On the Mordell–Weil ranks of supersingular abelian varieties over ${\mathbb{Z}}_{p}^{2}$-extensions

Let $p$ be an odd prime, $L$ be a number field and $L_{\mathrm{cyc}}$ be the cyclotomic ${\mathbb{Z}}_{p}$-extension of $L$. Let $L_{\mathrm{cyc}}=\cup_{n}L_{n}$ where ${\mathrm{Gal}}(L_{n}/L)\cong{\mathbb{Z}}/p^{n}{\mathbb{Z}}$. Let $A$ be an abelian variety over $L$. Suppose $A$ has good ordinary reduction at the primes of $L$ above $p$. Then it is conjectured by Mazur [1] that the classical $p$-primary Selmer group ${\mathrm{Sel}}_{p}(A/L_{\mathrm{cyc}})$ is cotorsion as a ${\mathbb{Z}}_{p}[[{\mathrm{Gal}}(L_{\mathrm{cyc}}/L)]]$-module (follows by the work of Kato [2] and Rohrlich [3] for elliptic curves defined over ${\mathbb{Q}}$ and $L/{\mathbb{Q}}$ abelian). Under this assumption Mazur's control theorem [1] on these classical $p$-primary Selmer groups implies that the Mordell–Weil rank $A(L_{n})$ is bounded independently of $n$. However, this technique only works under the ordinary assumption. If the abelian variety has supersingular reduction at one of the primes above $p$, then the ${\mathrm{Sel}}_{p}(A/L_{\mathrm{cyc}})$ is not cotorsion and Mazur's control theorem will not hold. Now suppose that $p$ is unramified in $L$ and $A$ has supersingular reduction at all primes of $L$ above $p$. Then Büyükboduk and Lei [4] constructed signed Selmer groups for $A$; these generalize plus and minus Selmer groups of Kobayashi [5] for $a_{p}=0$ and Sprung's $\sharp/\flat$-Selmer groups for $p\mid a_{p}\neq 0$ [6]. The former are proved to be cotorsion for elliptic curves over ${\mathbb{Q}}$ and the latter are conjectured to be cotorsion as well. In the case when $p\mid a_{p}\neq 0$, at least one of the $\sharp/\flat$-Selmer groups is known to be cotorsion [6]. In the case of $a_{p}=0$, Kobayashi also proved an analogue of the control theorem for $\pm$-Selmer groups. Combined, those results lead to a bound independent of $n$ for the growth of the rank for supersingular elliptic curves [5, Corollary 10.2]. Under the assumption that the signed Selmer groups of Büyükboduk and Lei are cotorsion, in [7], Lei and Ponsinet give an explicit sufficient condition in terms of Coleman maps attached to $A$ which ensures that the rank of the Mordell–Weil group of the dual abelian variety $A^{\vee}(L_{n})$ stays bounded as $n$ varies. Further they show that when the Frobenius operator on the Dieudonné module at $p$ can be expressed as certain block anti-diagonal matrix (cf. [7, Section 3.2.1]) then their explicit condition is satisfied; hence their result applies to abelian varieties of $GL_{2}$-type.

