2. Notation

We begin by introducing further notation, which will be used in later sections of the paper. We put $\varpi =\phi ({\mathfrak {p}})$ and $\varpi ^*=\phi ({\mathfrak {p}}^*)$ , and observe that these are generators of the respective ideals. For each pair of integers $m,n\geq 0$ , let $F_m$ denote the field $K(A[\varpi ^{*m+2}])$ and $K_{n,m}$ the field $F_m(A[\varpi ^{n+2}])$ . The extension $K_{n,m}$ over $F_m$ is totally ramified at the primes lying above $\mathfrak {p}$ , and $\mathfrak {p}$ is unramified in $F_m$ . From CM theory, it is well-known that the number of primes of $F_m$ lying above $\mathfrak {p}$ , which we denote by $r_m$ , is either equal to $1$ (in this case, $\mathfrak {p}$ is inert in $F_0/K$ ) or equal to twice of the index of the subgroup generated by $\pi$ in $(1+{\mathfrak {p}}^{*2})/(1+{\mathfrak {p}}^{*m+2})$ . Hence, there exists an integer $M$ such that $r_m=r_{0}2^m$ for $m\lt M$ and $r_m=r_02^M$ for $m\geq M$ . We choose and fix a prime ${\mathfrak {p}}_M$ of $F_M$ lying above $\mathfrak {p}$ , and let ${\mathfrak {p}}_m$ denote the unique prime of $F_m$ lying above (or below) ${\mathfrak {p}}_M$ .

We write ${\mathfrak {p}}_{n,m}$ for the unique prime of $K_{n,m}$ lying above ${\mathfrak {p}}_m$ . If $\omega$ is any prime of $F_m$ lying above $\mathfrak {p}$ , we let $\Xi _{n,m,\omega }$ be the completion of $K_{n,m}$ at the unique prime above $\omega$ , and we let $\Phi _{m,\omega }$ denote the completion of $F_m$ at $\omega$ . We shall write $\mathscr {I}_{m,\omega }$ for the ring of integers of $\Phi _{m,\omega }$ and we shall also write $\omega$ for the maximal ideal of $\mathscr {I}_{m,\omega }$ . For simplicity, we shall omit the subscript for the primes when referring to completions at or above ${\mathfrak {p}}_m$ . We denote by $K_{\mathfrak {p}}$ the completion of $K$ at $\mathfrak {p}$ , and we shall identify its ring of integers ${\mathcal {O}}_{\mathfrak {p}}$ with ${\mathbb {Z}}_2$ . Put

\begin{equation*}K_\infty =\bigcup _{n,m\geq 0} K_{n,m},\qquad F_\infty =\bigcup _{m\geq 0} F_m,\qquad \textrm {and}\qquad \Phi _\infty =\bigcup _{m\geq 0} \Phi _m.\end{equation*}

Let $\varphi$ denote the Artin symbol $({\mathfrak {p}},F_\infty /K)$ for the extension $F_\infty$ over $K$ . Then $\varphi$ also induces the Frobenius automorphism for the extension $\Phi _\infty$ over $K_{\mathfrak {p}}$ . Note that we always view our global fields as lying inside the complex numbers, and equipped with embeddings into their completions.

Write $\mathcal {G}$ for the Galois group of $K_\infty$ over $K$ . Let $\chi _{\mathfrak {p}}: {\mathcal {G}}\rightarrow {\mathbb {Z}}^\times _2$ and $\chi _{{\mathfrak {p}}^*}: {\mathcal {G}}\rightarrow {\mathbb {Z}}^\times _2$ be respectively the characters giving the actions of $\mathcal {G}$ on $A[\varpi ^\infty ]=A[{\mathfrak {p}}^\infty ]$ and $A[\varpi ^{*\infty }]=A[{\mathfrak {p}}^{*\infty }]$ . Observe that if $\sigma \in {\mathcal {G}}$ and $\alpha$ is an element of $\mathcal {O}$ such that $u^\sigma =\alpha u$ for all $u\in A[\varpi ^{*m+2}]$ , then $\chi _{{\mathfrak {p}}^*}(\sigma )$ is given modulo $2^{m+2}$ by a representative lying in $\mathbb {Z}$ of the coset of $\alpha$ modulo ${\mathfrak {p}}^{*m+2}$ . These rational integral representatives are precisely the rational integers belonging to the coset of $\overline {\alpha }$ modulo ${\mathfrak {p}}^{m+2}$ , and so, under our identification of ${\mathbb {Z}}_2$ with ${\mathcal {O}}_{\mathfrak {p}}$ , it follows that

(2.1) \begin{equation} \chi _{{\mathfrak {p}}^*}(\sigma )\equiv \overline {\alpha }{\ \mathrm {mod}\ } {\mathfrak {p}}^{m+2}. \end{equation}

By our assumption that $A$ has good reduction at $\mathfrak {p}$ and ${\mathfrak {p}}^*$ and the ramification consideration, we know that ${\mathcal {G}}=\Gamma \times \Delta$ , where $\Gamma ={\mathrm {Gal}}(K_\infty /K_{0,0})$ , and $\Delta =({\mathbb {Z}}/2{\mathbb {Z}})^2$ which can be identified with ${\mathrm {Gal}}(K_{0,0}/K)$ . We observe that the canonical characters $\chi _{\mathfrak {p}}$ and $\chi _{{\mathfrak {p}}^*}$ provide an isomorphism $(\chi _{\mathfrak {p}},\chi _{{\mathfrak {p}}^*}):{\mathcal {G}}\stackrel {\sim }{\longrightarrow }{\mathbb {Z}}^\times _2\times {\mathbb {Z}}^\times _2$ , and we deduce that $\Gamma \simeq {\mathbb {Z}}_2^2={\mathbb {Z}}_2\times {\mathbb {Z}}_2$ . We let $\eta _1$ and $\eta _2$ denote the restriction of $\chi _{\mathfrak {p}}$ and $\chi _{{\mathfrak {p}}^*}$ to $\Delta$ respectively. If $\mathfrak {M}$ is any ${\mathbb {Z}}_2[\Delta ]$ -module, we define ${\mathfrak {M}}^{(i_1,i_2)}$ to be the maximal quotient of $\mathfrak {M}$ on which $\Delta$ acts via $\eta ^{i_1}_1\eta ^{i_2}_2$ , where $i_1,i_2\in \{0,1\}$ . One can verify easily that ${\mathfrak {M}}^{(i_1,i_2)}={\mathfrak {M}}\otimes _{{\mathbb {Z}}_2[\Delta ]}{\mathbb {Z}}_2$ by viewing ${\mathbb {Z}}_2$ as a ${\mathbb {Z}}_2[\Delta ]$ -module via $\eta ^{i_1}_1\eta ^{i_2}_2$ .

Let $\Lambda$ be the ring of formal power series in the commuting indeterminates $T_1$ and $T_2$ with coefficients in ${\mathbb {Z}}_2$ . Choose a topological generator $u$ of $1+4{\mathbb {Z}}_2$ and let $\gamma _1$ and $\gamma _2$ be the elements of $\Gamma$ for which $\chi _{\mathfrak {p}}(\gamma _1)=\chi _{{\mathfrak {p}}^*}(\gamma _2)=u$ and $\chi _{\mathfrak {p}}(\gamma _2)=\chi _{{\mathfrak {p}}^*}(\gamma _1)=1$ . It is clear that $\gamma _1$ and $\gamma _2$ are a set of topological generators for $\Gamma$ . Any compact ${\mathbb {Z}}_2$ -module $\mathfrak {W}$ on which $\Gamma$ acts continuously can be endowed with a unique $\Lambda$ -module structure such that $\gamma _1x=(1+T_1)x$ and $\gamma _2 x=(1+T_2)x$ for all $x$ in $\mathfrak {W}$ .

The rings $\Xi _{n,m}=\prod _\omega \Xi _{n,m,\omega }$ and $\boldsymbol{\Phi }_m=\prod _{\omega }\Phi _{m,\omega }$ , where the product is taken over the set of primes $\omega$ of $F_m$ lying above $\mathfrak {p}$ , have a natural action of the Galois group $\mathcal {G}$ as follows. Let $\alpha _{k,\omega }(k=0,1,2,\cdots )$ be a Cauchy sequence of elements of $K_{n,m}$ (or $F_m$ ) which converge to $\alpha _\omega$ in $\Xi _{n,m,\omega }$ (or $\Phi _{m,\omega }$ ). Then the $\omega ^\sigma$ component of $(\alpha _\omega )^\sigma$ is the limit of the Cauchy sequence $a^\sigma _{k,\omega }(k=0,1,2,\cdots )$ in $\Xi _{n,m,\omega ^\sigma }$ (or $\Phi _{m,\omega ^\sigma }$ ). We embed $K_{n,m}$ and $F_m$ in these rings via the diagonal map, and it is easy to verify that the usual norm and trace maps on $\Xi _{n,m}$ , ${\boldsymbol{\Phi }}_m$ , $K_{n,m}$ and $F_m$ , as well as the Galois action, all commute with these embeddings.

Let $\widehat {A}$ be the formal group giving the kernel of reduction modulo $\mathfrak {p}$ on $A$ . The parameter of $\widehat {A}$ is $t=-\frac {x}{y}$ . Since $\widehat {A}$ is defined over ${\mathcal {O}}_{\mathfrak {p}}$ , we have the power series expansions

\begin{equation*}x=t^{-2}a(t),\qquad y=-t^{-3}a(t)\end{equation*}

where $a(t)$ has coefficients in ${\mathcal {O}}_{\mathfrak {p}}$ and constant term equal to $1$ . Denote by $\widehat {A}[\varpi ^n]$ the kernel of the endomorphism $[\varpi ^n]$ on $\widehat {A}$ , which we identify with $A[\varpi ^{n}]$ . We define $T_\varpi$ to be the Tate module $\varprojlim \widehat {A}[\varpi ^{n+2}]$ , where the limit is taken relative to the projection maps given by multiplication by powers of $\varpi$ . We fix a base $(u_n)$ of $T_{\varpi }$ , such that for each $n$ , $u_n$ corresponds to the point $P_n\in A[\varpi ^{n+2}]$ and $\varpi P_{n+1}=P_n$ .

Finally, we denote by $U'_{n,m,\omega }$ the units of $\Xi _{n,m,\omega }$ and by $U_{n,m,\omega }$ the subgroup consisting of those units which are congruent to $1$ modulo the maximal ideal. Put ${\mathbb {U}}'_{n,m}=\prod _{\omega }U'_{n,m,\omega }$ and ${\mathbb {U}}_{n,m}=\prod _\omega U_{n,m,\omega }$ , where the product is taken over the primes $\omega$ of $F_m$ lying above $\mathfrak {p}$ . Let ${\mathbb {U}}'_\infty$ and ${\mathbb {U}}_\infty$ denote the projective limits of the ${\mathbb {U}}'_{n,m}$ and ${\mathbb {U}}_{n,m}$ respectively relative to the norm maps on the $\Xi _{n,m}$ . We endow ${\mathbb {U}}_\infty$ with its natural structure as a ${\mathbb {Z}}_2[{\mathcal {G}}]$ -module. In particular, ${\mathbb {U}}_\infty$ is a compact $\Gamma$ -module, and thus also a $\Lambda$ -module.

3. Coleman power series and logarithmic derivatives

In this section, we first recall some basic facts about Coleman power series and associate a two-variable power series for each $\beta \in {\mathbb {U}}'_\infty$ , then we define the two-variable Coates-Wiles moment maps using these power series.

Recall that $T_{\varpi }$ denotes the Tate module $\varprojlim \widehat {A}[\varpi ^{n+2}]$ , and we have fixed a base $(u_n)$ of $T_{\varpi }$ . Let $\beta =(\beta _{n,m,\omega })$ be an element of ${\mathbb {U}}'_\infty$ .

Theorem 3.1 (Coleman [Reference Coleman9]).

