if we have chosen $v_1^\prime ,\ldots ,v_k^\prime$, choose $v_{k+1}^\prime \in \mathcal {V}$ as any vector so that

  $$\begin{aligned} v_{k+1}^\prime \notin \left( span (v_i^\prime : 1\le i\le k)+\mathcal {W}\right) \cup \left( span (v_i^\prime : 1\le i\le k)+\mathcal {W}^\prime \right) . \end{aligned}$$

  Note that we can do this as we have a union of two hyperplanes of dimension $k+r<r+s$.

• For any $0\le k<s$, suppose we have chosen $\mathfrak {B}_k=\{v_1^\prime ,\ldots ,v_{s+k}^\prime \}$ in a way that

  $$\begin{aligned} span \left( v_{i_j}^{\prime }: i_{j} \in I\right) \cap \left( \mathcal {W}\cup \mathcal {W}^\prime \right) =0 \end{aligned}$$

  for every subset $I\subset \{1,\ldots ,s+k\}$ of size s. (The first two steps will get us to this step with $k=0$.) Let $I^{(s-1)}$ denote all subsets of $I\subset \{1,\ldots ,s+k\}$ of size $s-1$ and let

  $$\begin{aligned} V^{(s-1)}=\bigcup _{J\in I^{(s-1)}}span \left( v_{i_j}: i_j \in J \right) . \end{aligned}$$

  This is a finite union of hyperplanes of dimension $s-1$. Now choose $v_{s+k+1}^\prime \in \mathcal {V}$ to be any element verifying

  $$\begin{aligned} v_{s+k+1}^\prime \notin \left( \mathcal {W}+V^{(s-1)}\right) \cup \left( \mathcal {W}^\prime +V^{(s-1)}\right) . \end{aligned}$$

  Note that the right side is a union of finitely many hyperplanes of dimension $r+s-1$ so an element $v_{s+k+1}^\prime$ does indeed exist. Set $\mathfrak {B}_{k+1}=\{v_1^\prime ,\ldots ,v_{s+k+1}^\prime \}$.

It is now easy to verify that the set $\mathfrak {B}_{s}$ is a strongly admissible basis. $\square$

Appendix 3: Coleman–Adapted Kolyvagin systems

Throughout this Appendix, let F be a totally real or a CM field as above. Let $\mathfrak {O}$ be the ring of integers of a finite extension $\Phi$ of $\mathbb {Q}_p$, with maximal ideal $\mathfrak {m}$, residue field k and uniformizer $\varpi$. Let T be a $G_F$-stable $\mathfrak {O}$-lattice inside $\mathcal {M}_p(\eta ^{-1})$, the twist of the p-adic realization of a motive $\mathcal {M}$ (of the sort considered in the main body of this article) by an even Dirichlet character $\eta$ of $\Delta$. Then T is a free $\mathfrak {O}$-module of finite rank which is equipped with a continuous $G_F$-action unramified outside a finite set of places $\Sigma$ of F. Set $\overline{T}=T/\mathfrak {m} T$. We assume that all places of F at infinity and above p are contained in $\Sigma$. We assume that T verifies the hypotheses (H1)–(H4) of [16, Section 3.5] as well as the following:

• (H.Tam) For every finite place $\lambda \in \Sigma$, the module $H^0(I_\lambda ,T\otimes \Phi /\mathfrak {O})$ is divisible. Here $I_\lambda$ stands for the inertia group at the prime $\lambda$.

• (H.nE) For every prime $\mathfrak {p}\mid p$ of F, we have

  $$\begin{aligned} H^0(F_\mathfrak {p},T)=H^2(F_{\mathfrak {p}},T)=0. \end{aligned}$$

In this appendix we let $F_\infty$ denote the cyclotomic $\mathbb {Z}_p$ extension of F and $\Gamma =Gal (F_\infty /F)$. Note that this is the pro-p part of the group considered in the main text. Let $\Lambda ^{(p)}=\mathfrak {O}[[\Gamma ]]$. Let $\mathbb {T}=T\otimes \Lambda ^{(p)}$ and fix $\underline{I}\in \mathfrak {I}_p$ as in the conclusion of Proposition 3.25. To ease notation, we will set $R=\Lambda ^{(p)}$ and $d=g_-$. We fix throughout an $\underline{I}\in \mathfrak {I}_p$ verifying the conclusion of Proposition 3.25 and associated to this choice, fix a signed Coleman map

$$\begin{aligned} \mathfrak {C}:=\mathrm {Col}_{\mathcal {M}_p}^{\underline{I},\eta }: H^1(F_p,\mathbb {T}){\longrightarrow }R^d. \end{aligned}$$

(26)

Here $\mathrm {Col}_{\mathcal {M}_p}^{\underline{I}}$ corresponds to the Coleman map denoted by $\mathrm {Col}_{T(\eta )}^{\underline{I}}$ in the main text and $\mathrm {Col}_{\mathcal {M}_p}^{\underline{I},\eta }$ is its restriction to $\eta$-isotypic component. Let $Z\subset R^d$ denote a R-submodule with the following properties:

• Z is free of rank d and $im (\mathfrak {C})\subset Z$.