In [8] (see also [9] for a less technical exposition), Lei and Sprung consider the special case of elliptic curves defined over ${\mathbb{Q}}$ and the base field $F=K$, an imaginary quadratic field in which $p$ splits. The Mordell–Weil rank of $E$ at the $n$-th layer of a ${\mathbb{Z}}_{p}$-extension of $K$ is know to be of the form $ap^{n}+O(1)$ where $a$ is an integer called the growth number. When $E$ has ordinary reduction at $p$, this growth number $a$ is equal to $0$ for the cyclotomic ${\mathbb{Z}}_{p}$-extension of $K$ by the works of Kato and Rohrlich. But works of [10], [11], [12], [13] and [14] exhibit cases where $a=1$ along the anticyclotomic extension of $K$ when the root number of $E/K$ is $-1$ as it is predicted by Mazur's growth number conjecture [15, §18, p.201]. In the supersingular $p\geq 5$ case, [16] also finds $a=1$ under some assumptions. So along the anticyclotomic tower, the Mordell–Weil rank may be unbounded. Let $K_{\infty}$ be the ${\mathbb{Z}}_{p}^{2}$-extension of $K$ obtained as a compositum of the cyclotomic and the anticyclotomic extension of $K$. Let $K_{n}$ be a subfield of $K_{\infty}$ such that ${\mathrm{Gal}}(K_{n}/K)\cong({\mathbb{Z}}/p^{n}{\mathbb{Z}})^{2}$. Suppose the elliptic curve $E$ has good supersingular reduction at $p$, Lei and Sprung constructed four $\sharp/\flat$-Selmer groups for elliptic curves over the $\mathbb{Z}_{p}^{2}$-extension $K_{\infty}$, generalizing works of Kim for the case $a_{p}=0$ [17]. Assume that $E$ is an elliptic curve over ${\mathbb{Q}}$ with conductor prime to $p$ and the class number of $K$ is coprime to $p$. When $p|a_{p}$, but $a_{p}\neq 0$, suppose that one of the four $\pm\pm$-Selmer groups defined by Kim [17] is cotorsion. Then, Lei and Sprung show that $\mathrm{rank}E(K_{n})=O(p^{n})$. In the case $a_{p}=0$, they show that $\mathrm{rank}E(K_{n})=O(p^{n})$ under the assumption that all of the four $\sharp/\flat\sharp/\flat$-Selmer groups they defined are cotorsion.

Kazim Büyükboduk and Antonio Lei jointly started a program to generalize the plus/minus theory of Kobayashi and Pollack in the case of motives crystalline at $p$. This led to their series of joint papers [4], [18] and [19]. Note that [4] and [18] (including the work of Lei–Ponsinet mentioned above) are over the cyclotomic ${\mathbb{Z}}_{p}$-extension and [19] is for representations coming from modular forms. In this paper our results add to that program giving us results for abelian varieties over a maximal abelian pro-$p$ extension of a number field that is unramified outside $p$.

We restrict ourselves to the case of imaginary quadratic field $K$ where $p$ splits. But instead of working with elliptic curves we work with abelian varieties with supersingular reduction at both primes above $p$. In this case, we have at our disposal the signed Selmer groups of Büyükboduk and Lei [4] which are defined only over the cyclotomic extension of $K$. The objectives in our paper are threefold. We

• •

  define multi-signed Selmer groups for $A$ over the ${\mathbb{Z}}_{p}^{2}$-extension $K_{\infty}$. In order to achieve this we construct multi-signed Coleman maps attached to the abelian variety $A$. This gives us the local condition at primes above $p$.

• •

  give an explicit sufficient condition in terms of Coleman maps (hypothesis (H-large)) mentioned in the text) attached to $A$ which ensures that the Mordell–Weil rank is bounded by a function which is $O(p^{n})$ along the tower $K_{n}$ (see theorem 5.5).

• •

  show that this explicit condition (H-large) is satisfied when the Frobenius on the Dieudonné module at primes above $p$ can be expressed in a certain block diagonal form. This special case can be thought of as an analogue for the case $a_{p}=0$ for supersingular elliptic curves and happens for abelian varieties of the $GL_{2}$-type (see section 4.3). When our signed Selmer groups for $A$ are cotorsion, we hence deduce $\mathrm{rank}A^{\vee}(K_{n})=O(p^{n})$ (see theorem 5.6).