Keep the notation as above, for each integer $m\geq 0$ and each prime $\omega$ of $F_m$ lying above $\mathfrak {p}$ , there exists a unique power series $c_{m,\omega, \beta }(T)\in {\mathscr {I}}_{m,\omega }[[T]]$ such that

(3.1) \begin{equation} \beta _{n,m,\omega }=c^{\varphi ^{-n}}_{m,\omega, \beta }(u_n)\qquad \textrm { for all}\,n\geq 0. \end{equation}

Moreover, these power series satisfy the functional equation

(3.2) \begin{equation} (c^{\varphi }_{m,\omega, \beta }\circ [\varpi ])(T)=\prod _{\eta \in \widehat {A}[\varpi ]}c_{m,\omega, \beta }(T\oplus \eta ), \end{equation}

where $T\oplus \eta$ denotes the sum of $T$ and $\eta$ under the addition on the formal group $\widehat {A}$ .

For the action of $\varphi ^{-n}$ in (3.1), we adopt the convention throughout this paper that an element of the Galois group written in this position acts only on coefficients of the power series. We denote the subring $\prod _\omega \mathscr {I}_{m,\omega }$ of $\boldsymbol{\Phi }_m$ by $\mathscr {I}_m$ . Then we denote by $c_{m,\beta }(T)$ the element $(c_{m,\omega, \beta }(T))\in \mathscr {I}_m[[T]]$ , with the Galois structure inherited from the structure on $\boldsymbol{\Phi }_m$ .

For $m'\geq m$ and $\omega '$ a prime of $F_{m'}$ lying above the prime $\omega$ of $F_m$ , let $N^n_{\omega ',\omega }$ denote the local norm map from $\Xi _{n,m',\omega '}$ to $\Xi _{n,m,\omega }$ . From (3.1), for each prime $\omega$ of $F_m$ lying above $\mathfrak {p}$ , we have

\begin{equation*}\prod _{\omega '\mid \omega }N^n_{\omega ',\omega }(c^{\varphi ^{-n}}_{m',\omega ',\beta }(u_n))=\beta _{n,m,\omega },\end{equation*}

where the product on the left is taken over all primes $\omega '$ of $F_{m'}$ lying above $\omega$ . Since $c_{m',\omega ',\beta }$ has coefficients in ${\mathscr {I}}_{m',\omega '}$ and $u_n\in \Xi _{n,m,\omega }$ , it is obvious that

(3.3) \begin{equation} \prod _{\omega '\mid \omega } \left (\left (\prod _{\sigma \in {\mathrm {Gal}}(\Phi _{m',\omega '}/\Phi _{m,\omega })}c^\sigma _{m',\omega ',\beta }\right )^{\varphi ^{-n}}(u_n)\right )=\beta _{n,m,\omega }. \end{equation}

We define $g_{m,\beta }(T)$ to be the vector in $\mathscr {I}_m[[T]]$ whose $\omega$ -component $(g_{m,\beta }(T))_\omega$ is given by

(3.4) \begin{equation} (g_{m,\beta })_\omega =\frac {1}{2}\log \left (\frac {c_{m,\omega, \beta }(T)^2}{c^\varphi _{m,\omega, \beta }([\varpi ](T))}\right ). \end{equation}

Here, we observe that $\varphi$ induces the Frobenius automorphism in ${\mathrm {Gal}}(\Phi _{m,\omega }/K_{\mathfrak {p}})$ , and

\begin{equation*}[\varpi ](T)\equiv T^2{\ \mathrm {mod}\ }\omega .\end{equation*}

Then we get

\begin{equation*}\frac {c_{m,\omega, \beta }(T)^2}{c^{\varphi }_{m,\omega, \beta }([\varpi ](T))}\equiv 1{\ \mathrm {mod}\ } \omega .\end{equation*}

Noting that $\mathfrak {p}$ is unramified in $F_m/K$ , we know $(g_{m,\beta }(T))_\omega \in \mathscr {I}_{m,\omega }[[T]]$ . Moreover, formula (3.2) shows that

(3.5) \begin{equation} \sum _{\eta \in \widehat {A}[\varpi ]}g_{m,\beta }(T\oplus \eta )=0. \end{equation}

Since the logarithm map takes multiplication to addition, for $m'\geq m$ , we let $\textrm {Tr}_{m',m}$ denote the trace map from $\mathscr {I}_{m'}[[T]]$ to $\mathscr {I}_m[[T]]$ .

Lemma 3.3. For each $\beta \in {\mathbb {U}}'_\infty$ , we have

\begin{equation*}g_{m,\beta }(T)=\textrm {Tr}_{m',m}(g_{m',\beta }(T)).\end{equation*}

We write $\varprojlim \mathscr {I}_m$ for the projective limit of the rings $\mathscr {I}_m$ relative to the trace maps. We put $\mathscr {I}_\infty =\cup _{m\geq 0}\mathscr {I}_m$ and denote the completion of $\mathscr {I}_\infty$ by $\widehat {\mathscr {I}}_\infty$ . The following theorem allows us to associate a power series with each element in $\varprojlim \mathscr {I}_m$ , and this theorem will play a key role for the rest of this paper.

Theorem 3.4. Let $b\in \varprojlim \mathscr {I}_m$ . Then there is a unique power series $h_b(T)\in \widehat {\mathscr {I}}_\infty [[T]]$ such that

\begin{equation*}h_b(T)\equiv \sum _{\sigma \in {\mathrm {Gal}}(F_\infty /K)}(b^\sigma )_{m,{\mathfrak {p}}_m}\cdot (1+T)^{\chi _{{\mathfrak {p}}^*}(\sigma )}{\ \mathrm {mod}\ } ((1+T)^{2^{m+2}}-1)\end{equation*}

for all $m\geq 0$ . Here, $(b^\sigma )_{m,{\mathfrak {p}}_m}$ denotes the ${\mathfrak {p}}_m$ -component of the projection onto $\mathscr {I}_m$ of the image of $b$ under the action of any element of $\mathcal {G}$ whose restriction to $F_m$ is $\sigma$ .

Proof. This is [Reference Yager27, Theorem 4]. Since it is important in later applications, we give a detailed proof below. Note that if $\theta \in {\mathrm {Gal}}(K_\infty /K)$ is trivial on $F_m$ , then $\chi _{{\mathfrak {p}}^*}(\theta )\equiv 1{\ \mathrm {mod}\ } 2^{m+2}$ , and hence $(1+T)^{\chi _{{\mathfrak {p}}^*}(\sigma )}$ is well-defined modulo $((1+T)^{2^{m+2}}-1)$ for all $\sigma \in {\mathrm {Gal}}(F_m/K)$ . Thus, it is suffice to check that appropriate compatibilities are satisfied. Let $m'\geq m$ . From the trace compatibility of an element of $\varprojlim \mathscr {I}_m$ , we have

\begin{equation*}(b^\sigma )_{m,{\mathfrak {p}}_m}=\sum _{\theta \in {\mathrm {Gal}}(F_{m'}/K)\,\, \theta |_{F_m}=\sigma }(b^\theta )_{m',{\mathfrak {p}}_{m'}}.\end{equation*}

Therefore, we get

\begin{equation*}\sum _{\theta \in {\mathrm {Gal}}(F_{m'}/K)\, \, \theta |_{F_m}=\sigma }(b^\theta )_{m',{\mathfrak {p}}_{m'}}\cdot (1+T)^{\chi _{{\mathfrak {p}}^*}(\theta )}\equiv (b^\sigma )_{m,{\mathfrak {p}}_m}\cdot (1+T)^{\chi _{{\mathfrak {p}}^*}(\sigma )}{\ \mathrm {mod}\ }(1+T)^{2^{m+2}}-1),\end{equation*}

and the theorem follows.

Now, applying the Theorem3.4 to the trace compatible element $(g_{m,\beta }(T))_m\in \varprojlim \mathscr {I}_m[[T]]$ , we get the following corollary.

Corollary 3.5. For each $\beta \in {\mathbb {U}}'_\infty$ , there is a unique power series $g_\beta (T_1,T_2)\in \widehat {\mathscr {I}}_\infty [[T_1,T_2]]$ such that

(3.6) \begin{equation} g_\beta (T_1,T_2)\equiv \sum _{\sigma \in {\mathrm {Gal}}(F_m/K)}(g^\sigma _{m,\beta }(T_1))_{{\mathfrak {p}}_m}\cdot (1+T_2)^{\chi _{{\mathfrak {p}}^*}(\sigma )}{\ \mathrm {mod}\ } ((1+T_2)^{2^{m+2}}-1). \end{equation}

Let $\lambda :\widehat {A}\stackrel {\sim }{\longrightarrow }{\mathbb {G}}_a$ be the logarithm map of $\widehat {A}$ , where ${\mathbb {G}}_a$ is the additive formal group. We may use $T_1$ for $t$ as the parameter of $\widehat {A}$ . For integers $k\geq 1$ and $j\geq 0$ , for each $\beta \in {\mathbb {U}}_\infty$ , we define

(3.7) \begin{equation} \delta _{k,j}(\beta )=\left (\lambda '(T_1)^{-1}\frac {\partial }{\partial T_1}\right )^k\left ((1+T_2)\frac {\partial }{\partial T_2}\right )^j g_\beta (T_1,T_2)\big |_{(T_1,T_2)=(0,0)}. \end{equation}

Lemma 3.6. Let $k\geq 1$ and $j\geq 0$ be two integers. Then $\delta _{k,j}$ is a homomorphism of ${\mathbb {Z}}_2$ -modules from ${\mathbb {U}}_\infty$ to $\widehat {\mathscr {I}}_\infty$ , and for all $\beta \in {\mathbb {U}}_\infty$ and all $\sigma \in {\mathcal {G}}$ , we have

(3.8) \begin{equation} \delta _{k,j}(\beta ^\sigma )=\chi _{\mathfrak {p}}(\sigma )^k\chi _{{\mathfrak {p}}^*}(\sigma )^{-j}\delta _{k,j}(\beta ). \end{equation}

Moreover, if $h(T_1,T_2)\in \Lambda$ , then

(3.9) \begin{equation} \delta _{k,j}(h(T_1,T_2)\beta )=h(u^k-1,u^{-j}-1)\delta _{k,j}(\beta ). \end{equation}

Proof. From the definition, it is clear that the map $\delta _{k,j}$ is a ${\mathbb {Z}}_2$ -homomorphism. For each $\sigma \in {\mathcal {G}}$ and $n\geq 0$ , from $u^\sigma _n=\chi _{\mathfrak {p}}(\sigma )u_n$ and the definition of the Coleman power series, we have

\begin{equation*}g_{m,\beta ^\sigma }(T)=g^\sigma _{m,\beta }([\chi _{\mathfrak {p}}(\sigma )](T)).\end{equation*}

Notice the formula (3.6), then we get

\begin{equation*}g_{\beta ^\sigma }(T_1,T_2)=g_\beta ([\chi _{\mathfrak {p}}(\sigma )](T_1),(1+T_2)^{\chi _{{\mathfrak {p}}^*}(\sigma )^{-1}}-1),\end{equation*}

thus, the formula (3.8) follows. For formula (3.9), it is (3.8) if we take $h(T_1,T_2)$ to be either $1+T_1$ or $1+T_2$ . Therefore, the formula (3.9) follows in general by linearity and continuity.

4. Elliptic units and their logarithmic derivatives

In this section, we shall define the groups of elliptic units in each field $K_{n,m}$ . These units were introduced by Robert in [Reference Robert21], and were used by Coates-Wiles, Yager and De Shalit to construct the $\mathfrak {p}$ -adic $L$ -functions over imaginary quadratic fields. Then we give the Coleman power series associated with these elliptic units and calculate the logarithmic derivatives of these power series.

Recall the integral ideal $\mathfrak {f}$ in $K$ which is the conductor of $\phi$ , where we recall that $\phi$ is the Hecke character over $K$ associated with $A$ . For any integral ideal $\mathfrak {b}$ in $K$ , we write $K({\mathfrak {b}})$ for the ray class field over $K$ with conductor $\mathfrak {b}$ .

