On the BDP Iwasawa main conjecture for modular forms

Abstract

Let K be an imaginary quadratic field where p splits, $p\ge 5$ a prime number and f an eigen-newform of even weight and level $N>3$ that is coprime to p. Under the Heegner hypothesis, Kobayashi–Ota showed that one inclusion of the Iwasawa main conjecture of f involving the Bertolini–Darmon–Prasanna p-adic L-function holds after tensoring by $\mathbb {Q}_p$. Under certain hypotheses, we improve upon Kobayahsi–Ota’s result and show that the same inclusion holds integrally. Our result implies the vanishing of the Iwasawa $\mu$-invariants of several anticyclotomic Selmer groups.

Appendix A: Integrality of specializations of the Perrin-Riou map

Notation note. The notation and hypotheses in the main body of the article continue to be in force. For a p-adic Lie group G and a p-adic ring R, we write $\Lambda _R(G)$ for the completed group ring of G with coefficients in R, and $\mathscr {H}_{\infty ,R}(G)$ for the algebra of $\textrm{Frac}(R)$-valued distributions on G. If $R=\mathcal {O}$, we often omit it from the notation.

1.1 A.1 Anticyclotomic Perrin-Riou regulator and Coleman maps

In this section, we review the construction of the anticyclotomic Perrin-Riou regulator and Coleman maps, by composing the two-variable counterparts with the projection map as done in [9, §5.1]. We begin by recalling the two-variable Perrin-Riou map from [26] and its decomposition given in [4]. Let $k_\infty$ be the $\mathbb {Z}_p^2$-extension of K, and fix an embedding $k_\infty \rightarrow \mathbb {C}_p$ such that the induced place on K is $\mathfrak {p}$; by an abuse of notation we will write this place of $k_\infty$ also as $\mathfrak {p}$. Since p is assumed to be split in K, there is a natural identification $K_\mathfrak {p}\simeq \mathbb {Q}_p$, and the local completion $k_{\infty ,\mathfrak {p}}$ can be identified with the compositum $F_\infty \mathbb {Q}_p^{\textrm{cyc}}$, where $F_\infty$ is the unramified $\mathbb {Z}_p$-extension of $\mathbb {Q}_p$ and $\mathbb {Q}_p^\textrm{cyc}$ is the cyclotomic $\mathbb {Z}_p$-extension of $\mathbb {Q}_p$. Put $U = {\text {Gal}}(F_\infty /\mathbb {Q}_p)$ and $F_m = F_\infty ^{U^{p^m}}$. When $m=0$, we shall write $F=F_0=\mathbb {Q}_p$.

The two-variable Perrin-Riou map constructed by Loeffler–Zerbes [26] is given by the projective limit

$$\begin{aligned} \mathcal {L}_{T,F_\infty }&= \varprojlim _m \mathcal {L}_{T,F_m}: \varprojlim _m H^1_\textrm{Iw}(F_m\mathbb {Q}_p^\textrm{cyc},T)\\&\rightarrow \varprojlim _m \mathscr {H}_{\infty }(\tilde{\Gamma }^{\textrm{cyc}})\otimes _{\mathbb {Q}_p} \mathbb {D}_{\textrm{cris}}(T)\otimes _{\mathbb {Z}_p} \mathcal {O}_{F_m}\\&\simeq \mathscr {H}_{\infty }(\tilde{\Gamma }^{\textrm{cyc}})\otimes _{\mathbb {Q}_p}\mathbb {D}_\textrm{cris}(T)\widehat{\otimes }_{\mathbb {Z}_p} S_{F_\infty /\mathbb {Q}_p}, \end{aligned}$$

where $\tilde{\Gamma }^{\textrm{cyc}} = {\text {Gal}}(F_\infty \mathbb {Q}_p^\textrm{cyc}/F_\infty )\subset {\text {Gal}}(F_\infty \mathbb {Q}_p^\textrm{cyc}/K_\mathfrak {p})$, $S_{F_\infty /\mathbb {Q}_p}$ is the Yager module [26, §3.2], and the transition maps in the inverse limits are given by corestrictions on the left, and trace maps on $\mathcal {O}_{F_m}$ on the right [26, Theorem 4.7]. As a $\Lambda _{\mathbb {Z}_p}(U)$-module, $S_{F_\infty /\mathbb {Q}_p}$ is contained in $\Lambda _{\mathcal {O}_{\hat{F}_\infty }}(U)$ and is free of rank one. We fix a basis $\mathcal {Y}$ of the $\Lambda _{\mathbb {Z}_p}(U)$-module $S_{F_\infty /\mathbb {Q}_p}$.