In order to state the main results more precisely, let us introduce some more notation first. Let $A^{\vee}$ be the dual abelian variety and let ${\mathrm{Sel}}_{\underline{J}}(A^{\vee}/K_{\infty})$ be the multi-signed Selmer group attached to $A$ constructed in section 3.3. Let $\mathcal{X}_{\underline{J}}(A^{\vee}/K_{\infty})$ denote the Pontryagin dual

Let $\Lambda$ be the completed group ring $\mathbb{Z}_{p}\llbracket{\mathrm{Gal}}(K_{\infty}/K)\rrbracket$. Then, the first of our main theorems is the following:

To state our next theorem, we need to introduce some further notation. Let $T_{p}(A)$ be the Tate module of $A$. Suppose that $p={\mathfrak{p}}{\mathfrak{p}}^{c}$ where ${\mathfrak{p}}$ and ${\mathfrak{p}}^{c}$ are prime ideals of $K$. Let $\mathbb{D}_{{\mathrm{cris}},{\mathfrak{p}}}(T_{p}(A))$ (reps. $\mathbb{D}_{{\mathrm{cris}},{\mathfrak{p}}^{c}}(T_{p}(A))$) be the Dieudonné module of $T_{p}(A)$ seen as a representation of ${\mathrm{Gal}}(\overline{K_{\mathfrak{p}}}/K_{\mathfrak{p}})$ (resp. ${\mathrm{Gal}}(\overline{K_{{\mathfrak{p}}^{c}}}/K_{{\mathfrak{p}}^{c}})$). These modules are naturally equipped with an action of a Frobenius operator. Let $C_{\mathfrak{p}}$ and $C_{{\mathfrak{p}}^{c}}$ be the matrices defined in section 2.2 arising from the Frobenius action on $\mathbb{D}_{{\mathrm{cris}},{\mathfrak{p}}}(T_{p}(A))$ and $\mathbb{D}_{{\mathrm{cris}},{\mathfrak{p}}^{c}}(T_{p}(A))$ respectively.

The Selmer groups appearing in theorem B are all multi-signed Selmer groups for particular choices of the indexing set $\underline{J}$ (see the discussion before proposition 4.14).

Let the abelian variety $A$ be of dimension $g$ and such that $C_{\mathfrak{p}}$ and $C_{{\mathfrak{p}}^{c}}$ are block anti-diagonal. The indexing set $\underline{J}:=(J_{\mathfrak{p}},J_{{\mathfrak{p}}^{c}})$ where $J_{\mathfrak{p}}$ and $J_{{\mathfrak{p}}^{c}}$ are subsets of $\{1,...,2g\}$ such that $|J_{\mathfrak{p}}|+|J_{{\mathfrak{p}}^{c}}|=2g$. Note that the Selmer groups ${\mathrm{Sel}}_{\underline{J}}(A^{\vee}/K_{\infty})$ depend not only on the indexing set $\underline{J}$ but also on the choice of Hodge-compatible bases of the Dieudonné modules $\mathbb{D}_{\mathrm{cris},{\mathfrak{p}}}(T_{p}(A))$ and $\mathbb{D}_{\mathrm{cris},{\mathfrak{p}}^{c}}(T_{p}(A))$ (see section 3.3 and remark 3.4). This is also the case for the multi-signed Selmer groups defined in [4] and [7] for supersingular abelian varieties in the cyclotomic case. When $2g=4$, upon fixing a Hodge-compatible bases, the indexing set $\underline{J}$ has cardinality $70$. But all of these $70$ multi-signed Selmer groups are not canonically defined. Defining multi-signed Selmer groups canonically is crucial in framing corresponding Iwasawa main conjectures relating our multi-signed Selmer groups to canonically defined $p$-adic $L$-functions in the analytic side. A new key observation made in this paper is that the specific four multi-signed Selmer groups mentioned in theorem B are independent of the choice of the Hodge-compatible bases and hence canonically defined (see proposition 4.18). ¹¹1We thank David Loeffler for suggesting this to us. Therefore, this gives a strong incentive for regarding these particular multi-signed choices as ``more fundamental and arithmetically interesting" than the other multi-signed Selmer groups. (In the case of elliptic curves over the ${\mathbb{Z}}_{p}^{2}$-extension $K_{\infty}$, they correspond to the usual four $\sharp/\flat$ signed Selmer groups of Lei and Sprung, generalizing four $+/-$ signed Selmer groups of Kim when $a_{p}=0$.) Apart from these four multi-signed Selmer groups, we also show that the two other multi-signed Selmer groups corresponding to the choices $\underline{J}=(\{1,...,2g\},\emptyset)$ and $\underline{J}=(\emptyset,\{1,...,2g\})$ are also canonically defined and independent of the choice of Hodge-compatible bases (see proposition 4.18). These two Selmer groups should correspond to BDP type $p$-adic $L$-functions (see section 6).