• The R-module $Z/im (\mathfrak {C})$ is pseudo-null.

The existence of such Z is guaranteed by Corollary 2.22.

We now fix an arbitrary rank-one direct summand $\mathbb {L} \subset Z$.

Definition 3.50

Let $\mathcal {F}_{\mathbb {L}}$ denote the Selmer structure on $\mathbb {T}$ given with the following data:

• $H^1_{\mathcal {F}_{\mathbb {L}}}(F_\lambda ,\mathbb {T})=H^1(F_{\lambda }, \mathbb {T})$ for primes $\lambda \not \mid p$,

• $H^1_{\mathcal {F}_{\mathbb {L}}}(F_p,\mathbb {T})=\ker \left( H^1(F_p,\mathbb {T})\mathop {\longrightarrow }\limits ^{\mathfrak {C}}Z/\mathbb {L}\right) .$

Let $\mathcal {P}$ be the set of places of F that does not contain the archimedean places, primes at which T is ramified and primes above p. Finally let $\overline{\mathbf {KS}}(\mathbb {T},\mathcal {F}_{\mathbb {L}},\mathcal {P})$ be the R-module of generalized Kolyvagin systems defined as in [6, Section 3.2.2]. An element of this module will be called an $\mathbb {L}$ -restricted Kolyvagin system.

We also let $\mathcal {F}_{\mathbb {L}}^*$ denote the dual Selmer structure on the Cartier dual $\mathbb {T}^\dagger$, in the sense of [16, Definition 1.3.1 and §2.3].

As in the main body of this text, we assume the truth of the weak Leopoldt conjecture for T. Our goal in this appendix is to give a proof of Theorem 3.53 below.

Lemma 3.51

Suppose R is any commutative ring and M, N, Q are finitely generated R-modules such that we have an exact sequence

$$\begin{aligned} 0\longrightarrow M\mathop {\longrightarrow }\limits ^{\iota } N\longrightarrow Q \end{aligned}$$

and the quotient $N/\iota (M)$ is R-torsion-free. For any ideal I of R, let $X_I=X\otimes _R R/I$ for $X=M,N,R$. Then the following sequence of $R_I$-modules is exact:

$$\begin{aligned} 0\longrightarrow M_I \mathop {\longrightarrow }\limits ^{\iota _I} N_I\longrightarrow Q_I. \end{aligned}$$

Proof

Suppose $m \in M$ is such that $\iota (m) \in I\cdot N$, say $\iota (m)=r\cdot n_0$ for some $r\in I$ and $n_0 \in N$. As the quotient $N/\iota (M)$ is R-torsion-free, it follows that $n_0 \in \iota (M)$; say $n_0=\iota (m_0)$. Thus $\iota (m)=\iota (r\cdot m_0)$ and since $\iota$ is injective, $m\in I\cdot M$. We just proved that $I\cdot M=\ker \left( M\mathop {\longrightarrow }\limits ^{\iota } N_I\right)$ which is equivalent to the assertion of our lemma. $\square$

Lemma 3.52

The R-module $H^1_{\mathcal {F}_{\mathbb {L}}}(F_p,\mathbb {T})$ is free of rank $g_+$+1.

Proof

Let L denote the image of $\mathbb {L}$ (resp., $\overline{Z}$ the image of Z) under the augmentation map $\mathfrak {A}:R\twoheadrightarrow \mathfrak {O}$. Observe the commutative diagram