From the lemma 4.1, we let $f_m=f\varpi ^{*m+2}$ , where $\varpi ^*=\phi ({\mathfrak {p}}^*)$ , and it is easily seen that $f_m$ generates ${\mathfrak {f}}_m$ . Let $\mathscr {J}$ denote the set of all $\alpha \in {\mathcal {O}}$ with $\alpha \neq 0, \pm 1$ and $\alpha$ is prime to $6{\mathfrak {f}}$ . For each $\alpha \in \mathscr {J}$ , we define the rational function $r_{\alpha }(P)$ on $A$ over $K$ by

(4.1) \begin{equation} r_{\alpha }(P) = \prod _{V}(x(P) - x(V))^{-1}, \end{equation}

where $V$ runs over any set of representatives of the set of non-zero elements of the Galois module $A[\alpha ]$ modulo $\pm 1$ . As is shown in the appendix of [Reference Coates3], there exists a unique non-zero $c_A(\alpha ) \in K$ such that the normalized function

(4.2) \begin{equation} R_{\alpha }(P) = c_A(\alpha )r_{\alpha, A}(P) \end{equation}

satisfies

(4.3) \begin{equation} R_{\alpha }(\beta (P)) = \prod _{W \in A[\beta ]}R_{\alpha }(P+ W) \end{equation}

for all non-zero $\beta$ in $\mathcal {O}$ with $(\beta, \alpha ) = 1$ .

Note that we have fixed an embedding of $K$ in $\mathbb {C}$ such that ${\mathcal {L}} = \Omega {\mathcal {O}}$ is the period lattice of the Néron differential of the global minimal Weierstrass equation (1.1) for $A$ over $K$ . Then we have the Weierstrass isomorphism ${\mathcal {W}}(z, {\mathcal {L}}) : {\mathbb {C}}/{\mathcal {L}} \to A({\mathbb {C}})$ given by

(4.4) \begin{equation} {\mathcal {W}}(z, {\mathcal {L}}) = \left (\wp (z, {\mathcal {L}}) - b_2/12, \frac {1}{2}(\wp '(z, {\mathcal {L}}) - a_1(\wp (z, {\mathcal {L}}) - b_2/12)) - a_3\right ), \end{equation}

where $b_2 = a_1^2 + 4a_2$ , and $\wp (z, {\mathcal {L}})$ is the classical Weierstrass $\wp$ -function of the lattice $\mathcal {L}$ .

Now, for each integer $m\geq 0$ , we define a primitive $f_m$ -division point $V_m$ on $A$ given by

(4.5) \begin{equation} V_m = {\mathcal {W}}\left (\frac {\Omega }{f_m}, {\mathcal {L}}\right ). \end{equation}

It is clear from definition that $\varpi ^* V_{m+1}=V_m$ for all $m\geq 0$ . By Lemma 4.1, we get the following diagram of field extensions over $K$

(4.6)

By the ramification consideration, $K({\mathfrak {f}})$ and $F_m$ are linearly disjoint over $K$ , furthermore, ${\mathrm {Gal}}(K({\mathfrak {f}}_m)/F_m)$ is isomorphic to ${\mathrm {Gal}}(K({\mathfrak {f}})/K)$ by the restriction map. Then, we define

(4.7) \begin{equation} {\mathfrak {R}}_{\alpha, V_m}(P) = \prod _{\tau \in {\mathrm {Gal}}(K(A[f_m])/F_m)} R_{\alpha }(P +V_m^\tau ). \end{equation}

Obviously ${\mathfrak {R}}_{\alpha, V_m}(P)$ is a rational function on $A$ with coefficients in $F_m$ . It is easy to show that ${\mathfrak {R}}_{\alpha, V_m}(P)$ depends only on the orbit of $V_m$ under the action of ${\mathrm {Gal}}(K(A[f_m])/F_m)$ . For any ideal $\mathfrak {b}$ of $K$ primes to ${\mathfrak {f}}{\mathfrak {p}}^*$ , we write $\sigma _{\mathfrak {b}}$ for the Artin symbol of $\mathfrak {b}$ in ${\mathrm {Gal}}(K({\mathfrak {f}}_m)/K)$ . We list several basic properties of ${\mathfrak {R}}_{\alpha, V_m}(P)$ in the following proposition, whose proof can be found in [Reference Kang16].

We recall that $P_n$ is the $\varpi ^{n+2}$ -division point of $A$ corresponding to $u_n$ of $\widehat {A}[\varpi ^{n+2}]$ , and choose $\epsilon _n\in {\mathcal {O}}$ such that

\begin{equation*}\epsilon _n\varpi ^*\equiv 1{\ \mathrm {mod}\ } {\mathfrak {p}}^{n+2}.\end{equation*}

Observe that $\varpi ^*$ is a unit in ${\mathbb {Z}}_2$ , and that we have $[\varpi ^{*-(m+2)}](u_n)$ corresponding to $\epsilon ^{m+2}_nP_n$ . It is well-known from [Reference Robert21] (or [Reference de Shalit12]) that ${\mathfrak {R}}_{\alpha, V_m}(\epsilon ^{m+2}_n P_n)$ is a unit of $K_{n,m}$ . Following [Reference Coates and Wiles7] and [Reference Yager27], we define $C'_{n,m}$ to be the subgroup of $K_{n,m}^\times$ which is generated by all units ${\mathfrak {R}}_{\alpha, V_m}(\epsilon ^{m+2}_n P_n)$ and their conjugates under the action of the Galois group of $K_{n,m}/K$ , where $\alpha$ runs over all elements of $\mathscr {J}$ . We call $C'_{n,m}$ the group of elliptic units in $K_{n,m}$ , and one can easily show that $C'_{n,m}$ is stable under the action of $\mathcal {G}$ . By Proposition 4.2, we can easily deduce the following lemma.

Noting Theorem3.1, we get the Coleman power series for $e(\alpha )$ in the following

Proposition 4.4. For every $\alpha \in \mathscr {I}$ , the Coleman power series $c_{m,e(\alpha )}(T)\in \mathscr {I}_m[[T]]$ attached to $e(\alpha )\in {\mathbb {U}}'_\infty$ is given by

\begin{equation*}c_{m,e(\alpha )}(T)={\mathfrak {R}}_{\alpha, V_m}([\varpi ^{*-(m+2)}](T)).\end{equation*}

Let $\overline {C}'_{n,m}$ be the closure of $C'_{n,m}$ in ${\mathbb {U}}'_{n,m}$ . We write $\overline {C}'_\infty$ for the projective limits of the $\overline {C}'_{n,m}$ with respect to the norm maps, it is clear that $e(\alpha )\in \overline {C}'_\infty$ for all $\alpha \in \mathscr {I}$ . We will use $\langle \beta \rangle$ to denote the projection from ${\mathbb {U}}'_\infty$ to ${\mathbb {U}}_\infty$ for each $\beta \in {\mathbb {U}}'_\infty$ .

We recall the homomorphism $\delta _{k,j}$ defined in (3.7). Now, we turn to consider the value of $\delta _{k,j}$ at $\langle e(\alpha )\rangle$ . To do this, we shall need to introduce some further notation. Recall the lattice ${\mathcal {L}}=\Omega {\mathcal {O}}$ in the complex plane. Then, for each integer $k\geq 1$ , the complex function

\begin{equation*}H_k(z,s,{\mathcal {L}})=\sum _{w\in {\mathcal {L}},\, w\neq -z}\frac {(\overline {z}+\overline {w})^k}{|z+w|^{2s}},\qquad {\mathrm {Re}}(s)\gt \frac {k}{2}+1,\end{equation*}

can be analytically continued to the whole complex plane as a function of $s$ . Following Weil and Yager (see [Reference Yager28]), for integers $k\gt j\geq 0$ , we set

(4.8) \begin{equation} E_{j,k}(z)=(k-1)!\cdot \left (\frac {2\pi }{\sqrt {7}}\right )^j\cdot \frac {H_{k+j}(z,k,{\mathcal {L}})}{|\Omega |^{2j}} \end{equation}

and

(4.9) \begin{equation} E_k(z)=E_{0,k}(z). \end{equation}

Obviously both $H_{k}(z,s,{\mathcal {L}})$ and $E_{j,k}(z)$ are periodic in $z$ with period lattice $\mathcal {L}$ . As we shall see in the following, the values of $E_{j,k}(z)$ at division points of ${\mathbb {C}}/{\mathcal {L}}$ are closely related to the values of Hecke $L$ -series, while the values of $E_k(z)$ at division points of ${\mathbb {C}}/{\mathcal {L}}$ are closely related to the logarithmic derivatives of elliptic units by the work of Coates-Wiles [Reference Coates and Wiles7].

Let us denote the class $\frac {\Omega }{f_m}{\ \mathrm {mod}\ }{\mathcal {L}}$ in ${\mathbb {C}}/{\mathcal {L}}$ by $\rho _m$ , then $\rho _m$ is a primitive $f_m$ -division point of ${\mathbb {C}}/{\mathcal {L}}$ . For any integral ideal $\mathfrak {b}$ in $K$ prime to ${\mathfrak {f}}{\mathfrak {p}}^*$ , recall that we have defined $\sigma _{\mathfrak {b}}$ to be the Artin symbol of $\mathfrak {b}$ in ${\mathrm {Gal}}(K({\mathfrak {f}}_m)/K)$ . For an integral ideal $\mathfrak {c}$ in $K$ , we let $N{\mathfrak {c}}$ be the norm of $\mathfrak {c}$ . Recall also that $\phi$ has conductor $\mathfrak {f}$ .

Proof. For (1), we write $E_{j,k}(z)$ as following

\begin{equation*}E_{j,k}(z)=(k-1)!\left (\frac {1}{\sqrt {-7}}\right )^j\cdot \left (\frac {\Omega \cdot \overline {\Omega }}{2\pi \sqrt {-1}}\right )^{-j}H_{k+j}(z,k,{\mathcal {L}}).\end{equation*}

Note [Reference Goldstein and Schappacher14, Corollary (2.3)], the assertion follows from $\left (\frac {\Omega \cdot \overline {\Omega }}{2\pi \sqrt {-1}}\right )^{-j}H_{k+j}(\rho _m,k,{\mathcal {L}})\in K({\mathfrak {f}}_m)$ and $\sqrt {-7}\in K$ . The second assertion is [Reference Yager28, Lemma (2.8)]. One can also find the detailed proof for (2) in [Reference Kang16, Lemma (4.1)], the same method can be applied to the even prime case, during the proof, we need to note that $K(A_{{\mathfrak {f}}_m})=K({\mathfrak {f}}_m)$ and that $\mathfrak {f}$ is a nontrivial ideal in $K$ which is prime to $2$ .

From the diagram (4.6), since $F_m$ and $K(A[{\mathfrak {f}}])$ are linearly disjoint over $K$ , and $F_m.K(A[{\mathfrak {f}}])=K(A[{\mathfrak {f}}_m])$ , then ${\mathrm {Gal}}(K(A[{\mathfrak {f}}_m])/K)$ is isomorphic to the direct sum of

\begin{equation*}{\mathrm {Gal}}(K(A[{\mathfrak {f}}_m])/K(A[{\mathfrak {f}}]))\quad \textrm {and}\quad {\mathrm {Gal}}(K(A[{\mathfrak {f}}_m])/F_m).\end{equation*}

We denote by ${\mathfrak {B}}_{\mathfrak {f}}$ (resp. ${\mathfrak {B}}_m$ ) a set of representatives of integral ideals of $K$ , prime to ${\mathfrak {f}}{\mathfrak {p}}^*$ , such that ${\mathrm {Gal}}(K(A[{\mathfrak {f}}_m])/F_m)=\{\sigma _{\mathfrak {x}}:{\mathfrak {x}}\in {\mathfrak {B}}_{\mathfrak {f}}\}$ (respectively, ${\mathrm {Gal}}(K(A[{\mathfrak {f}}_m])/K(A[{\mathfrak {f}}])=\{\sigma _{\mathfrak {y}}:{\mathfrak {y}}\in {\mathfrak {B}}_m\}$ ). The following lemma relates the Kronecker-Eisenstein series to the logarithmic derivatives of the Coleman power series given in Proposition 4.4.