We fix an $\mathcal {O}$-basis $\omega$ of $\textrm{Fil}^0\mathbb {D}_{\textrm{cris}}(T)$, then $v_1=\omega ,v_2=\varphi (\omega )$ form an $\mathcal {O}$-basis of $\mathbb {D}_{\textrm{cris}}(T)$ (see, e.g., [22, Lemma 3.1]). Furthermore, for any intermediate extension $F_m$, we have Coleman maps

$$\begin{aligned} \textrm{Col}_{\sharp /\flat ,F_m}:H^1_{\textrm{Iw}}(F_m\mathbb {Q}_p^\textrm{cyc},T)\rightarrow \Lambda ({\text {Gal}}(F_m\mathbb {Q}_p^{\textrm{cyc}}/F_m))\otimes _{\mathbb {Z}_p}\mathcal {O}_{F_m}, \end{aligned}$$

such that [4, §2.3]

$$\begin{aligned} \mathcal {L}_{T,F_m} = (v_1,v_2)M_{\log }\begin{pmatrix} \textrm{Col}_{\sharp ,F_m}\\ \textrm{Col}_{\flat ,F_m}\\ \end{pmatrix}, \end{aligned}$$

where $M_{\log }$ is the logarithm matrix, a $2\times 2$ matrix valued in $\mathscr {H}_\infty ({\text {Gal}}(F_m\mathbb {Q}_p^{\textrm{cyc}}/F_m))\simeq \mathscr {H}_\infty (\tilde{\Gamma }^{\textrm{cyc}})$ that is independent of m. Taking the inverse limit, we thus obtain a factorization of the big logarithm:

$$\begin{aligned} \mathcal {L}_{T,F_\infty } = (v_1\ v_2)M_{\log } \begin{pmatrix} \textrm{Col}_{\sharp ,F_\infty }\\ \textrm{Col}_{\flat ,F_\infty }\\ \end{pmatrix}, \end{aligned}$$

(A.1)

where, for $\star \in \{\flat ,\sharp \}$,

$$\begin{aligned} \textrm{Col}_{\star ,F_\infty }:H^1_\textrm{Iw}(k_{\infty ,\mathfrak {p}},T)\rightarrow \Lambda (\tilde{\Gamma }^{\textrm{cyc}})\widehat{\otimes } \mathcal {Y}\Lambda (U)\subset \Lambda _{\mathcal {O}_{\hat{F}_\infty }}(\tilde{\Gamma }^{\textrm{cyc}}\times U) \end{aligned}$$

is the limit of $\textrm{Col}_{\star ,F_m}$.

Remark A.1

Since $\mathcal {L}_{T,F_\infty }$ is $\Lambda (U)$-linear [26, paragraph after Definition 4.6], so are the Coleman maps defined above. As such, for $\star \in \{\sharp ,\flat \}$, a consideration of the determinants of $\mathcal {L}_{T,F_\infty }$ and $M_{\log }$ shows that $\textrm{im}(\textrm{Col}_{\star ,F_m})= \textrm{im}(\textrm{Col}_{\star ,F})\otimes \mathcal {O}_{F_m}\subseteq \Lambda ({\text {Gal}}(F_m\mathbb {Q}_p^{\textrm{cyc}}/F_m))\otimes _{\mathbb {Z}_p}\mathcal {O}_{F_m}$. Thus

$$\begin{aligned} \textrm{im}(\textrm{Col}_{\star ,F_\infty })= \textrm{im}(\textrm{Col}_{\star ,F})\otimes \mathcal {Y}\Lambda (U)\subseteq \Lambda (\tilde{\Gamma }^{\textrm{cyc}})\otimes \Lambda _{\mathcal {O}_{\hat{F}_\infty }}(U). \end{aligned}$$