where $\mathfrak {C}_\mathfrak {A}:=\mathfrak {C}\otimes _\mathfrak {A}\mathfrak {O}$ is the induced map on $H^1_{\mathrm {Iw}}(F_p,{T})\otimes _{\mathfrak {A}}\mathfrak {O}\mathop {\longrightarrow }\limits ^{\sim }H^1(F_p,{T})$. As the cokernel of $\mathfrak {C}$ is finite so is the cokernel of $\mathfrak {C}_\mathfrak {A}$ and it follows that $\ker (\mathfrak {C}_\mathfrak {A})$ is a free $\mathfrak {O}$-module of rank $g_++1$ and by Nakayama’s lemma that the R-module $H^1_{\mathcal {F}_{\mathbb {L}}}(F_p,\mathbb {T})$ is generated by at most $g_++1$ elements. On the other hand, the first row of the diagram above shows that the generic fiber of $H^1_{\mathcal {F}_{\mathbb {L}}}(F_p,\mathbb {T})$ has rank $g_++1$ hence, together with our the discussion above, we conclude that the R-module $H^1_{\mathcal {F}_{\mathbb {L}}}(F_p,\mathbb {T})$ is generated by exactly $g_++1$ elements. It is not hard to see (using the fact that R is a UFD) that these generators cannot satisfy a non-trivial R-linear relation. $\square$

Theorem 3.53

Let $\mathcal {P}_{1,\bar{1}} \subset \mathcal {P}$ be as in Definition 3.55 below.

• (i) The R-module $\overline{\mathbf {KS}}(\mathbb {T},\mathcal {F}_{\mathbb {L}},\mathcal {P})$ is free of rank one, generated by any Kolyvagin system $\pmb {\kappa }$ whose image $\overline{\pmb {\kappa }} \in {\mathbf {KS}}(\overline{T},\mathcal {F}_{\mathbb {L}},\mathcal {P}_{1,\bar{1}})$ is non-zero.

• (ii) For an arbitrary generator $\{\kappa _n\}=\pmb {\kappa }$, the leading term $\kappa _1\in H^1_{\mathcal {F}_{\mathbb {L}}}(F,\mathbb {T})$ is non-vanishing.

• (iii) Suppose $\{\kappa _n\}=\pmb {\kappa }\in \overline{\mathbf {KS}}(\mathbb {T},\mathcal {F}_{\mathbb {L}},\mathcal {P}_X)$ is a generator. Then,

  $$\begin{aligned} char \left( H^1_{\mathcal {F}_{\mathbb {L}}}(F,\mathbb {T})/\Lambda \cdot \kappa _1\right) = char \left( H^1_{\mathcal {F}_{\mathbb {L}}^*}(F,\mathbb {T}^\dagger )^{\vee }\right) . \end{aligned}$$

It is the statement of Theorem 3.53(iii) that is key to all our results towards Perrin-Riou’s main conjectures.

Proof of the parts (i) and (iii) of Theorem 3.53 is identical to the proof of [7, Theorem A.12]² once we verify (a) that the analogous statement to Definition/Theorem A.9 in loc.cit. holds true in our setting and (b) that the core Selmer rank $\chi (\overline{T},\mathcal {F}_{\mathbb {L}})$ (in the sense of [16, Definition 4.1.11]) of the Selmer structure $\mathcal {F}_{\mathbb {L}}$ on $\overline{T}$ is 1. The first of these is achieved in Theorem 3.57 below and the second in Proposition 3.59. The main difficulty is that the images of the Coleman maps are not necessarily free.

We first provide a proof of (ii) here.

Proof of Theorem 6.4(ii)

Thanks to our choice of $\underline{I}\in \mathfrak {I}_p$ and Proposition 3.28, note that the modified Selmer group $\mathrm {Sel}_{\underline{I}}(T^\dagger /F(\mu _{p^\infty }))^\Delta$ is R-cotorsion. Thus the R-module $H^1_{\mathcal {F}_{\mathbb {L}}^*}(F,\mathbb {T}^\dagger ) \subset \mathrm {Sel}_{\underline{I}}(T^\dagger /F(\mu _{p^\infty }))^\Delta$ is cotorsion as well. We may now conclude the proof using [6, Theorem 5.10]. $\square$

Before settling Theorem 3.53 in full, we introduce the necessary terminology that we mostly borrow from [16]. Fix a topological generator $\gamma$ of the group $\Gamma$. We then have a (non-canonical) isomorphism $R\cong \mathfrak {O}[[\gamma -1]]$.

Definition 3.54

For $k, \alpha \in \mathbb {Z}^+$, set

$$\begin{aligned} R_{k,\alpha }:= & {} R/(\varpi ^k,(\gamma -1)^{\alpha }),\\ \mathbb {T}_{k,\alpha }:= & {} \mathbb {T}\otimes _{R}R_{k,\alpha }=\mathbb {T}/(\varpi ^k,(\gamma -1)^{\alpha }) \end{aligned}$$

and define the collection

$$\begin{aligned} Quot (\mathbb {T}):=\{\mathbb {T}_{k,\alpha }: k,\alpha \in \mathbb {Z}^+\}. \end{aligned}$$

The propagation of the Selmer structure $\mathcal {F}_{\mathbb {L}}$ (in the sense of [16, Example 1.1.2]) to the quotients $\mathbb {T}_{k,\alpha }$ will still be denoted by the symbol $\mathcal {F}_{\mathbb {L}}$ (as well as its propagation to the quotient T).