Lemma 4.6. Let $\alpha \in \mathscr {I}$ . For any ${\mathfrak {y}}\in {\mathfrak {B}}_m$ , ${\mathfrak {x}}\in {\mathfrak {B}}_{\mathfrak {f}}$ and integer $k\geq 1$ , we have

(4.11) \begin{equation} \begin{aligned} &\left (\frac {d}{dz}\right )^k\log {\mathfrak {R}}_{\alpha, V^{\sigma _{\mathfrak {y}}}_m}\left ([\varpi ^{*-(m+2)}]\left ({\mathcal {W}}(z,{\mathcal {L}})\right )\right )\big |_{z=0}\\ =&(\!-1)^{k-1}\cdot \varpi ^{*-k(m+2)}\cdot \sum _{{\mathfrak {x}}\in {\mathfrak {B}}_{\mathfrak {f}}}\left (N\alpha \cdot E_k(\phi ({\mathfrak {x}}{\mathfrak {y}})\rho _m)-\phi ^k((\alpha ))\cdot E_k(\phi (\alpha {\mathfrak {x}}{\mathfrak {y}})\rho _m)\right ). \end{aligned} \end{equation}

Recall that $A$ has good ordinary reduction at $\mathfrak {p}$ , then classical theory shows that there is an isomorphism $j$ between the multiplicative formal group ${\mathbb {G}}_m$ and $\widehat {A}$ over $\mathscr {I}$ , such that $t=j(w)$ with leading coefficient $\Omega _{\mathfrak {p}}\in \mathscr {I}^\times$ . The non-zero number $\Omega _{\mathfrak {p}}$ is called $\mathfrak {p}$ -adic period in literature. We postpone the proof of the following Theorem of Yager and De Shalit to the next section.

In our later calculation, the integers $k,j$ will be fixed, and the integer $m$ will vary, so the above congruence formula (4.12) is enough for the construction of the $2$ -adic measure. We give some remarks on Theorem4.7. In fact, Yager found in [Reference Yager28] the description for $\Omega _{\mathfrak {p}}$ in (1) up to some finite prime powers, later on, De Shalit showed in [Reference de Shalit12] these prime powers do not appear. Although Yager dealt Theorem4.7 for odd primes, we will see in next section that the same method can be applied to the even prime case (except that we need to use modulo ${\mathfrak {p}}^{m+2-j}$ in (4.12)). For (2), we need to note that since $\sigma _{\mathfrak {x}}$ fixes $A[{\mathfrak {p}}^{*(m+2)}]$ so $\chi _{{\mathfrak {p}}^*}(\sigma _{\mathfrak {x}})=1$ .

With the help of Yager-De-Shalit Theorem, now we can calculate the value of $\delta _{k,j}$ at $\langle e(\alpha )\rangle$ for all $\alpha \in \mathscr {I}$ .

Theorem 4.8. Let $f$ be the integer in $K$ generated $\mathfrak {f}$ which we fixed before. Let $k$ and $j$ be integers such that $k\gt j\geq 0$ . Write $e(\alpha )$ (with $\alpha \in \mathscr {I}$ ) for the elliptic units we constructed in Lemma 4.3 . Then

(4.13) \begin{equation} \delta _{k,j}(\langle e(\alpha )\rangle ) =(\!-1)^{k-1}(k-1)!f^k\cdot (N\alpha -\phi ^k((\alpha ))\overline {\phi }^{-j}((\alpha )))\cdot \left (\frac {2\pi }{\sqrt {7}}\right )^j\Omega ^j_{\mathfrak {p}}\cdot \left (1-\frac {\phi ^{k+j}({\mathfrak {p}})}{2^{j+1}}\right )\cdot \frac {L^{({\mathfrak {f}}{\mathfrak {p}}^*)}(\overline {\phi }^{k+j},k)}{\Omega ^{k+j}}. \end{equation}

We recall that for any $\beta \in {\mathbb {U}}'_\infty$ , we have defined power series $g_{m,\beta }$ associated with the Coleman power series $c_{m,\beta }$ given in (3.4). Recall also that $\varphi =({\mathfrak {p}},F_\infty /K)$ , then we can restrict $\varphi$ to $F_m$ , and we denote this element simply by $\varphi$ . In order to prove the theorem, we need the following lemma.

Lemma 4.9. We have

(4.14) \begin{equation} \begin{aligned} &\sum _{{\mathfrak {y}}\in {\mathfrak {B}}_m}\chi _{{\mathfrak {p}}^*}(\sigma _{\mathfrak {y}})^j\cdot \left (\frac {d}{dz}\right )^k\frac {1}{2} \log \left (\frac {{\mathfrak {R}}^{\sigma _{\mathfrak {y}}}_{\alpha, V_m}([\varpi ^{*-(m+2)}]({\mathcal {W}}(z,{\mathcal {L}})))^2}{{\mathfrak {R}}^{\sigma _{\mathfrak {y}}\varphi }_{\alpha, V_m}([\varpi \varpi ^{*-(m+2)}]({\mathcal {W}}(z,{\mathcal {L}})))}\right )\bigg |_{z=0}\\ =&\sum _{{\mathfrak {y}}\in {\mathfrak {B}}_m}\chi _{{\mathfrak {p}}^*}(\sigma _{\mathfrak {y}})^j\cdot \left (1-\frac {\varpi ^{k+j}}{2^{j+1}}\right )\cdot \left (\frac {d}{dz}\right )^k\log {\mathfrak {R}}^{\sigma _{\mathfrak {y}}}_{\alpha, V_m}([\varpi ^{*-(m+2)}]({\mathcal {W}}(z,{\mathcal {L}})))\big |_{z=0}. \end{aligned} \end{equation}

Proof. This follows from a direct calculation. One needs to use the formula (2.1), which implies the following relation

\begin{equation*}\chi _{{\mathfrak {p}}^*}(\varphi )=\overline {\phi }({\mathfrak {p}})=\varpi ^*.\end{equation*}

Now, we come back to the proof of Theorem4.8.

Proof of Theorem 4.8 . We write $\beta$ for $\langle e(\alpha )\rangle$ and use $T_1$ for $t$ as the parameter of $\widehat {A}$ in the following proof. From definition of $\delta _{k,j}$ , we have

\begin{equation*}\delta _{k,j}(\beta )=\left (\lambda '(T_1)\frac {\partial }{\partial T_1}\right )^k\left ((1+T_2)\frac {\partial }{\partial T_2}\right )^j g_\beta (T_1,T_2)\big |_{(T_1,T_2)=(0,0)}.\end{equation*}

Here, $\lambda :\widehat {A}\rightarrow {\mathbb {G}}_a$ is the logarithm map from $\widehat {A}$ to the formal additive group ${\mathbb {G}}_a$ satisfying $\lambda '(T_1)\frac {\partial }{\partial T_1}=\frac {d}{dz}$ , where $z$ the parameter for ${\mathbb {G}}_a$ . We recall that $g_\beta (T_1,T_2)$ is the power series given in Corollary 3.5. Note the formula (3.6), by modulo $2^{m+2}\mathscr {I}$ , $\delta _{k,j}(\beta )$ becomes to be

(4.15) \begin{equation} \delta _{k,j}(\beta )\equiv \sum _{{\mathfrak {y}}\in {\mathfrak {B}}_m}\left (\lambda '(T_1)\frac {\partial }{\partial T_1}\right )^k ((g^{\sigma _{\mathfrak {y}}}_{m,\beta }(T_1))_{{\mathfrak {p}}_m})\big |_{T_1=0}\cdot \left ((1+T_2)\frac {\partial }{\partial T_2}\right )^j ((1+T_2)^{\chi _{{\mathfrak {p}}^*}(\sigma _{\mathfrak {y}})})\big |_{T_2=0}{\ \mathrm {mod}\ } 2^{m+2}\mathscr {I}. \end{equation}

Note the Coleman power series associated with $\beta$ given in Proposition 4.4 and formula (3.4), then

\begin{equation*}g_{m,\beta }(T_1)=\frac {1}{2}\log \left (\frac {{\mathfrak {R}}_{\alpha, V_m}([\varpi ^{*-(m+2)}](T_1))^2}{{\mathfrak {R}}^\varphi _{\alpha, V_m}([\varpi \varpi ^{*-(m+2)}](T_1))}\right ).\end{equation*}

Since the inverse of $\lambda$ is given by ${\mathcal {W}}(z,{\mathcal {L}})$ , $\frac {d}{dz}=\lambda '(T_1)^{-1}\frac {\partial }{\partial T_1}$ and Lemma 4.9, then the formula (4.15) arrives at

(4.16) \begin{equation} \delta _{k,j}(\beta )\equiv \sum _{{\mathfrak {y}}\in {\mathfrak {B}}_m}\chi _{{\mathfrak {p}}^*}(\sigma _{\mathfrak {y}})^j\cdot \left (1-\frac {\varpi ^{k+j}}{2^{j+1}}\right )\cdot \left (\frac {d}{dz}\right )^k \log {\mathfrak {R}}^{\sigma _{\mathfrak {y}}}_{\alpha, V_m}([\varpi ^{*-(m+2)}]({\mathcal {W}}(z,{\mathcal {L}})))\big |_{z=0}{\ \mathrm {mod}\ }2^{m+2}\mathscr {I}. \end{equation}

Noting Lemma 4.6, then the right-hand side of formula (4.16) is equal to

(4.17) \begin{equation} \begin{aligned} &(-1)^{k-1}\varpi ^{*-k(m+2)}\cdot \left (1-\frac {\varpi ^{k+j}}{2^{j+1}}\right )\sum _{{\mathfrak {y}}\in {\mathfrak {B}}_m,\, {\mathfrak {x}}\in {\mathfrak {B}}_{\mathfrak {f}}}\chi _{{\mathfrak {p}}^*}(\sigma _{\mathfrak {y}})^j\cdot (N\alpha \cdot E_k(\phi ({\mathfrak {x}}{\mathfrak {y}})\rho _m)-\phi ^k((\alpha ))\cdot E_k(\phi (\alpha {\mathfrak {x}}{\mathfrak {y}})\rho _m))\\ =&(-1)^{k-1}\varpi ^{*-k(m+2)}\cdot \left (1-\frac {\varpi ^{k+j}}{2^{j+1}}\right )\sum _{{\mathfrak {y}}\in {\mathfrak {B}}_m,\, {\mathfrak {x}}\in {\mathfrak {B}}_{\mathfrak {f}}}(N\alpha -(\phi ^k\overline {\phi }^{-j})((\alpha )))\cdot \chi _{{\mathfrak {p}}^*}(\sigma _{\mathfrak {y}})^j\cdot E_k(\phi ({\mathfrak {x}}{\mathfrak {y}})\rho _m)\\ =&(-1)^{k-1}f^k\Omega ^{j}_{\mathfrak {p}}\cdot \left (1-\frac {\varpi ^{k+j}}{2^{j+1}}\right )(N\alpha -(\phi ^k\overline {\phi }^{-j})((\alpha )))\sum _{{\mathfrak {x}}\in {\mathfrak {B}}_{\mathfrak {f}},\, {\mathfrak {y}}\in {\mathfrak {B}}_m}\Omega ^{-j}_{\mathfrak {p}}\cdot \chi _{{\mathfrak {p}}^*}(\sigma _{\mathfrak {y}})^j f^{-k}_m\cdot E_k(\phi ({\mathfrak {x}}{\mathfrak {y}})\rho _m). \end{aligned} \end{equation}

Here, the first equality in (4.17) follows easily by a change of variables in the second term in the sum of the first line in (4.17). Now, we can apply the Yager-De-Shalit formula (4.12) in Theorem4.7, by modulo $2^{m+2-j}\mathscr {I}$ , to the last term in the formula (4.17). We get

\begin{equation*}\begin{aligned} &\delta _{k,j}(\beta )\\ \equiv &(\!-1)^{k-1}f^k\Omega ^{j}_{\mathfrak {p}}\cdot \left (1-\frac {\varpi ^{k+j}}{2^{j+1}}\right )(N\alpha -(\phi ^k\overline {\phi }^{-j})((\alpha ))) \sum _{{\mathfrak {x}}\in {\mathfrak {B}}_{\mathfrak {f}},\, {\mathfrak {y}}\in {\mathfrak {B}}_m} \left (\frac {N{\mathfrak {f}}^j_m}{f^{k+j}_m}\right )\cdot E_{j,k}(\phi ({\mathfrak {x}}{\mathfrak {y}})\rho _m) {\ \mathrm {mod}\ } 2^{m+2-j}\mathscr {I}\\ \equiv & (\!-1)^{k-1}(k-1)!\cdot \left (\frac {2\pi }{\sqrt {7}}\right )^jf^k\Omega ^{j}_{\mathfrak {p}}\cdot \left (1-\frac {\varpi ^{k+j}}{2^{j+1}}\right ) (N\alpha -(\phi ^k\overline {\phi }^{-j})((\alpha )))\cdot \frac {L^{({\mathfrak {f}}{\mathfrak {p}}^*)}(\overline {\phi }^{k+j},k)}{\Omega ^{k+j}} {\ \mathrm {mod}\ } 2^{m+2-j}\mathscr {I}.\end{aligned}\end{equation*}

Here the last equality comes from the formula (4.10). Now, note that the integer $m$ can be chosen to be arbitrarily large, so, by letting $m\to +\infty$ , the formula (4.13) follows.