We now project the factorization (A.1) to the anticyclotomic subextension, following [9, Theorem 5.1]. Let $K^\textrm{ac}$ denote the anticyclotomic $\mathbb {Z}_p$-extension of K,³ and denote by $\Gamma ^\textrm{ac} = {\text {Gal}}(K^\textrm{ac}/K)\simeq {\text {Gal}}(K^\textrm{ac}_\mathfrak {p}/K_\mathfrak {p})\simeq \mathbb {Z}_p$.

Theorem A.2

By quotienting out $\mathfrak {J}=\ker (\Lambda (U\times \tilde{\Gamma }^{\textrm{cyc}})\rightarrow \Lambda (\Gamma ^\textrm{ac}))$, the map $\mathcal {L}_{T,F_\infty }$ descends to a $\Lambda (\Gamma ^\textrm{ac})$-linear map

$$\begin{aligned} \mathcal {L}_{T}^{\textrm{ac}}: H^1_\textrm{Iw}(K^\textrm{ac}_\mathfrak {p}, T) \rightarrow \mathscr {H}_{\infty ,{\hat{F}}_\infty }(\Gamma ^\textrm{ac})\otimes \mathbb {D}_\textrm{cris}(T). \end{aligned}$$

Therefore, there exist Coleman maps $\textrm{Col}_{\sharp }^\textrm{ac},\textrm{Col}_{\flat }^\textrm{ac}: H^1_\textrm{Iw}(K^\textrm{ac}_\mathfrak {p},T)\rightarrow \Lambda (\Gamma ^\textrm{ac})$, being the projections of $\textrm{Col}_{\sharp ,F_\infty },\textrm{Col}_{\flat ,F_\infty }$ in $\Lambda (\Gamma ^\textrm{ac})$, such that the following factorization holds for all $z\in H^1_\textrm{Iw}(K^\textrm{ac}_\mathfrak {p},T)$

$$\begin{aligned} \mathcal {L}_{T}^{\textrm{ac}}(z) = (v_1\ v_2)\overline{M}_{\log } \begin{pmatrix} \textrm{Col}_{\sharp }^{\textrm{ac}}(z)\\ \textrm{Col}_{\flat }^{\textrm{ac}}(z)\\ \end{pmatrix}. \end{aligned}$$

Here, $\overline{M}_{\log }$ denotes the image of $M_{\log }$ in $M_2(\mathscr {H}_{k-1,\mathfrak {F}}(\tilde{\Gamma }^\textrm{cyc})/\mathfrak {J})=M_2(\mathscr {H}_{k-1,\mathfrak {F}}(\Gamma ^\textrm{ac}))$.

Proof

Recall from [18, Lemma 2.7] that $V^{{\text {Gal}}(K^\textrm{ac}_\mathfrak {p}/K)}=0$. Thus, the theorem follows from [9, Theorem 5.1]. $\square$

Remark A.3

As explained in [24, Remark 5.2], we may identify ${\text {Gal}}(k_{\infty ,\mathfrak {p}}/K_\mathfrak {p})$ with $\mathbb {Z}_p^2$ such that U is the first component and $\tilde{\Gamma }^{\textrm{cyc}}$ is the second component. As such, $\Gamma ^\textrm{ac}$ corresponds to anti-diagonal elements

$$\begin{aligned} \{(a,-a)\in \mathbb {Z}_p^2: a\in \mathbb {Z}_p\}. \end{aligned}$$