Definition 3.55

For $k, \alpha \in \mathbb {Z}^+$ define

• (i) $H_{k,\alpha }=\ker \left( G_F\rightarrow Aut (\mathbb {T}_{k,\alpha })\oplus Aut (\pmb {\mu }_{p^{k}})\right)$,

• (ii) $L_{k,\alpha }=\overline{F}^{H_{k,\alpha }}$,

• (iii) $\mathcal {P}_{k,\alpha }=\{\hbox {Primes }\lambda \in \mathcal {P}_{X} : \lambda \hbox { splits completely in } L_{k,\alpha }/F\}$.

The collection $\mathcal {P}_{k,\alpha }$ is called the collection of Kolyvagin primes for $\mathbb {T}_{k,\alpha }$. Define $\mathcal {N}_{k,\alpha }$ to be the set of square free products of primes in $\mathcal {P}_{k,\alpha }$.

Definition 3.56

• (i) Given $\lambda \in \mathcal {P}_{k,\alpha }$ fix once and for all an abelian extension $F^\prime /F_\lambda$ which is totally and tamely ramified, and moreover is a maximal such extension. As in [16, Definition 1.1.6(iv)], the transverse local condition at $\lambda$ is defined to be

  $$\begin{aligned} H^1_{tr }(F_\lambda ,T_{k,\alpha })=\ker \{H^1(F_\lambda ,T_{k,\alpha }) \longrightarrow H^1(F^\prime ,T_{k,\alpha })\}. \end{aligned}$$

• (ii) For $\mathfrak {n} \in \mathcal {N}_{k,\alpha }$, define the Selmer structure $\mathcal {F}_{\mathbb {L}}(\mathfrak {n})$ on $T_{k,\alpha }$ by setting

  $$\begin{aligned} H^1_{\mathcal {F}_{\mathbb {L}}(\mathfrak {n})}(F_\lambda ,T_{k,\alpha })=\left\{ \begin{array}{cr} H^1_{\mathcal {F}_{\mathbb {L}}}(F_\lambda ,T_{k,\alpha }), &{} \quad \hbox { if }\, \lambda \not \mid \mathfrak {n},\\ \\ H^1_{tr }(F_\lambda ,T_{k,\alpha }), &{}\quad \hbox { if }\,\lambda \mid \mathfrak {n}. \end{array} \right. \end{aligned}$$

The following list of properties is key in proving Theorem 3.53.

Theorem 3.57

For any $\mathfrak {n} \in \mathbb {N}_{k,\alpha }$ the Selmer structure $\mathcal {F}_{\mathbb {L}}(\mathfrak {n})$ is cartesian on the collection $Quot (\mathbb {T})$ in the following sense. Let $\lambda$ be any prime of F.

• $\mathbf{(C1)}$ (Functoriality) For $\alpha \le \beta$ and $k\le k^\prime , H^1_{\mathcal {F}_{\mathbb {L}}(\mathfrak {n})}(F_{\lambda },\mathbb {T}_{k,\alpha })$ is the exact image of $H^1_{\mathcal {F}_{\mathbb {L}}(\mathfrak {n})}(F_\lambda ,\mathbb {T}_{k^{\prime },\beta })$ under the canonical map $H^1(F_{\lambda },\mathbb {T}_{k^\prime ,\beta }) \rightarrow H^1(F_{\lambda },\mathbb {T}_{k,\alpha }).$

• $\mathbf{(C2)}$ (Cartesian property along the cyclotomic tower)

  $$\begin{aligned} H^1_{\mathcal {F}_{\mathbb {L}}(\mathfrak {n})}(F_{\lambda },\mathbb {T}_{k,\alpha })=\ker \left( H^1(F_{\lambda },\mathbb {T}_{k,\alpha })\longrightarrow \frac{H^1(F_{\lambda },\mathbb {T}_{k,\alpha +1})}{H^1_{\mathcal {F}_{\mathbb {L}}(\mathfrak {n})}(F_{\lambda },\mathbb {T}_{k,\alpha +1})}\right) . \end{aligned}$$

  Here the arrow is induced from the injection $\mathbb {T}_{k,\alpha }\mathop {\longrightarrow }\limits ^{[\gamma -1]}\mathbb {T}_{k,\bar{\alpha +1}}$ and $[\gamma -1]$ is the multiplication by $\gamma -1$ map.