5. Congruence formula of Yager and De Shalit

In this section, we shall give a detailed proof for the Theorem4.7 in section §4. Let $\mathfrak {g}$ be any ideal in $\mathcal {O}$ prime to $\mathfrak {p}$ but not ${\mathfrak {p}}^*$ . We write $g$ for a generator of $\mathfrak {g}$ . Recall that $\mathcal {L}$ is the periods lattice of $A$ , and $\Omega$ is a fixed non-zero complex number which generates $\mathcal {L}$ . We set $\rho =\frac {\Omega }{g}$ . Recall that we have defined the modified Eisenstein series $E_{j,k}(z)$ (with $k\gt j\geq 0$ ) in (4.8) and $E_k(z)$ in (4.9). Recall also that $\phi$ is the Hecke character over $K$ associated with $A$ with conductor $\mathfrak {f}$ . For integers $k\gt j\geq 0$ and any integral ideal $\mathfrak {b}$ in $\mathcal {O}$ prime to ${\mathfrak {f}}{\mathfrak {g}}$ , by Lemma 4.5, we know that $E_{j,k}(\phi ({\mathfrak {b}})\rho )\in K({\mathfrak {g}})$ . Furthermore, for the Artin symbol $\tau _{\mathfrak {b}}\in {\mathrm {Gal}}(K(A[{\mathfrak {g}}])/K)$ , we have

(5.1) \begin{equation} E_{j,k}(\rho )^{\tau _{\mathfrak {b}}}=E_{j,k}(\phi ({\mathfrak {b}})\rho ). \end{equation}

Denote by ${\mathcal {O}}_{\mathfrak {g}}$ the ring of integers of $K(A[{\mathfrak {g}}])$ , and by ${\mathfrak {I}}_{\mathfrak {g}}$ the ring ${\mathcal {O}}_{\mathfrak {g}}\otimes _{\mathcal {O}}{\mathbb {Z}}_p$ . We list the integrality properties of these Eisenstein numbers in the following lemma.

Lemma 5.1. Keep the notation as above, then the following numbers are all contained in ${\mathfrak {I}}_{\mathfrak {g}}$ :

1. (i) $(N{\mathfrak {g}})^j E_{j,k}(\phi ({\mathfrak {b}})\rho )$ with $k\gt 1$ ,

2. (ii) $(N{\mathfrak {g}})E_1(\phi ({\mathfrak {b}})\rho )$ , and

3. (iii) $\overline {\phi }({\mathfrak {b}})E_1(\rho )-E_1(\phi ({\mathfrak {b}})\rho )$ with $\mathfrak {b}$ is prime to $6{\mathfrak {f}}{\mathfrak {g}}$ .

Furthermore, if ${\mathfrak {p}}^{*m}$ divides $\mathfrak {g}$ , then for all $k\gt 1$ and $j\leq m$ , we have

(5.2) \begin{equation} (N{\mathfrak {g}})^j E_{j,k}(\phi ({\mathfrak {b}})\rho )\equiv (-N{\mathfrak {g}}\cdot E_1(\phi ({\mathfrak {b}})\rho ))^j\cdot E_k(\phi ({\mathfrak {b}})\rho ){\ \mathrm {mod}\ }{\mathfrak {p}}^{m-j}{\mathfrak {I}}_{\mathfrak {g}}. \end{equation}

Proof. For odd prime case, this is [Reference Yager28, Theorem (4.1)] (or [Reference de Shalit12, Proposition (3.3) and Lemma (3.4)] for a generalization), we remark that the same method can be applied to the even prime case, however, the original congruence formula of Yager in [Reference Yager28, (4.2) in Theorem (4.1)] needs a modification at the even prime case. A detailed proof for (i) and (ii) are given in [Reference de Shalit12, (iv) Proposition (3.3)]. For (iii), recall the rational function $R_\alpha (P\oplus V_g)$ given in (4.2), where $V_g={\mathcal {W}}(\rho, {\mathcal {L}})$ . Then a similar calculation as in Lemma 4.6 shows that

(5.3) \begin{equation} \left (\frac {d}{dz}\right )\log R_\alpha \left ({\mathcal {W}}(z+\phi ({\mathfrak {b}})\rho, {\mathcal {L}})\right )\big |_{z=0}=N\alpha \cdot E_1(\phi ({\mathfrak {b}})\rho )-\phi ((\alpha ))\cdot E_1(\phi (\alpha {\mathfrak {b}})\rho ). \end{equation}

By the formula (4.7) and the definition of Coleman power series in Proposition 4.4, $R_\alpha ({\mathcal {W}}(z+\phi ({\mathfrak {b}})\rho, {\mathcal {L}}))$ is a rational function on $A$ with coefficients in $K(A[{\mathfrak {g}}])$ , and the $t$ -expansion of $R_\alpha ({\mathcal {W}}(z+\phi ({\mathfrak {b}})\rho, {\mathcal {L}}))$ is a unit in ${\mathfrak {I}}_{\mathfrak {g}}[[t]]$ . Thus,

\begin{equation*}N\alpha \cdot E_1(\phi ({\mathfrak {b}})\rho )-\phi ((\alpha ))\cdot E_1(\phi (\alpha {\mathfrak {b}})\rho )\in {\mathfrak {I}}_{\mathfrak {g}}.\end{equation*}

Since $\alpha$ is prime to $6{\mathfrak {f}}$ , by our assumption on $\mathfrak {b}$ , we take $\alpha$ such that $\alpha {\mathcal {O}}={\mathfrak {b}}$ in the above formula, then we get

\begin{equation*}N{\mathfrak {b}}\cdot E_1(\phi ({\mathfrak {b}})\rho )-\phi ({\mathfrak {b}})\cdot E_1(\phi ({\mathfrak {b}}^2)\rho )\in {\mathfrak {I}}_{\mathfrak {g}}.\end{equation*}

By applying the Galois action of $\tau ^{-1}_{{\mathfrak {b}}}$ on the above formula, then (iii) follows by $N{\mathfrak {b}}=\phi ({\mathfrak {b}})\cdot \overline {\phi }({\mathfrak {b}})$ and that $\phi ({\mathfrak {b}})$ is a $\mathfrak {p}$ -adic unit. For the formula (5.2), we use the following equality due to Weil, for all $k\gt j\geq 0$ , there is a polynomial $P_{j,k}(x_1,\cdots, x_{k+j})\in {\mathbb {Z}}[x_1,\cdots, x_{k+j}]$ of degree $j+1$ and the highest power of $x_1$ occurring is at most $j-1$ , such that, for all $z\notin {\mathcal {L}}$ ,

(5.4) \begin{equation} E_{j,k}(z)=(\!-E_1(z))^j E_k(z)+2^{-j}P_{j,k}(E_1(z),\cdots, E_{k+j}(z)). \end{equation}

Put $z=\phi ({\mathfrak {b}})\rho$ and multiply both sides of (5.4) by $N{\mathfrak {g}}^j$ , then (5.2) follows.

Now, we take $\mathfrak {g}$ to be an integral ideal of $K$ prime to $2$ and let $g$ be a generator of $\mathfrak {g}$ . For any integer $m\geq 0$ , we write ${\mathfrak {g}}_m={\mathfrak {g}}{\mathfrak {p}}^{*(m+2)}$ . Let $g_m=g\varpi ^{*(m+2)}$ be a generator of ${\mathfrak {g}}_m$ , where $\varpi ^*=\phi ({\mathfrak {p}}^*)$ . We set $\rho _m=\frac {\Omega }{g_m}$ . Let ${\mathcal {O}}_{{\mathfrak {g}}_m}$ be the ring of integers of $K({\mathfrak {g}}_m)$ , we denote by ${\mathfrak {I}}_{m}={\mathcal {O}}_{{\mathfrak {g}}_m}\otimes {\mathbb {Z}}_p$ and write ${\mathfrak {I}}=\cup _{m\geq 0}{\mathfrak {I}}_m$ . Since ${\mathfrak {g}}_m$ is prime to $\mathfrak {p}$ , so by Lemma 5.1, $\overline {g}_m E_1(\rho _m)$ is an element in ${\mathfrak {I}}_m$ . Furthermore, the following lemma shows that the sequence $\overline {g}_m E_1(\rho _m)(m\geq 0)$ is a Cauchy sequence in $\mathfrak {I}$ , and the limit of this sequence plays a key role in Yager’s definition for the $\mathfrak {p}$ -adic period.

Lemma 5.2. As $m$ varies, the sequence $\overline {g}_m E_1(\rho _m)\in {\mathfrak {I}}$ is a Cauchy sequence, therefore, the limit

\begin{equation*}\lim _{m\rightarrow \infty }\overline {g}_m E_1(\rho _m)\end{equation*}

exists, and lies in $\mathfrak {I}$ .

Proof. For positive integers $m\gt n$ , we let $B$ be a set of integral ideals in $K$ prime to $6{\mathfrak {f}}{\mathfrak {g}}_0$ such that $\{\phi ({\mathfrak {b}}):{\mathfrak {b}}\in B\}$ is a set of representatives for the quotient of $1+{\mathfrak {g}}_n$ by ${\mathfrak {g}}_m$ . By [Reference Yager28, Lemma (2.8)], we have the following two formulas on partial $L$ -series

\begin{equation*}\frac {L^{({\mathfrak {g}}_0)}(\sigma _{\mathfrak {b}},\overline {\phi },1)}{\Omega }=\frac {1}{g_m}E_1(\phi ({\mathfrak {b}})\rho _m)\qquad \forall \sigma _{\mathfrak {b}}\in {\mathrm {Gal}}(K({\mathfrak {g}}_m)/K),\end{equation*}

and

\begin{equation*}\frac {L^{({\mathfrak {g}}_0)}(\sigma '_{\mathfrak {b}},\overline {\phi },1)}{\Omega }=\frac {1}{g_n}E_1(\phi ({\mathfrak {b}})\rho _n)\qquad \forall \sigma '_{\mathfrak {b}}\in {\mathrm {Gal}}(K({\mathfrak {g}}_n)/K).\end{equation*}

Here, for any $\sigma \in {\mathrm {Gal}}(K({\mathfrak {G}})/K)$ , we define the partial $L$ -series $L^{({\mathfrak {G}})}(\sigma, \overline {\phi },s)$ to be the analytic continuation of

\begin{equation*}L^{({\mathfrak {G}})}(\sigma, \overline {\phi },s)=\sum _{\tau _{\mathfrak {a}}=\sigma }\overline {\phi }({\mathfrak {a}})N{\mathfrak {a}}^{-s},\qquad {\mathrm {Re}}(s)\gt 3/2,\end{equation*}

where the sum is taken over all ideals $\mathfrak {a}$ in the ray class whose Artin symbol $\tau _{\mathfrak {a}}$ is $\sigma$ . By the below relation

\begin{equation*}\sum _{\sigma _{\mathfrak {c}},\ \sigma _{\mathfrak {c}}|_{K({\mathfrak {g}}_n)}=\sigma '_{\mathfrak {b}}}\frac {L^{({\mathfrak {g}}_0)}(\sigma _{\mathfrak {c}},\overline {\phi },1)}{\Omega }=\frac {L^{({\mathfrak {g}}_0)}(\sigma '_{\mathfrak {b}},\overline {\phi },1)}{\Omega },\end{equation*}

where the sum runs over all ideals $\mathfrak {c}$ such that the restriction of $\sigma _{\mathfrak {c}}$ to $K({\mathfrak {g}}_n)$ is $\sigma '_{\mathfrak {b}}$ , then

\begin{equation*}g^{-1}_nE_1(\rho _n)=\sum _{{\mathfrak {b}}\in B}g^{-1}_mE_1(\phi ({\mathfrak {b}})\rho _m),\end{equation*}

or equivalently,

(5.5) \begin{equation} (N{\mathfrak {g}}_m)g^{-1}_nE_1(\rho _n)= g^{-1}_m\sum _{{\mathfrak {b}}\in B}(N{\mathfrak {g}}_m)E_1(\phi ({\mathfrak {b}})\rho _m). \end{equation}