1.2 A.2 Evaluating the logarithm matrix

In what follows, we fix a family of primitive $p^n$-th roots of unity $\zeta _{p^n}$ and write $\epsilon _n = \zeta _{p^n}-1$. We may choose topological generators $\gamma ^{\textrm{ur}}\in U$ and $\gamma ^\textrm{cyc}\in \tilde{\Gamma }^\textrm{cyc}$ resulting in a topological generator $\gamma$ of $\Gamma ^\textrm{ac}$ given by

$$\begin{aligned} \gamma = (\gamma ^\textrm{cyc}/\gamma ^\textrm{ur})^{1/2}, \end{aligned}$$

which is possible by Remark A.3 and that $p\ne 2$. By these choices, for $G\in \{U,\tilde{\Gamma }^{\textrm{cyc}},\Gamma ^\textrm{ac}\}$, we regard $\mathscr {H}_{\infty }(G)$ as the set of power series convergent on the open unit disc centered at 0 with variable $X_G$. To simplify notation, we write $Y=X_U$, $Z = X_{\tilde{\Gamma }^\textrm{cyc}}$ and $X=X_{\Gamma ^\textrm{ac}}$. As $\gamma = (\gamma ^\textrm{cyc}/\gamma ^\textrm{ur})^{1/2}$, the natural projection

$$\begin{aligned} \mathscr {H}_\infty (U\times \tilde{\Gamma }^\textrm{cyc}) = \mathscr {H}_\infty (U)\hat{\otimes }\mathscr {H}_\infty (\tilde{\Gamma }^\textrm{cyc}) \rightarrow \mathscr {H}_\infty (\Gamma ^\textrm{ac}) \end{aligned}$$

is given by sending f(Y, Z) to $f((1+X)^{-1}-1,X)$.

We now turn to the explicit description of $M_{\log }$ and thus $\overline{M}_{\log }$, following [3, §2]. Denote by $\Phi _{p^n}(Z)$ the $p^n$-th cyclotomic polynomial $\frac{(Z+1)^{p^n}-1}{(Z+1)^{p^{n-1}}-1}$. Additionally, recall the matrices

$$\begin{aligned} A= \begin{pmatrix} 0 &{} -1/p\\ 1 &{} a_p/p \end{pmatrix}\quad and \quad Q_n(Z) = \begin{pmatrix} a_p &{} 1\\ -\Phi _{p^{n+1}}(Z) &{} 0\\ \end{pmatrix}\ (n\ge 0). \end{aligned}$$

Proposition 2.5 ibid. tells us that

$$\begin{aligned} M_{\log }(Y,Z) = M_{\log }(Z) = \lim _{n\rightarrow \infty } A(Z)^{n+1}Q_{n-1}(Z)\cdots Q_0(Z). \end{aligned}$$

Consequently,

$$\begin{aligned} \overline{M}_{\log }(X) = M_{\log }(X) = \lim _{n\rightarrow \infty } A(X)^{n+1}Q_{n-1}(X)\cdots Q_0(X). \end{aligned}$$

Henceforward we shall not distinguish $\overline{M}_{\log }$ from $M_{\log }$.

Next, given $\phi =\begin{pmatrix} a &{} b \\ c &{} d\\ \end{pmatrix}$ with entries valued in $\overline{\mathbb {Q}_p}$, following the notation introduced in [31, Definition 4.4], we write

$$\begin{aligned} \textrm{ord}_p(\phi ) = \begin{pmatrix} \textrm{ord}_p(a) &{} \textrm{ord}_p(b)\\ \textrm{ord}_p(c) &{} \textrm{ord}_p(d)\\ \end{pmatrix}. \end{aligned}$$

Further, we denote $\textrm{ord}_p(a_p)$ by v. To state the result below, we recall from [8, Theorem 2.1] that the two roots of $X^2 - a_p X + p$ are distinct since f is of weight 2.