• $\mathbf{(C3)}$ (Cartesian property as powers of p vary)

  $$\begin{aligned} H^1_{\mathcal {F}_{\mathbb {L}}(\mathfrak {n})}(F_\lambda ,\mathbb {T}_{k,\alpha })=\ker \left( H^1(F_\lambda ,\mathbb {T}_{k,\alpha })\mathop {\longrightarrow }\limits ^{[\varpi ]} \frac{H^1(F_\lambda ,\mathbb {T}_{k+1,\alpha })}{H^1_{\mathcal {F}_{\mathbb {L}}(\mathfrak {n})}(F_\lambda ,\mathbb {T}_{k+1,\alpha })}\right) , \end{aligned}$$

  where the arrow is induced from the injection $\mathbb {T}_{k,\alpha } \mathop {\longrightarrow }\limits ^{[\varpi ]} \mathbb {T}_{k+1,\alpha }.$

Proof

For the primes $\lambda \not \mid \mathfrak {n}p$, the asserted properties may be verified as in [5, §2.3.1]. The key points are the fact that the inertia group $I_\lambda \subset G_F$ acts trivially on $\Lambda ^{(p)}$ and that we assumed (H.Tam). For the primes $\lambda \mid \mathfrak {n}$, they may be proved as in [6, §4.1.4] (which itself, in this particular case of interest, is a slight generalization of [5, Proposition 2.21]).

It therefore remains to verify the claimed properties at primes above p. The property (C1) is a direct consequence of our definitions. Using Lemma 3.51, one has the following identification for every $k,\alpha \in \mathbb {Z}^+$:

$$\begin{aligned} H^1_{\mathcal {F}_{\mathbb {L}}(\mathfrak {n})}(F_p,\mathbb {T}_{k,\alpha })=H^1_{\mathcal {F}_{\mathbb {L}}}(F_p,\mathbb {T})\otimes _{R}R_{k,\alpha } \end{aligned}$$

(i.e. $H^1_{\mathcal {F}_{\mathbb {L}}(\mathfrak {n})}(F_p,\mathbb {T}_{k,\alpha })$ is the image of $H^1_{\mathcal {F}_{\mathbb {L}}}(F_p,\mathbb {T})$ under the obvious map). Note that Lemma 3.51 applies with $M=H^1_{\mathcal {F}_{\mathbb {L}}}(F_p,\mathbb {T})$ and $N=H^1(F_p,\mathbb {T})$ as the quotient $H^1(F_p,\mathbb {T})/H^1_{\mathcal {F}_{\mathbb {L}}}(F_p,\mathbb {T})$ is R-torsion free by construction. The properties (C2) and (C3) follow now at once using the fact that the R-module $H^1_{\mathcal {F}_{\mathbb {L}}}(F_p,\mathbb {T})$ is free (of rank $g_++1$) by Lemma 3.52. $\square$

Let $\mathcal {F}_{null }$ denote the Selmer structure on $\mathbb {T}$ given with the following data:

• $H^1_{\mathcal {F}_{null }}(F_\lambda ,\mathbb {T})=H^1(F_{\lambda }, \mathbb {T})$ for primes $\lambda \not \mid p$,

• $H^1_{\mathcal {F}_{null }}(F_p,\mathbb {T})=H^1_{\underline{I}}(F_p,T):=\ker \left( H^1(F_p,\mathbb {T})\mathop {\longrightarrow }\limits ^{\mathfrak {C}}Z\right) .$

The assertion concerning the Selmer structure $\mathcal {F}_{\mathbb {L}}$ in the following Corollary follows immediately by Theorem 3.57. We need the statement on $\mathcal {F}_{null }$ in our companion article [4] and it follows easily by modifying Lemma 3.52 appropriately.

Corollary 3.58

Propagations of both Selmer structures $\mathcal {F}_{\mathbb {L}}$ and $\mathcal {F}_{null }$ on T verify the hypothesis (H6) of [16].

Proposition 3.59

The core Selmer rank $\chi (\overline{T},\mathcal {F}_\mathbb {L})$ of the Selmer structure $\mathcal {F}_\mathbb {L}$ on $\overline{T}$ equals one, whereas $\chi (\overline{T},\mathcal {F}_{null })$ equals.