By (ii) and (iii) in Lemma 5.1, and $(N{\mathfrak {g}}_m)g^{-1}_m=\phi ({\mathfrak {p}})^{m+2}\overline {g}\in {\mathfrak {p}}^{m+2}{\mathfrak {I}}_m$ , then

(5.6) \begin{equation} g^{-1}_m\sum _{{\mathfrak {b}}\in B}(N{\mathfrak {g}}_m)E_1(\phi ({\mathfrak {b}})\rho _m)\equiv g^{-1}_m\sum _{{\mathfrak {b}}\in B}\overline {\phi }({\mathfrak {b}})(N{\mathfrak {g}}_m)E_1(\rho _m){\ \mathrm {mod}\ } {\mathfrak {p}}^{m+2} {\mathfrak {I}}_m. \end{equation}

Next, from the set of all elements $\phi ({\mathfrak {b}})$ modulo ${\mathfrak {g}}_m$ for ${\mathfrak {b}}\in B$ is equal to the set of elements $1+g\overline {\phi }({\mathfrak {p}})^{n+2} s$ modulo ${\mathfrak {g}}_m$ with $s\in {\mathcal {O}}$ runs over a set of representatives of ${\mathcal {O}}/({\mathfrak {p}}^*)^{m-n}$ , we get

(5.7) \begin{equation} \sum _{{\mathfrak {b}}\in B}\overline {\phi }({\mathfrak {b}})\equiv 2^{m-n}{\ \mathrm {mod}\ } {\mathfrak {p}}^{m+1}{\mathbb {Z}}_p. \end{equation}

Then by formulas (5.5), (5.6) and (5.7), we get

\begin{equation*}(N{\mathfrak {g}}_m) g^{-1}_n E_1(\rho _n)2^{n-m}\equiv g^{-1}_m N{\mathfrak {g}}_m E_1(\rho _m){\ \mathrm {mod}\ } {\mathfrak {p}}^{n+1}{\mathfrak {I}}_m.\end{equation*}

By $(N{\mathfrak {g}}_m)g^{-1}_n\cdot p^{n-m}=\overline {g}_n$ and $g^{-1}_mN{\mathfrak {g}}_m=\overline {g}_m$ , thus, we get the following key relation that for all $m\gt n\gt 0$ ,

(5.8) \begin{equation} \bar {g}_m\cdot E_1(\rho _m)\equiv \overline {g}_n\cdot E_1(\rho _n){\ \mathrm {mod}\ } {\mathfrak {p}}^{n+1}{\mathfrak {I}}_m. \end{equation}

Therefore, noting that ${\mathfrak {p}}^{n+1}{\mathfrak {I}}_m$ is contained in ${\mathfrak {p}}^{n+1}{\mathfrak {I}}$ , the Lemma follows.

Now, we can prove the following theorem of Yager and De Shalit.

Theorem 5.3. We can choose the $\mathfrak {p}$ -adic period $\Omega _{\mathfrak {p}}$ in $\mathfrak {I}$ such that

(5.9) \begin{equation} \Omega ^{-1}_{\mathfrak {p}}=-\lim _{m\rightarrow \infty } \overline {g}_m E_1(\rho _m). \end{equation}

Moreover, $\Omega _{\mathfrak {p}}$ is independent of the choice of the ideal $\mathfrak {g}$ .

We prove Theorem5.3 by showing the following proposition holds. In fact, by adapting the isomorphism $j:{\mathbb {G}}_m\rightarrow \widehat {A}$ by an automorphism, we can let $\beta =1$ in the following proposition, and then Theorem5.3 follows.

Proposition 5.4. There exists $\beta \in {\mathbb {Z}}^\times _2$ , not depending on the choice of the ideal $\mathfrak {g}$ , such that for all integer $m\gt 0$ ,

(5.10) \begin{equation} -\overline {g}_m\cdot E_1(\rho _m)\equiv \beta \cdot \Omega ^{-1}_{\mathfrak {p}}{\ \mathrm {mod}\ } {\mathfrak {p}}^{m+2}{\mathfrak {I}}. \end{equation}

In fact, Yager showed that (5.10) holds with $\beta \in {\mathbb {Z}}_p$ (for $p$ an odd prime), however, he cannot prove that $\beta$ is a unit at $p$ . Later on, De Shalit in [Reference de Shalit12] proved that $\beta$ is always a $p$ -adic unit. In the following, we will prove Proposition 5.4 by generalizing Yager’s argument to the prime $2$ , and sketch the proof of De Shalit showing that $\beta$ is a $2$ -adic unit. For these reasons, we call Theorem5.3 the Yager-De Shalit Theorem.

Proof of Proposition 5.4 . Recall that $\chi _{{\mathfrak {p}}^*}$ can be viewed as the character of ${\mathrm {Gal}}(K^{\textrm {ab}}/K)$ giving the action on $A[{\mathfrak {p}}^{*\infty }]$ , and for all ideals $\mathfrak {b}$ prime to ${\mathfrak {f}}\bar {{\mathfrak {p}}}$ in $K$ , the value of $\chi _{{\mathfrak {p}}^*}$ on the Artin symbol of $\mathfrak {b}$ in ${\mathrm {Gal}}(K(A[{\mathfrak {f}}{\mathfrak {p}}^{*\infty }])/K)$ is given by $\overline {\phi }({\mathfrak {b}})$ . For any $\sigma \in {\mathrm {Gal}}(K(A[{\mathfrak {f}}{\mathfrak {p}}^{*\infty }])/K)$ , from (5.1) and (iii) in Lemma 5.1, we have

\begin{equation*}\overline {g}_m E_1(\rho _m)^\sigma \equiv \chi _{{\mathfrak {p}}^*}(\sigma )\overline {g}_m E_1(\rho _m){\ \mathrm {mod}\ } {\mathfrak {p}}^{m+2}{\mathfrak {I}}.\end{equation*}

Recall the isomorphism $j:{\mathbb {G}}_m\rightarrow \widehat {A}$ and that $\Omega _{\mathfrak {p}}$ is the coefficient of $w$ in $t=j(w)$ , we also have $\Omega ^\sigma _{\mathfrak {p}}=\chi ^{-1}_{{\mathfrak {p}}^*}(\sigma )\Omega _{\mathfrak {p}}$ (see [Reference Yager28, P335]). Therefore,

\begin{equation*}\beta _{\mathfrak {g}}=-(\!\lim _{m\rightarrow \infty }\overline {g}_m E_1(\rho _m))\Omega _{\mathfrak {p}}\end{equation*}

is invariant under ${\mathrm {Gal}}(K(A[{\mathfrak {f}}{\mathfrak {p}}^{*\infty }])/K)$ . Notice that $\Omega _{\mathfrak {p}}$ is a $\mathfrak {p}$ -adic unit, then $\beta _{\mathfrak {g}}\in {\mathbb {Z}}_2$ . A key observation made by Yager in [Reference Yager28] implies that $\beta _{\mathfrak {g}}$ is independent of $\mathfrak {g}$ . Here we recall that $\mathfrak {g}$ is any ideal in $\mathcal {O}$ prime to $\mathfrak {p}$ but not ${\mathfrak {p}}^*$ . To prove this observation, note that for any $\mathfrak {b}$ -division point (with the conductor of $\phi ^{k+j}$ divides $\mathfrak {b}$ ), we still have the formula (4.10) in Lemma 4.5, we then apply the formula (4.10) by taking $j=0$ in the following. If ${\mathfrak {q}}=(q)$ is a prime ideal not dividing ${\mathfrak {g}}_0$ , for integer $\ell \geq 1$ , we can view $\chi _{{\mathfrak {p}}^*}(\sigma ){\ \mathrm {mod}\ } 2^{m+2}$ as a Galois character on ${\mathrm {Gal}}(K(A[{\mathfrak {q}}^\ell {\mathfrak {g}}_m])/K)$ giving the action on $A[{\mathfrak {p}}^{*(m+2)}]$ , which we simply denote by $\chi _m$ , for integer $k\gt 1$ , by (i) in Lemma 5.1 and formula (4.10), we get

(5.11) \begin{equation} \begin{aligned} \sum _{\sigma \in {\mathrm {Gal}}(K(A[{\mathfrak {q}}^\ell {\mathfrak {g}}_m])/K)}\chi ^j_m(\sigma )\cdot (q^\ell g_m)^{-k}\cdot E_k\left (\frac {\Omega }{q^\ell g_m}\right )^\sigma = (k-1)!\Omega ^{-k}L^{({\mathfrak {g}}_0{\mathfrak {q}})}(\chi _m^j\bar {\phi }^k,k), \end{aligned} \end{equation}

and

(5.12) \begin{equation} \sum _{\sigma \in {\mathrm {Gal}}(K(A[{\mathfrak {g}}_m])/K)}\chi ^j_m(\sigma )\cdot (g_m)^{-k}\cdot E_k\left (\frac {\Omega }{g_m}\right )^\sigma = (k-1)!\Omega ^{-k}L^{({\mathfrak {g}}_0)}(\chi ^j_m\bar {\phi }^k,k). \end{equation}

Therefore, if $k+j\equiv 0{\ \mathrm {mod}\ } 2$ (note that $2$ is the number of ${\mathcal {O}}^\times$ ), recall that $\chi _{{\mathfrak {p}}^*}(\sigma )\equiv \chi _m(\sigma ){\ \mathrm {mod}\ } {\mathfrak {p}}^{m+2}{\mathfrak {I}}$ , from equalities (5.11), (5.12), we get

(5.13) \begin{equation} \begin{aligned} &\sum _{\sigma \in {\mathrm {Gal}}(K(A[{\mathfrak {q}}^\ell {\mathfrak {g}}_m])/K)}\chi ^j_{{\mathfrak {p}}^*}(\sigma )\cdot (q^\ell g_m)^{-k}\cdot E_k\left (\frac {\Omega }{q^\ell g_m}\right )^\sigma \\ \equiv & \left (1-\frac {\overline {\phi }^{k+j}({\mathfrak {q}})}{N{\mathfrak {q}}^k}\right )\cdot \sum _{\sigma \in {\mathrm {Gal}}(K(A[{\mathfrak {g}}_m])/K)}\chi ^j_{{\mathfrak {p}}^*}(\sigma )\cdot ( g_m)^{-k}\cdot E_k\left (\frac {\Omega }{ g_m}\right )^\sigma {\ \mathrm {mod}\ } {\mathfrak {p}}^{m+2}{\mathfrak {I}}, \end{aligned} \end{equation}

since we will take $k$ to be sufficiently large, so we may assume that $k\gt 1$ , thus by (i) in Lemma 5.1, the values of $E_k$ on the above torsion points are all $\mathfrak {p}$ -integral. For $k+j\equiv 0{\ \mathrm {mod}\ } 2$ , by taking ${\mathfrak {f}}_m$ to be ${\mathfrak {g}}_m$ in (4.10), from formula (5.2) and (iii) in Lemma 5.1, we have

(5.14) \begin{equation} \begin{aligned}&(k-1)!\left (\frac {2\pi }{\sqrt {7}}\right )^j \Omega ^{-k}L^{({\mathfrak {g}}_0)}(\overline {\phi }^{k+j},k)\\ \equiv &\sum _{\sigma \in {\mathrm {Gal}}(K(A[{\mathfrak {g}}_m])/K) }\frac {1}{g^{k+j}_m}\left (-N{\mathfrak {g}}_m\cdot \chi _{{\mathfrak {p}}^*}(\sigma )\cdot E_1(\rho _m)\right )^j E_k(\rho _m)^\sigma {\ \mathrm {mod}\ } {\mathfrak {p}}^{m+2-j}{\mathfrak {I}}\\ \end{aligned} \end{equation}