Lemma A.4

Let $\alpha \ne \beta$ be the two roots of the Hecke polynomial $X^2 - a_p X + p$. Also let S denote the matrix

$$\begin{aligned} S = \begin{pmatrix} 1 &{} 1\\ -\alpha &{} -\beta \end{pmatrix}. \end{aligned}$$

Then $M_{\log }(\epsilon _n)$ is of the form

$$\begin{aligned}(\alpha -\beta )^{-1}S \begin{pmatrix} -s_1/\beta ^n &{} -s_2/\beta ^n\\ s_1/\alpha ^n &{} s_2/\alpha ^n \end{pmatrix}, \end{aligned}$$

for some $s_1,s_2\in \overline{\mathbb {Q}_p}$ of p-adic valuations $v\varvec{1}_{2\not \mid n}+\sum _{1\le i\le n/2}p^{-2i+1}$ and $v\varvec{1}_{2\mid n} + \sum _{1\le i< n/2}p^{-2i}$ respectively.

Proof

Note that for $i\ge n$, we have $\Phi _{p^{i+1}}(\epsilon _n) = p$, which implies that $A = Q_i(\epsilon _n)^{-1}$. This implies that $M_{\log }(\epsilon _n) = A^{n+1}Q_{n-1}\cdots Q_0(\epsilon _n)$. By [22, Proposition 4.6], we have

$$\begin{aligned} \text {ord}_p(Q_{n-1}\cdots Q_0(\epsilon _n))= \left\{ \begin{array}{ll} \begin{pmatrix} v+\sum _{i=1}^{\frac{n-1}{2}} \frac{1}{p^{2i-1}} &{}{} \sum _{i=1}^{\frac{n-1}{2}} \frac{1}{p^{2i}}\\ \infty &{}{} \infty \\ \end{pmatrix} &{}{} \text{ if } n \text{ is } \text{ odd, }\\ \\ \begin{pmatrix} \sum _{i=1}^{\frac{n}{2}} \frac{1}{p^{2i-1}} &{}{} v+\sum _{i=1}^{\frac{n}{2}-1} \frac{1}{p^{2i}}\\ \infty &{}{} \infty \\ \end{pmatrix} &{}{} \text{ if } n \text{ is } \text{ even }.\\ \end{array}\right. \end{aligned}$$

For the matrix A, we have the diagonalization

$$\begin{aligned} A = p^{-1}S\begin{pmatrix} \alpha &{} 0\\ 0 &{} \beta \\ \end{pmatrix}S^{-1}. \end{aligned}$$

Write $Q_{n-1}\cdots Q_0(\epsilon _n) = \begin{pmatrix} s_1 &{} s_2 \\ 0 &{} 0\\ \end{pmatrix}$, we have

$$\begin{aligned} S^{-1}A^{n+1}Q_{n-1}\cdots Q_0(\epsilon _n)&= p^{-n-1} \begin{pmatrix} \alpha ^{n+1} &{} 0 \\ 0 &{} \beta ^{n+1} \end{pmatrix} S^{-1} \begin{pmatrix} s_1 &{} s_2 \\ 0 &{} 0\\ \end{pmatrix}\\&=p^{-n-1} \begin{pmatrix} \alpha ^{n+1} &{} 0 \\ 0 &{} \beta ^{n+1} \end{pmatrix} \begin{pmatrix} -\beta s_1 &{} -\beta s_2\\ \alpha s_1 &{} \alpha s_2 \end{pmatrix} \frac{1}{(\alpha -\beta )}\\&=\frac{p^{-n-1}}{\alpha -\beta } \begin{pmatrix} -p\alpha ^n &{} 0\\ 0 &{} p\beta ^n\\ \end{pmatrix} \begin{pmatrix} s_1 &{} s_2\\ s_1 &{} s_2\\ \end{pmatrix}\\&=\frac{1}{\alpha -\beta } \begin{pmatrix} -s_1/\beta ^n &{} -s_2/\beta ^n\\ s_1/\alpha ^n &{} s_2/\alpha ^n\\ \end{pmatrix}. \end{aligned}$$

$\square$

1.3 A.3. Evaluation of Coleman maps

We shall evaluate the images of the Coleman maps at $\epsilon _n$ using [21, Proposition 2.2]. Write $\overline{\mathcal {Y}}$ as the anticyclotomic projection of $\mathcal {Y}$, and we define $\underline{\textrm{Col}} = (\textrm{Col}_\sharp ^\textrm{ac},\textrm{Col}_\flat ^\textrm{ac}): H^1_{\textrm{Iw}}(K_\mathfrak {p}^\textrm{ac},T)\rightarrow \Lambda (\Gamma ^{\textrm{ac}})^{\oplus 2}$.