Proof

The proof of this proposition is similar to the proof of Proposition 9.2 in [4]. Let $\mathcal {F}_{can }$ denote the canonical Selmer structure on $\mathbb {T}$ given with the data $H^1_{\mathcal {F}_{can }}(F_\lambda ,\mathbb {T})=H^1(F_{\lambda }, \mathbb {T})$ for every prime $\lambda \in \Sigma$. Using the global duality argument in [24, Proposition 1.6] and Corollary 3.58 we conclude that

$$\begin{aligned} \chi (\overline{T},\mathcal {F}_can )-\chi (\overline{T},\mathcal {F})=rank _R\, H^1_{\mathrm {Iw}}(F_p,{T})-rank _R\,H^1_{\mathcal {F}}(F_p,\mathbb {T}) \end{aligned}$$

for $\mathcal {F}=\mathcal {F}_\mathbb {L}$ or ${\mathcal {F}_{null }}$. Since

$$\begin{aligned} rank _R\, H^1_{\mathrm {Iw}}(F_p,T)=g \hbox { and } \chi (\overline{T},\mathcal {F}_can )=g_- \end{aligned}$$

(c.f., [16, Theorem 5.2.15]) we infer that

$$\begin{aligned} \chi (\overline{T},\mathcal {F})=rank _R\,H^1_{\mathcal {F}}(F_p,\mathbb {T})-g_+. \end{aligned}$$

The first assertion in our proposition follows from Lemma 3.52 and the second using its appropriate version to apply with ${\mathcal {F}_{null }}$. $\square$

1.1 The module of Kolyvagin determinants

Let $B=\{\phi _1,\ldots , \phi _{d-1}\}$ be a basis of the free R-module $Hom _{R}\left( Z{/}\mathbb {L},\,R\right) .$ We then have an isomorphism

$$\begin{aligned} \bigoplus _{i=1}^{d-1}\phi _i:R^d/\mathbb {L} \mathop {\longrightarrow }\limits ^{\sim } R^{d-1}. \end{aligned}$$

Let $\widetilde{\phi }_i\in Hom _{R}\left( Z,R\right)$ denote the pullback of $\phi _i$ with respect to the obvious projection. Note that the map $\phi := \bigoplus _{i=1}^{d-1} \widetilde{\phi }_i:Z \rightarrow R^{d-1}$ is surjective with kernel $\mathbb {L}$. Define

$$\begin{aligned} \Phi :=\widetilde{\phi }_1\wedge \cdots \wedge \widetilde{\phi }_{d}\in \wedge ^{d} Hom _{R}\left( Z,R\right) , \end{aligned}$$

where the exterior product is taken in the category of R-modules. Let

$$\begin{aligned} \Psi \in \wedge ^{d}\,Hom _{R}\left( H^1(F_p,\mathbb {T}),R\right) \end{aligned}$$

be the pullback of $\Phi$ with respect to the Coleman map $\mathfrak {C}$.

Proposition 3.60

• (i) The map $\Phi$ maps $\wedge ^d Z$ isomorphically onto $\mathbb {L}$.

• (ii) For every $c \in \wedge ^d H^1(F_p,\mathbb {T})$ we have $\Psi (c) \in H^1_{\mathcal {F}_\mathbb {L}}(F_p,\mathbb {T})$.

• (iii) The map $\Psi$ induces a map (which we still denote by $\Psi$)

  $$\begin{aligned} \Psi :\,H^1(F_p,\mathbb {T})/H^1_{\underline{I}}(F_p,T)\longrightarrow H^1_{\mathcal {F}_\mathbb {L}}(F_p,\mathbb {T})/H^1_{\underline{I}}(F_p,T). \end{aligned}$$

Proof

Linear Algebra. $\square$

Proposition 3.60 may be summarized via the following commutative diagram:

(27)

As a consequence of the proposition below, it follows that the map $\Psi$ on the third row and the map $loc _p^{\otimes {d}}$ are both surjective.

Proposition 3.61

Under our running assumptions both R-modules $\wedge ^d H^1(F,\mathbb {T})$ and $H^1_{\mathcal {F}_\mathbb {L}}(F,\mathbb {T})$ are free of rank one.

Proof

It follows from the weak Leopoldt conjecture for T (which we assume) that the R-module $H^1_{\mathcal {F}_{can }^*}(F,\mathbb {T}^*)^\vee$ is torsion, where the canonical Selmer structure ${\mathcal {F}_{can }}$ of Mazur and Rubin is given in the proof of Proposition 3.59. By control theorem (which holds true for this Selmer structure), we may find a specialization $\pi : R\twoheadrightarrow \mathfrak {O}$ (whose kernel is necessarily principal, say generated by $\varpi \in R$) such that $H^1_{\mathcal {F}_{can }^*}(F,{T}_\pi ^*)$ has finite cardinality, where $T_\pi :=\mathbb {T}\otimes _\pi \mathfrak {O}$. By [16, Theorem 5.2.15], it follows that $H^1_{\mathcal {F}_{can }}(F,{T}_\pi )$ is an $\mathfrak {O}$-module of rank g, which is also torsion-free (hence free) by our running assumptions.