We now give an outline of the paper. In section 2, we introduce our notations and recall the basic preliminaries on $p$-adic Hodge theory that we will need throughout our article. The definition of multi-signed Selmer groups and Coleman maps are given in section 3. In section 4, under the hypothesis (H-large), we use multi-signed Coleman maps and logarithmic matrices to study the growth of

where $H^{1}_{f}(K_{n,v},T)$ is the Bloch-Kato local condition defined using the kernel of the dual exponential map. To achieve this, we generalized the work of [7] to the setting of $\mathbb{Z}_{p}^{2}$-extensions by using inputs from [8]. We find that $\mathrm{rank}_{\mathbb{Z}_{p}}(\mathcal{Y}_{n}^{\prime})=O(p^{n})$. In section 5, we consider the short exact sequence

where $\mathcal{X}_{n}$ is the Pontryagin dual of the classical $p$-Selmer group of $A^{\vee}$ over $K_{n}$, $\mathcal{X}_{n}^{0}$ is the Pontryagin dual of the fine Selmer group of $A^{\vee}$ and

Under the assumption that at least one of the Selmer groups ${\mathrm{Sel}}_{\underline{J}}(A^{\vee}/K_{\infty})$ is $\Lambda$-cotorsion, we show that $\mathrm{rank}_{\mathbb{Z}_{p}}(\mathcal{X}_{n}^{0})$ is also $O(p^{n})$. Using the natural surjection $\mathcal{Y}_{n}^{\prime}\to\mathcal{Y}_{n}$, we deduce that the growth of $\mathcal{Y}_{n}$ is at most $O(p^{n})$. We are then able to conclude that the Mordell-Weil rank of $A^{\vee}$ is $O(p^{n})$ along $K_{\infty}$.

One expects that there should be analytic analogues of all of the above, corresponding to our (algebraic) results under appropriate Iwasawa main conjectures. In section 6, written by Chris Williams, we give some speculative remarks on the shape of the analytic theory.

The above concludes the outline. Now, before going to the next section, we would like to make further remarks. In this paper we stick to the case of ${\mathbb{Z}}_{p}^{2}$-extension of an imaginary quadratic field instead of working more generally over ${\mathbb{Z}}_{p}^{d}$-extension of a number field. Our guess is that the growth condition will be $O(p^{(d-1)n})$, but we will need to make the hypotheses that the multi-signed Selmer groups are cotorsion over the corresponding Iwasawa algebra. Such hypotheses are currently only known for elliptic curves under non-vanishing of signed $p$-adic $L$-functions over ${\mathbb{Z}}_{p}^{2}$-extension of an imaginary quadratic field (see remark 4.3). The analytic side for general abelian varieties are even more mysterious for $d>2$ (see section 6), and in this case (the Euler system machinery is also unavailable) as per our knowledge, there are currently no results in the literature towards cotorsion-ness of signed Selmer groups. Even if we assume that the signed Selmer groups are cotorsion, the present techniques in this paper will not generalize verbatim for $d>2$. We also faced technical difficulty generalizing (4.5) for general ${\mathbb{Z}}_{p}^{d}$-extensions.