Proposition A.5

Let

$$\begin{aligned} I_v = \{(G_1,G_2)\in \overline{\mathcal {Y}}\Lambda (\Gamma ^{\textrm{ac}})^{\oplus 2}\subseteq \Lambda _{\mathcal {O}_{\hat{F}_\infty }}(\Gamma ^{\textrm{ac}})^{\oplus 2}: (p-1)G_1(0) = (2-a_p)G_2(0)\}. \end{aligned}$$

Then $\textrm{im}(\underline{\textrm{Col}})=I_v$.

Proof

It follows from [21, Proposition 2.2] that

$$\begin{aligned} \textrm{im}(\textrm{Col}_{\sharp ,F},\textrm{Col}_{\flat ,F})&= \{(G_1,G_2)\in \Lambda ({\text {Gal}}(\mathbb {Q}_p^{\textrm{cyc}}/\mathbb {Q}_p))^{\oplus 2}: (p-1)G_1(0)\\&=(2-a_p)G_2(0)\}. \end{aligned}$$

Thus, the affirmation on $\textrm{im}(\underline{\textrm{Col}})$ follows from Remark A.1. $\square$

Lemma A.6

The period $\mathcal {Y}\in \Lambda _{\mathcal {O}_{\hat{F}_\infty }}(U)$ is invertible, and thus so is $\overline{\mathcal {Y}}\in \Lambda _{\mathcal {O}_{\hat{F}_\infty }}(\Gamma ^{\textrm{ac}})$.

Proof

The period $\mathcal {Y}$ is constructed from choosing a compatible system of integral normal basis generator $(x_{F_m})_{m\ge 0}\in \varprojlim _m \mathcal {O}_{F_m}$, which is identified with an element of $\Lambda _{\mathcal {O}_{\hat{F}_\infty }}(U)$ via the maps

$$\begin{aligned} y_{F_m/F}: \mathcal {O}_{F_m} \rightarrow \mathcal {O}_{F_m}[{\text {Gal}}(F_m/F)],\quad x\mapsto \sum _{\sigma \in {\text {Gal}}(F_m/F)} x^{\sigma }[\sigma ^{-1}], \end{aligned}$$

(see [26, §3.2]). Thus, the constant term of $\mathcal {Y}$ as a power series is $\lim _m {\text {Tr}}_{F_m/F}(x_{F_m})=x_F$, which is a unit of $\mathcal {O}_F$ since $\mathcal {O}_F \cdot x_F = \mathcal {O}_F$, from which the lemma follows. $\square$

Henceforth, we shall use the same notation for $\mathcal {Y}$ and $\overline{\mathcal {Y}}$ for presentational simplicity.

Corollary A.7

There exists a $\Lambda (\Gamma ^\textrm{ac})$-basis $(z_1,z_2)$ of $H^1_{\textrm{Iw}}(K_\mathfrak {p},T)$ such that

$$\begin{aligned} \underline{\textrm{Col}}(z_1) = \mathcal {Y}(X,0), \ \underline{\textrm{Col}}(z_2) = \mathcal {Y}(a',1), \end{aligned}$$

where $a'=\frac{2-a_p}{p-1}$.

Proof

It can be checked that $X\oplus 0,$ and $a'\oplus 1$ form a $\Lambda$-basis of the image of $\mathcal {Y}^{-1}\underline{\textrm{Col}}$. Thus, the result follows from the injectivity of $\underline{\textrm{Col}}$ (see [4, Proof of Corollary 4.6]). $\square$

Next we compare the maps $\mathcal {L}_{T}^\textrm{ac}$ and $\tilde{\Omega }^{\epsilon }_{V,1}$ constructed by Kobayashi [19, §10], using the explicit reciprocity law.