Consider the natural injection $H^1(F,\mathbb {T})/\varpi H^1(F,\mathbb {T}) \hookrightarrow H^1_{\mathcal {F}_{can }}(F,T_\pi )$. Using Nakayama’s lemma, we see that $H^1(F,\mathbb {T})$ may be generated by the lifts of a basis of $H^1_{\mathcal {F}_{can }}(F,T_\pi )$. Relying on the fact that R is a UFD, one may further verify that these generators may not satisfy a non-trivial R-linear relation. This completes the proof of the assertion that $\wedge ^d H^1(F,\mathbb {T})$ is free of rank 1. The rest is proved in an identical manner. $\square$

Definition 3.62

• (i) Define the $\Lambda$-module of Kolyvagin leading terms $\mathfrak {L}(T)$ as the module

  $$\begin{aligned} \mathfrak {L}(T)=\left\{ \sum _{\chi \in \widehat{\Delta }^+}{\kappa _1^{\chi }\cdot e_\chi } \in H^1_{Iw ,S}(F,T):\,\, \{\kappa _{\mathfrak {n}}^{\chi }\}=\pmb {\kappa }^\chi \in \overline{\mathbf {KS}}(\mathbb {T}(\chi ),\mathcal {F}_{\mathbb {L}},\mathcal {P})\right\} . \end{aligned}$$

  Here $\widehat{\Delta }^+$ denotes the set of even characters of $\Delta$ and $e_\chi \in \mathbb {Z}_p[\Delta ]$ the idempotent corresponding to $\chi$. It is not hard to see using Theorem 3.53 (for each twist $T(\chi )$) that the $\Lambda$-module $\mathfrak {L}(T)$ is free of rank 1.

• (ii) The $\Lambda$-module of Kolyvagin determinants $\mathfrak {K}(T)$ is defined as

  $$\begin{aligned} \mathfrak {K}(T)=\left\{ \Xi \in \wedge ^d H^1_{Iw ,S}(F,T):\,\Psi (\Xi ) \in \mathfrak {L}(T)\right\} . \end{aligned}$$

Remark 3.63

The diagram (27) above and the fact that $Z/im (\mathfrak {C})$ is pseudo-null together show that $\mathfrak {K}(T)\ne 0$. One may also prove that this module does not depend on any of the choices made above and depends only on T. A suitable extension of the theory of higher rank Kolyvagin systems (as studied in [17]) over coefficient rings of dimension larger than one would yield a more natural definition of $\mathfrak {K}(T)$. We plan to get back to this point in the future.

Appendix 4: Comparison with works of Kobayashi and Pollack

We shall compare the signed Selmer groups that we denoted by $Sel _{\underline{I}}$ in the main body of the article to the ±-Selmer groups of Kobayashi [12]; and the $\underline{I}$-signed p-adic L-functions to ±-p-adic L-functions of Pollack [18]. In particular, we shall justify that our theory offers a natural generalization of their work.

Throughout this appendix, we assume that the motive $\mathcal {M}=h^1(E)(1)$ is associated to an elliptic curve $E/\mathbb {Q}$ that has good supersingular reduction at p and that $a_p(E)=0$, so that the p-adic realization T of $\mathcal {M}$ will be the p-adic Tate module of E and the Pontryagin dual $T^\dagger$ is the p-divisible group $E[p^\infty ]$. Note that in this case $g_-=1$ and we no longer fix an admissible basis. As it shall be clear from the discussion below, Lemma 3.16 follows already from the work of Kobayashi and the second named author even if the basis of the Dieudonné module is no longer strongly admissible.