Note the relation

\begin{equation*}\beta _{{\mathfrak {g}}}\Omega ^{-1}_{{\mathfrak {p}}}\equiv -\overline {g}_m\cdot E_1(\rho _m){\ \mathrm {mod}\ } {\mathfrak {p}}^{m+2}{\mathfrak {I}}, \end{equation*}

the right-hand side of congruence (5.14) is equal to

(5.15) \begin{equation} (\beta _{\mathfrak {g}} \Omega ^{-1}_{\mathfrak {p}})^j \sum _{\sigma \in {\mathrm {Gal}}(K(A[{\mathfrak {g}}_m])/K)}\chi ^j_{{\mathfrak {p}}^*}(\sigma ) g^{-k}_mE_k(\rho _m)^\sigma {\ \mathrm {mod}\ }{\mathfrak {p}}^{m+2-j}{\mathfrak {I}}. \end{equation}

Now, we successively apply the formula (5.13) to the formula (5.15), noting the formula (5.14), we then get

(5.16) \begin{equation} \begin{aligned} &(k-1)!\left (\frac {2\pi }{\sqrt {7}}\right )^j \Omega ^{-k}L^{({\mathfrak {g}}_0)}(\overline {\phi }^{k+j},k)\\ \equiv & (\beta _{\mathfrak {g}}\Omega ^{-1}_{\mathfrak {p}})^j \cdot \prod _{{\mathfrak {q}}\mid {\mathfrak {g}}}\left (1-\frac {\overline {\phi }({\mathfrak {q}})^{k+j}}{N{\mathfrak {q}}^k}\right )\cdot \sum _{\sigma \in {\mathrm {Gal}}(K(A[{\mathfrak {p}}^{*m}])/K)}\chi ^j_{{\mathfrak {p}}^*}(\sigma ) \varpi ^{*-(m+2)k}E_k\left (\frac {\Omega }{\varpi ^{*(m+2)}}\right )^\sigma {\ \mathrm {mod}\ } {\mathfrak {p}}^{m+2-j}{\mathfrak {I}}.\end{aligned} \end{equation}

Here, the product is taken over the prime ideals $\mathfrak {q}$ dividing $\mathfrak {g}$ . We should also note that from $i+j\equiv 0{\ \mathrm {mod}\ }2$ , the conductor of $\phi ^{k+j}$ is $\mathcal {O}$ . On the other hand,

\begin{equation*}L^{({\mathfrak {g}}_0)}(\overline {\phi }^{k+j},k)=\prod _{{\mathfrak {q}}\mid {\mathfrak {g}}}(1-\overline {\phi }^{k+j}({\mathfrak {q}})/N{\mathfrak {q}}^k)L^{({\mathfrak {p}}^*)}(\overline {\phi }^{k+j},k),\end{equation*}

and $L^{({\mathfrak {p}}^*)}(\overline {\phi }^{k+j},k)\neq 0$ for all $k$ sufficiently larger than $j$ , and so it follows from (5.16) that $\beta _{\mathfrak {g}}=\beta _{\mathcal {O}}$ is independent of $\mathfrak {g}$ . Finally, by a delicate choice of the rational functions for the Weil pairing on $A_{2^{m+2}}\times A_{2^{m+2}}$ and a careful study on the $\mathfrak {p}$ -integrality for this function as given in [Reference de Shalit12, (4.15)], De Shalit shows that $\beta _{\mathfrak {g}}$ is indeed a $\mathfrak {p}$ -adic unit, and the result follows.

Finally, take ${\mathfrak {g}}_m$ to be ${\mathfrak {f}}_m$ in the following corollary, then (2) in Theorem4.7 follows.

Corollary 5.5. Assume integers $k\gt j\geq 0$ and $m\geq j-2$ , for any $\sigma \in {\mathrm {Gal}}(K(A_{{\mathfrak {g}}_m})/K)$ , then

\begin{equation*}\frac {N{\mathfrak {g}}_m^j}{g^{k+j}_m}E_{j,k}(\rho _m)^\sigma \equiv \Omega ^{-j}_{\mathfrak {p}}\chi ^j_{{\mathfrak {p}}^*}(\sigma )\cdot g^{-k}_m E_k(\rho _m)^\sigma {\ \mathrm {mod}\ } {\mathfrak {p}}^{m+2-j}{\mathfrak {I}}.\end{equation*}

6. Two-variable canonical measures attached to elliptic units

In this section, we shall show the existence of the power series giving $2$ -adic interpolations of the following numbers

\begin{equation*}\left (\frac {2\pi }{\sqrt {7}}\right )^j\cdot \frac { L^{({\mathfrak {f}}{\mathfrak {p}}^*)}(\overline {\phi }^{k+j},k)}{\Omega ^{k+j}},\end{equation*}

and prove Theorem1.1.

We write ${\mathbb {Z}}^2_2$ for ${\mathbb {Z}}_2\times {\mathbb {Z}}_2$ . Recall that $\mathscr {I}$ is the ring of integers of the completion of the maximal unramified extension of $K_{\mathfrak {p}}$ . We denote by ${\mathbb {M}}({\mathbb {Z}}^2_2,\mathscr {I})$ the $\mathscr {I}$ -valued measure on ${\mathbb {Z}}^2_2$ , then by Mahler theorem, ${\mathbb {M}}({\mathbb {Z}}^2_2,\mathscr {I})$ is isomorphic to the two-variable power series ring $\mathscr {I}[[T_1,T_2]]$ by the following map

(6.1) \begin{equation} {\mathbb {M}}({\mathbb {Z}}^2_2,\mathscr {I})\stackrel {\sim }{\longrightarrow }\mathscr {I}[[T_1,T_2]]\qquad \mu \mapsto M(\mu )(T_1,T_2), \end{equation}

where

(6.2) \begin{equation} M(\mu )(T_1,T_2)=\int _{{\mathbb {Z}}^2_2}(1+T_1)^x(1+T_2)^yd\mu (x,y). \end{equation}

Conversely, for any $f(T_1,T_2)\in \mathscr {I}[[T_1,T_2]]$ , we write $\mu _f$ for the $\mathscr {I}$ -valued measure on ${\mathbb {Z}}^2_2$ determined by the above isomorphism. The following lemma can be easily deduced from the isomorphism (6.1).

Lemma 6.1. Assume that $n\geq 0$ . For $k$ and $j$ modulo $2^n$ , there are uniquely determined elements $b_{k,j}\in \mathscr {I}$ such that

\begin{equation*}f(T_1,T_2)\equiv \sum _{k,j{\ \mathrm {mod}\ } 2^n}b_{k,j}(1+T_1)^k(1+T_2)^j{\ \mathrm {mod}\ } \left ((1+T_1)^{2^n}-1,(1+T_2)^{2^n}-1\right ).\end{equation*}

Then $\mu _f$ is the unique measure such that

\begin{equation*}\int _{(k+2^n{\mathbb {Z}}_p)\times (j+2^n{\mathbb {Z}}_p)}d\mu _f=b_{k,j}\end{equation*}

holds for all $k,j$ .

We say that a measure $\mu$ is supported on a measure subset $\mathfrak {A}$ of ${\mathbb {Z}}^2_2$ , if for all measurable function $\psi :{\mathbb {Z}}^2_2\rightarrow \mathscr {I}$ , we have $\int _{{\mathbb {Z}}^2_2}\psi d\mu =\int _{\mathfrak {A}}\psi d\mu$ .

Recall that $j:{\mathbb {G}}_m\rightarrow \widehat {A}$ is the isomorphism defined over $\mathscr {I}$ such that $j(T_1)$ is the power series in $\mathscr {I}[[T_1]]$ with the first term given by $\Omega _{\mathfrak {p}} T_1$ . For any $\beta \in {\mathbb {U}}_\infty$ , we have defined the two-variable power series $g_\beta$ given in Corollary 3.5 associated with $\beta$ . We then define $h_\beta (T_1,T_2)=g_\beta (j(T_1),T_2)$ , where we can view both variables $T_i(i=1,2)$ as parameters of ${\mathbb {G}}_m$ .

Lemma 6.2. For each $\beta \in {\mathbb {U}}_\infty$ , the $\mathscr {I}$ -valued measure on ${\mathbb {Z}}^2_2$ corresponding to $h_\beta (T_1,T_2)$ is supported on ${\mathbb {Z}}^\times _2\times {\mathbb {Z}}^\times _2$ . Furthermore, let $D_i$ be the operator $(1+T_i)\frac {\partial }{\partial T_i}$ on $\mathscr {I}[[T_1,T_2]]$ for $i=1,2$ , and let $k\gt j\geq 0$ be two integers. Then we have

(6.3) \begin{equation} D^k_1D^j_2h_\beta (T_1,T_2)\big |_{(T_1,T_2)=(0,0)}=\Omega ^k_{\mathfrak {p}}\cdot \delta _{k,j}(\beta ), \end{equation}

where $\delta _{k,j}$ is defined in (3.7) .

Proof. For each prime $\omega$ of $F_m$ over $\mathfrak {p}$ , note the formulas (3.4) and (3.2), by changing the variable of $g_{m,\beta }$ to $j(T_1)$ , we have

\begin{equation*}(g_{m,\beta }(j(T_1)))_\omega =\log c_{m,\omega, \beta }(j(T_1))-\frac {1}{2}\sum _{\zeta \in \mu _2}\log c_{m,\omega, \beta }(\zeta (1+j(T_1))-1).\end{equation*}

Thus, an easy calculation shows that

\begin{equation*}\sum _{\zeta \in \mu _2}(g_{m,\beta }(\zeta (1+j(T_1))-1))_\omega =0,\end{equation*}

therefore, by a well-known fact (refer to [Reference Coates and Li5, formula(5.11)] or [Reference Yager27, Lemma (19)]), we know that the one-variable $2$ -adic measure corresponding to $(g_{m,\beta }(j(T_1)))_\omega$ is supported on ${\mathbb {Z}}^\times _2$ . Next, from Corollary 3.5 and $\chi _{{\mathfrak {p}}^*}$ takes values in ${\mathbb {Z}}^\times _2$ , by Lemma 6.1, we know that the two-variable $p$ -adic measure on ${\mathbb {Z}}^2_2$ corresponding to $h_\beta (T_1,T_2)$ is supported on ${\mathbb {Z}}^\times _2\times {\mathbb {Z}}^\times _2$ . Finally, let $\iota (T)\in \mathscr {I}[[T]]$ be the inverse of $j(T)$ , by the uniqueness of the exponential map on $\widehat {A}$ (refer to [Reference Coates and Li5, (5.23)]), then $\iota (T)=\exp (\Omega ^{-1}_{\mathfrak {p}}\lambda (T))-1$ , thus, we get

\begin{equation*}(\Omega _{\mathfrak {p}}\lambda '(T)^{-1})\frac {d}{dT}f(\iota (T))\big |_{T=j(T_1)}=(1+T_1)\frac {d}{dT_1}f(T_1).\end{equation*}

Then the formula (6.3) follows.

Recall that ${\mathcal {G}}={\mathrm {Gal}}(K_\infty /K)$ where $K_\infty =K(A_{2^\infty })$ . We can identify the Iwasawa algebra $\Lambda _{\mathscr {I}}({\mathcal {G}})$ with the $\mathscr {I}$ -valued measure on $\mathcal {G}$ . Since $\mathcal {G}$ is isomorphic to ${\mathbb {Z}}^\times _2\times {\mathbb {Z}}^\times _2$ , and Lemma 6.2 shows that the measure associated with $h_\beta (T_1,T_2)$ is supported on ${\mathbb {Z}}^\times _2\times {\mathbb {Z}}^\times _2$ , then we can get a two-variable $2$ -adic measure $\mu _\beta \in \Lambda _{\mathscr {I}}({\mathcal {G}})$ corresponding to $\beta \in {\mathbb {U}}_\infty$ .