Lemma A.8

There exists a unit in $u^\epsilon _T\in \Lambda _{\mathcal {O}_{\hat{F}_\infty }}(\Gamma ^\textrm{ac})$ such that

$$\begin{aligned} \mathcal {L}^\textrm{ac}_T\circ \tilde{\Omega }^{\epsilon }_{V,1}= u^\epsilon _T\ell _0, \end{aligned}$$

where $\ell _0=\log (\gamma )/\log (\kappa (\gamma ))$ and $\kappa$ is the Lubin–Tate character attached to the extension $K^\textrm{ac}_\mathfrak {p}/\mathbb {Q}_p$.

Proof

Let $\textrm{Col}^\epsilon _e:H^1_{\textrm{Iw}}(K_\mathfrak {p}^\textrm{ac},T)\rightarrow \mathscr {H}_\infty (\Gamma ^\textrm{ac})\otimes \mathbb {D}_\textrm{cris}(T)$ be the map defined in [10, Definition 3.3], where e is some fixed unit in $\Lambda (\Gamma ^{\textrm{ac}})$ (note that we are taking $F=\mathbb {Q}_p$ in loc. cit.). Upon comparing the interpolation formulae given in Theorem 3.4 of op. cit. and [9, Theorem 5.1], we see that $\mathcal {L}_T^\textrm{ac}$ and $\textrm{Col}^\epsilon _e$ agree up to a unit. Therefore, it is enough to study the composition $\textrm{Col}^\epsilon _e$ and $\tilde{\Omega }_{V,1}^\epsilon$.

Let $[-,-]_V$ and $\langle -,-\rangle _{F_\infty }$ be the pairings defined in [10, §3.3]. Then (3,7) in op. cit. says that

$$\begin{aligned}{}[\textrm{Col}_e^\epsilon (z),\eta ]_V=\langle z,\Omega _{V,1}^\epsilon (\eta \otimes e)\rangle _{F_\infty } \end{aligned}$$

for all $z\in H^1_{\textrm{Iw}}(K_\mathfrak {p}^\textrm{ac},T)$ and $\eta \in \mathbb {D}_\textrm{cris}(T)$. Thus, given any $x\in \mathscr {H}_\infty (\Gamma ^\textrm{ac})\otimes \mathbb {D}_\textrm{cris}(T)$, we deduce from the explicit reciprocity law (see [19, Theorem 10.13])

$$\begin{aligned} {[}\textrm{Col}_e^\epsilon \circ \Omega _{V,1}^\epsilon (x),\eta ]_V&=\langle \Omega _{V,1}^\epsilon (x),\Omega _{V,1}^\epsilon (\eta \otimes e)\rangle _{F_\infty }\\&=\langle \ell _0 \Omega _{V,0}^\epsilon (x),\Omega _{V,1}^\epsilon (\eta \otimes e)\rangle _{F_\infty }\\&=[ \ell _0 x,\eta \otimes e]_V\\&=[ \ell _0e^\iota x,\eta ]_V. \end{aligned}$$

(Note that the image of $\delta _{-1}$ in loc. cit. is sent to the trivial element in $\Gamma ^\textrm{ac}$.) Therefore, the result follows from the non-degeneracy of the pairing $[-,-]_V$. $\square$

Corollary A.9

Let $e_{\alpha },e_{\beta }$ be a $\varphi$-eigenbasis of $\mathbb {D}_{\textrm{cris}}(V)$ given by

$$\begin{aligned} (e_\alpha ,e_\beta ) = (\omega ,\varphi (\omega ))S = (\omega ,\varphi (\omega )) \begin{pmatrix} 1 &{} 1\\ -\alpha &{} -\beta \\ \end{pmatrix} \end{aligned}$$

(with $\varphi (e_{\lambda })=\lambda p^{-1}e_{\lambda }$ for $\lambda \in \{\alpha ,\beta \}$). The matrix of $\mathcal {L}_{T}^\textrm{ac}$ with respect to the bases $(z_1,z_2)$ and $(e_{\alpha },e_{\beta })$ is given by

$$\begin{aligned}\mathcal {Y}S^{-1} M_{\log }\cdot \begin{pmatrix} X&{}a'\\ 0&{}1 \end{pmatrix}. \end{aligned}$$