1.1 Kobayashi’s ±-Selmer groups

Kobayashi in [12] defined the ±-Selmer groups $\mathrm {Sel}_p^\pm (E/\mathbb {Q}(\mu _{p^\infty }))$ by properly modifying the Bloch–Kato conditions at p. This is exactly what we do in Definition 3.26, except that we used as our local conditions at p the submodules $H^1_{\underline{I}}(\mathbb {Q}_p(\mu _{p^\infty }),T^\dagger )$ in place of Kobayashi’s submodules $E^\pm (\mathbb {Q}_p(\mu _{p^\infty }))\subset E(\mathbb {Q}_p(\mu _{p^\infty }))$ given by some “jumping” trace conditions. Furthermore, as proved in [13, §4], Kobayashi’s submodules may be realized as the orthogonal complements of the kernel of some ±-Coleman maps $\mathrm {Col}^\pm :H^1_{\mathrm {Iw}}(\mathbb {Q}_p,T)\rightarrow \Lambda$, in the same way that the local conditions $H^1_{\underline{I}}(\mathbb {Q}_p(\mu _{p^\infty }),T^\dagger )$ in Definition 3.26 are defined as the orthogonal complement of $\ker (\mathrm {Col}_{\underline{I}})$. Therefore, in order to compare our $\mathrm {Sel}_{\underline{I}}$ with Kobayashi’s $\mathrm {Sel}_p^\pm$, it is enough to compare our Coleman maps $\mathrm {Col}_{\underline{I}}$ with the ±-Coleman maps defined in [12]. Note that these were already rewritten in the language of Dieudonné modules in [13].

Let $\mathbb {D}_\mathrm{cris}(T)=\mathbb {D}_{\mathbb {Q}_p}(T)$. We fix a basis $v_1\in {\text {Fil}}^0\mathbb {D}_\mathrm{cris}(T)$ and we extend it to a basis $v_1,v_2=\varphi (v_1)$ of $\mathbb {D}_\mathrm{cris}(T)$. The matrix of $\varphi$ with respect to this basis is given by

$$\begin{aligned} C_\varphi =\begin{pmatrix} 0&{} -1/p\\ 1&{}0 \end{pmatrix}=\begin{pmatrix} 0&{}-1\\ 1&{}0 \end{pmatrix}\begin{pmatrix} 1&{}0\\ 0&{}1/p \end{pmatrix}. \end{aligned}$$

Therefore, under the notation of Proposition 2.5, we find that the logarithmic matrix $M_T$ with respect to the same basis is given by

$$\begin{aligned} M_T=\begin{pmatrix} 0&{}-\log ^+\\ \log ^-&{}0 \end{pmatrix}, \end{aligned}$$

where $\log ^\pm$ are Pollack’s ±-logarithms defined by the formulae

$$\begin{aligned} \log ^+&=\frac{1}{p}\prod _{n\ge 1}\frac{\Phi _{p^{2n}}(1+X)}{p},\\ \log ^-&=\frac{1}{p}\prod _{n\ge 1}\frac{\Phi _{p^{2n-1}}(1+X)}{p}. \end{aligned}$$

Let $\mathrm {Col}_1,\mathrm {Col}_2$ be the two Coleman maps corresponding to this matrix as in Theorem 2.13. We have the relation

$$\begin{aligned} \mathcal {L}_{T,1}=-\log ^+\mathrm {Col}_2\quad \text {and}\quad \mathcal {L}_{T,2}=\log ^-\mathrm {Col}_1. \end{aligned}$$

On combining this with (4), we may compare our Coleman maps with the ±-Coleman maps defined in [13, §3.4] and see that they differ simply by a minus sign, namely

$$\begin{aligned} \mathrm {Col}^+=-\mathrm {Col}_2\quad \text {and}\quad \mathrm {Col}^-=\mathrm {Col}_1. \end{aligned}$$

(28)

In particular they have the same kernels.

Remark 3.64

Note that this choice of basis of $\mathbb {D}_\mathrm{cris}(T)$ is not admissible in the sense of Definition 3.2. As noted in Remark 3.4, this means that the images of our Coleman maps would not be pseudo-isomorphic to $\mathbb {Z}_p[[X]]$. Indeed, as shown in [12, Propositions 8.23 and 8.24], $\mathrm {Col}^+$ is surjective, while the isotypic component of $\mathrm {Im}(\mathrm {Col}^-)$ at a non-trivial character is $X\mathbb {Z}_p[[X]]$. This is consistent with our Propositions 2.20 and 2.21.

1.2 Pollack’s ±-p-adic L-functions

In [13, §3.4] as well as [12, Theorem 6.3], it has been showed that the Pollack’s ±-p-adic L functions in [18] is the image of the Beilinson–Kato elements along the cyclotomic tower (as constructed in [11]) under the ±-Coleman maps, up to a sign. Note in particular that the tower of Beilinson–Kato elements does satisfy Conjecture 3.11. Furthermore, the $\underline{I}$-signed p-adic L-functions given as in Definition 3.17 are simply the image of the Beilinson–Kato elements under $\mathrm {Col}_1$ and $\mathrm {Col}_2$. Therefore, thanks to (28), they agree with Pollack’s ±-p-adic L functions, up to a sign.