Proposition 6.3. Let $i_1$ and $i_2$ be integers which are either $0$ or $1$ . Assume that $\beta \in {\mathbb {U}}_\infty$ . Then there exists a unique power series $\mathscr {R}^{(i_1,i_2)}_\beta (T_1,T_2)\in \mathscr {I}[[T_1,T_2]]$ such that for all integers $k\gt j\geq 0$ satisfying $(k,j)\equiv (i_1,i_2){\ \mathrm {mod}\ } 2$ , we have

(6.4) \begin{equation} \mathscr {R}^{(i_1,i_2)}_\beta (u^k-1,u^{-j}-1)=\Omega ^k_{\mathfrak {p}}\cdot \delta _{k,j}(\beta ). \end{equation}

Here we recall that $u$ is a topological generator of $1+4{\mathbb {Z}}_2$ . Moreover, the map from ${\mathbb {U}}_\infty$ to $\mathscr {I}[[T_1,T_2]]$ sending $\beta$ to $\mathscr {R}^{(i_1,i_2)}_\beta$ is a $\mathscr {I}[[T_1,T_2]]$ -module homomorphism.

Proof. If $x$ is any unit in ${\mathbb {Z}}_2$ , we can write $x=\omega (x)\langle x\rangle$ , where $\omega (x)\in \mu _2$ and $\langle x\rangle \equiv 1{\ \mathrm {mod}\ } 4$ . Let $f$ be a power series in $\mathscr {I}[[T_1,T_2]]$ corresponding to a measure $\mu _f$ on ${\mathbb {Z}}^2_2$ , we define a $\Gamma$ -transform $\Gamma ^{(i_1,i_2)}_f:{\mathbb {Z}}^2_2\rightarrow \mathscr {I}$ by

(6.5) \begin{equation} \Gamma ^{(i_1,i_2)}_f(s_1,s_2)=\int _{{\mathbb {Z}}^\times _2\times {\mathbb {Z}}^\times _2}\omega ^{i_1}(x)\omega ^{i_2}(y)\cdot \langle x\rangle ^{s_1}\langle y\rangle ^{s_2}d\mu _f(x,y). \end{equation}

We claim that $\Gamma ^{(i_1,i_2)}_f(s_1,s_2)$ can be written as $G^{(i_1,i_2)}(u^{s_1}-1,u^{s_2}-1)$ for some power series $G^{(i_1,i_2)}\in \mathscr {I}[[T_1,T_2]]$ . In fact, since $u$ is a topological generator of $1+4{\mathbb {Z}}_2$ , we define a homomorphism $\ell :{\mathbb {Z}}^\times _2\rightarrow {\mathbb {Z}}_2$ by $\langle x\rangle =u^{\ell (x)}$ for all $x\in {\mathbb {Z}}^\times _2$ . By the binomial theorem, we can write

\begin{equation*}\begin{aligned}\Gamma ^{(i_1,i_2)}_f(s_1,s_2)=&\int _{{\mathbb {Z}}^\times _2\times {\mathbb {Z}}^\times _2}(1+u^{s_1}-1)^{\ell (x_1)}(1+u^{s_2}-1)^{\ell (x_2)} \omega ^{i_1}(x_1)\omega ^{i_2}(x_2)d\mu _f\\ =&\sum _{n,m\geq 0}(u^{s_1}-1)^n(u^{s_2}-1)^m\int _{{\mathbb {Z}}^\times _2\times {\mathbb {Z}}^\times _2}C^n_{\ell (x_1)}\cdot C^m_{\ell (x_2)}\omega ^{i_1}(x_1)\omega ^{i_2}(x_2)d\mu _f, \end{aligned}\end{equation*}

where $C^k_x$ denotes the binomial coefficient function $\frac {x(x-1)\cdots (x-k+1)}{k!}$ (taking values in ${\mathbb {Z}}_2$ if $x\in {\mathbb {Z}}_2$ ). Thus, we can take $G^{(i_1,i_2)}(T_1,T_2)$ to be the power series for which the coefficient of $T^n_1T^m_2$ is given by

\begin{equation*}\int _{{\mathbb {Z}}^\times _2\times {\mathbb {Z}}^\times _2}C^n_{\ell (x_1)}C^m_{\ell (x_2)}\omega ^{i_1}(x_1)\omega ^{i_2}(x_2)d\mu _f,\end{equation*}

and then the claim follows.

Now, for any integers $k\gt j\geq 0$ which satisfy $(k,j)\equiv (i_1,i_2){\ \mathrm {mod}\ } 2$ . For $\beta \in {\mathbb {U}}_\infty$ , note that $\mu _{h_\beta }$ is supported on ${\mathbb {Z}}^\times _2\times {\mathbb {Z}}^\times _2$ and the formula (6.3) in Lemma 6.2, we have

(6.6) \begin{equation} \Gamma ^{(i_1,i_2)}_{h_\beta }(k,j)=\int _{{\mathbb {Z}}_2\times {\mathbb {Z}}_2}x^ky^jd\mu _{h_\beta }=D^k_1D^j_2h_\beta (T_1,T_2)\big |_{(T_1,T_2)=(0,0)} =\Omega ^k_{\mathfrak {p}}\delta _{k,j}(\beta ). \end{equation}

We denote by $G^{(i_1,i_2)}_\beta$ the power series in $\mathscr {I}[[T_1,T_2]]$ corresponding to $\Gamma ^{(i_1,i_2)}_{h_\beta }$ given by the above claim. Let us define

(6.7) \begin{equation} \mathscr {R}^{(i_1,i_2)}_\beta (T_1,T_2)=G^{(i_1,i_2)}(T_1,(1+T_2)^{-1}-1), \end{equation}

then the result in the Proposition is followed by the claim and (6.6).

We state the main theorem in this section in the following

Theorem 6.4. For each $\alpha \in \mathscr {J}$ , there exists a $2$ -adic measure $\mathscr {L}_\alpha \in \Lambda _{\mathscr {I}}({\mathcal {G}})$ such that, for all integers $k,j$ with $k\gt j\geq 0$ , we have

(6.8) \begin{equation} \Omega ^{-(k+j)}_{\mathfrak {p}}\int _{\mathcal {G}} \chi _{\mathfrak {p}}^k\chi ^{-j}_{{\mathfrak {p}}^*}d\mathscr {L}_\alpha =b_k(\alpha )\cdot (k-1)!\left (\frac {2\pi }{\sqrt {7}}\right )^j\cdot \left (1-\frac {\phi ^{k+j}({\mathfrak {p}})}{2^{j+1}}\right )\cdot \frac {L^{({\mathfrak {f}}{\mathfrak {p}}^*)}(\overline {\phi }^{k+j},k)}{\Omega ^{k+j}}. \end{equation}

Here, we denote by $b_k(\alpha )=(\!-1)^{k-1}f^k\cdot (N\alpha -(\phi ^k\overline {\phi }^{-j})((\alpha )))$ .

Proof. Recall that $\Delta ={\mathrm {Gal}}(K_{0,0}/K)$ and $\Gamma ={\mathrm {Gal}}(K_\infty /K_{0,0})$ , we also have the isomorphism ${\mathcal {G}}\simeq \Delta \times \Gamma$ . As we have chosen the two topological generators $\gamma _1,\gamma _2$ of $\Gamma \simeq {\mathbb {Z}}^2_p$ , then we have the isomorphism ${\mathfrak {S}}: \mathscr {I}[[{\mathcal {G}}]]\stackrel {\sim }{\longrightarrow }\mathscr {I}[\Delta ][[T_1,T_2]]$ via sending $\gamma _i$ to $1+T_i$ for $i=1,2$ . Since our identification of $\mathscr {I}[[{\mathcal {G}}]]$ with the $\mathscr {I}$ -valued measure on $\mathcal {G}$ , we write ${\mathfrak {S}}(\mathscr {L}_\alpha )=\sum _{m,n\geq 0}a_{m,n}T^m_1T^n_2$ with $a_{m,n}\in \mathscr {I}[\Delta ]$ , then the left-hand side of (6.8) is equal to

(6.9) \begin{equation} \Omega _{\mathfrak {p}}^{-(k+j)}\sum _{m,n}\eta ^k_1\eta ^{-j}_2(a_{m,n})(\chi ^k_{\mathfrak {p}}(\gamma _1)-1)^m(\chi ^{-j}_{{\mathfrak {p}}^*}(\gamma _2)-1)^n, \end{equation}

here, we denote by $\eta _1$ (resp. $\eta _2$ ) the restriction of $\chi _{\mathfrak {p}}$ (resp. $\chi _{{\mathfrak {p}}^*}$ ) to $\Delta$ . As $(k,j)$ varies in all integers satisfying $k\gt j\geq 0$ , the characters $\chi ^k_{\mathfrak {p}}\chi ^{-j}_{{\mathfrak {p}}^*}$ form a dense subset of the continuous functions on $\mathcal {G}$ , therefore, to determine $\mathscr {L}_\alpha \in \mathscr {I}[[{\mathcal {G}}]]$ , it is suffice to determine (6.9) for all such $k,j$ ’s.

Let us denote ${\mathcal {R}}=\mathscr {I}[[T_1,T_2]]$ . Note that the $\Delta$ -action on $\mathscr {I}[\Delta ][[T_1,T_2]]={\mathcal {R}}[\Delta ]$ is given by multiplication on group elements in $\Delta$ , for any pair $(i_1,i_2)$ with $i_k\in \{0,1\}$ , we have

\begin{equation*}\eta ^{i_1}_1\eta ^{i_2}_2(\sigma f)=(\eta ^{i_1}\eta ^{i_2}_2)(\sigma )\cdot (\eta ^{i_1}_1\eta ^{i_2}_2)(f)\qquad \forall f\in \mathscr {I}[\Delta ][[T_1,T_2]],\, \sigma \in \Delta .\end{equation*}

Now, there are four possible cases for the pairs $(i_1,i_2)$ . So, for any $k\gt j\geq 0$ , (6.9) is non-vanishing only for the pair $(i_1,i_2)$ with $i_1\equiv k{\ \mathrm {mod}\ } 2$ and $i_2\equiv j{\ \mathrm {mod}\ } 2$ . Thus, to determine $\mathscr {L}_\alpha$ , we only need to determine the four component $\mathscr {L}^{(i_1,i_2)}_\alpha$ for $i_j\in \{0,1\}(j=1,2)$ . Each $\mathscr {L}^{(i_1,i_2)}\in \Lambda _{\mathscr {I}}(\Gamma )$ is characterized by the following property: for integers $k\gt j\geq 0$ with $(k,j)\equiv (i_1,i_2)$ , we have

\begin{equation*}\int _{\mathcal {G}} \chi ^{k}_{\mathfrak {p}}\chi ^{-j}_{{\mathfrak {p}}^*}d\mathscr {L}_\alpha =\int _\Gamma \chi ^k_{{\mathfrak {p}},\Gamma }\chi ^{-j}_{{\mathfrak {p}}^*,\Gamma }d\mathscr {L}^{(i_1,i_2)}_\alpha .\end{equation*}

Here, for a character $\xi$ on $\mathcal {G}$ , we write $\xi _\Gamma$ for the restriction of $\xi$ to $\Gamma$ .

After these remarks, we can come back to the proof of the theorem. We let $\beta =\langle e(\alpha )\rangle$ be the elliptic units which we defined in §4, we define $\mathscr {L}^{(i_1,i_2)}_\alpha \in \Lambda _{\mathscr {I}}(\Gamma )$ which send to the power series $\mathscr {R}^{(i_1,i_2)}_\beta$ under the isomorphism $\Lambda _{\mathscr {I}}(\Gamma )\simeq \mathscr {I}[[T_1,T_2]]$ , since $\chi _{\mathfrak {p}}(\gamma _1)=\chi _{{\mathfrak {p}}^*}(\gamma _2)=u$ , (6.9) and Lemma 6.4, for $k\gt j\geq 0$ with $(k,j)\equiv (i_1,i_2){\ \mathrm {mod}\ }2$ , we have

\begin{equation*}\Omega _{\mathfrak {p}}^{-(k+j)}\int _\Gamma \chi ^k_{{\mathfrak {p}},\Gamma }\chi ^{-j}_{{\mathfrak {p}}^*,\Gamma }d\mathscr {L}^{(i_1,i_2)}_\alpha =\Omega _{\mathfrak {p}}^{-(k+j)}\mathscr {R}^{(i_1,i_2)}_\beta (u^k-1,u^{-j}-1)=\Omega _{\mathfrak {p}}^{-j}\delta _{k,j}(\langle e(\alpha )\rangle ).\end{equation*}

Therefore, the assertion in the theorem follows from Theorem4.8.