The matrix of $\tilde{\Omega }^{\epsilon }_{V,1}$ with respect to the same bases is given by

$$\begin{aligned} \frac{u^\epsilon _T \ell _0}{\mathcal {Y}X}\begin{pmatrix} 1 &{} -a' \\ 0 &{} X\\ \end{pmatrix} M_{\log }^{-1}S. \end{aligned}$$

Proof

By Corollary A.7, we have a basis $z_1,z_2$ of $H^1_\textrm{Iw}(K_\mathfrak {p},T)$ such that

$$\begin{aligned} (\mathcal {L}_T^\textrm{ac}(z_1),\mathcal {L}_T^\textrm{ac}(z_2)) = (\omega , \varphi (\omega ))M_{\log }\mathcal {Y}\begin{pmatrix} X &{} a'\\ 0 &{} 1\\ \end{pmatrix}. \end{aligned}$$

The affirmation regarding $\mathcal {L}_T^{\textrm{ac}}$ now follows from the change of variable formula in the definition of $e_\alpha ,e_\beta$. Taking the inverse of $\mathcal {L}^\textrm{ac}_T$ in Lemma A.8 gives the matrix for $\tilde{\Omega }^\epsilon _{V,1}$.

Corollary A.10

For large enough even integers m, the specialization of $\tilde{\Omega }^\epsilon _{V,1}$ at $\mathfrak {P}_m=(\Theta _m(X))$ has p-adic valuation matrix

$$\begin{aligned} \begin{pmatrix} -m\cdot \textrm{ord}_p(\alpha ) + \displaystyle \sum _{i=1}^{m/2}\frac{1}{p^{2i-1}} &{} \displaystyle -m\cdot \textrm{ord}_p(\beta ) + \sum _{i=1}^{m/2}\frac{1}{p^{2i-1}} \\ \displaystyle -m\cdot \textrm{ord}_p(\alpha ) + \sum _{i=1}^{ m/2}\frac{1}{p^{2i-1}}+\frac{1}{p^{m-1}(p-1)} &{} \displaystyle -m\cdot \textrm{ord}_p(\beta ) + \sum _{i=1}^{m/2}\frac{1}{p^{2i-1}}+ \frac{1}{p^{m-1}(p-1)}\\ \end{pmatrix} \end{aligned}$$

Proof

By [3, Proposition 2.5], $\det (M_{\log })$ and $\ell _0/X$ differ by a unit in $\Lambda (\Gamma ^\textrm{ac})$. It follows that $\frac{\ell _0}{X} M_{\log }^{-1}S$ is the adjugate matrix of $\det (S)S^{-1}M_{\log } = (\alpha -\beta )S^{-1}M_{\log }$, up to a unit. By Lemmas A.4 and A.6, we see that, up to a unit, the matrix of $\tilde{\Omega }^\epsilon _{V,1}$ given in Corollary A.10 specialized at $\Theta _m$, is of the form

$$\begin{aligned} \begin{pmatrix} 1 &{} -a' \\ 0 &{} \epsilon _m \end{pmatrix} \begin{pmatrix} s_2/\alpha ^m &{} s_2/\beta ^m\\ -s_1/\alpha ^m &{} -s_1/\beta ^m \end{pmatrix}. \end{aligned}$$

It follows from our assumption that $v\ge \frac{1}{p+1}$, for an even integer m that is sufficiently large,

$$\begin{aligned} \textrm{ord}_p(s_2) = v+ \sum _{1\le i<m/2}\frac{1}{p^{2i}} = v + \frac{1-1/p^{m}}{p^2-1} > \frac{p-1/p^{m-1}}{p^2-1} = \textrm{ord}_p(s_1). \end{aligned}$$

Since $a'$ is a p-adic unit, we have $\textrm{ord}_p(s_2+a's_1) = \textrm{ord}_p(s_1)$. Hence, the result follows. $\square$