Let $p\geq 3$ be a fixed prime number for the rest of the paper. Let $K$ be an imaginary quadratic field in which $p$ splits into the primes ${\mathfrak{p}}$ and ${\mathfrak{p}}^{c}$ of $K$. Here, $c$ denotes the complex conjugation. We will always use ${\mathfrak{q}}$ to mean an element of the set $\{{\mathfrak{p}},{\mathfrak{p}}^{c}\}$. Let $K_{\infty}$ be the unique $\mathbb{Z}_{p}^{2}$-extension of $K$ with Galois group $\Gamma\colonequals{\mathrm{Gal}}(K_{\infty}/K)$. If $\mathfrak{a}$ is an ideal of $\mathcal{O}_{K}$, $K(\mathfrak{a})$ will denote the ray class field of $K$ of conductor $\mathfrak{a}$. If $n\geq 0$ is an integer, write $\Gamma_{n}\colonequals\Gamma^{p^{n}}$ and $K_{n}\colonequals K_{\infty}^{\Gamma_{n}}=K(p^{n+1})^{{\mathrm{Gal}}(K(1)/K)}$. We make the hypothesis that $K(1)\bigcap K_{\infty}=K$. It follows that the Galois group $\Gamma$ is isomorphic to $G_{{\mathfrak{p}}}\times G_{{\mathfrak{p}}^{c}}$ where $G_{\mathfrak{q}}$ is the Galois group of the extension $K({\mathfrak{q}}^{\infty})\bigcap K_{\infty}/K$. We fix topological generators $\gamma_{\mathfrak{p}}$ and $\gamma_{{\mathfrak{p}}^{c}}$ respectively for these groups. Let $\Lambda\colonequals\mathbb{Z}_{p}\llbracket\Gamma\rrbracket\cong\mathbb{Z}_{p}\llbracket\gamma_{\mathfrak{p}}-1,\gamma_{{\mathfrak{p}}^{c}}-1\rrbracket$. Sometimes, we shall also use the notation $\Lambda(\Gamma)$ for $\Lambda$ when we want to emphasize that $\Lambda$ is the Iwasawa algebra of the group $\Gamma$. More generally, if $G$ is any profinite group, we denote by $\Lambda(G)$ the completed group ring $\mathbb{Z}_{p}\llbracket G\rrbracket$. Write $\Gamma_{\mathfrak{q}}$ for the decomposition group of ${\mathfrak{q}}$ in $\Gamma$. Let $\mu_{p^{n}}$ denote the set of $p^{n}$th roots of unity and let $\mu_{p^{\infty}}\colonequals\bigcup_{n\geq 1}\mu_{p^{n}}$.

We let $F_{\infty}$ and $k_{\text{cyc}}$ be the unramified $\mathbb{Z}_{p}$-extension and the cyclotomic $\mathbb{Z}_{p}$-extension of $\mathbb{Q}_{p}$ respectively. Let $k_{\infty}$ be the compositum of $F_{\infty}$ and $k_{\text{cyc}}$. For $n\geq 0$, $k_{n}$ and $F_{n}$ denote the subextension of $k_{\text{cyc}}$ and $F_{\infty}$ such that $[k_{n}:\mathbb{Q}_{p}]=p^{n}$ and $[F_{n}:\mathbb{Q}_{p}]=p^{n}$. Write $\Gamma_{p}\colonequals{\mathrm{Gal}}(k_{\infty}/\mathbb{Q}_{p})\cong\mathbb{Z}_{p}^{2}$, $\Gamma_{\text{ur}}\colonequals{\mathrm{Gal}}(F_{\infty}/\mathbb{Q}_{p})=\overline{\langle\sigma\rangle}\cong\mathbb{Z}_{p}$ and $\Gamma_{\text{cyc}}\colonequals{\mathrm{Gal}}(k_{\text{cyc}}/\mathbb{Q}_{p})=\overline{\langle\gamma\rangle}\cong\mathbb{Z}_{p}$. We identify $K_{\mathfrak{q}}$ with $\mathbb{Q}_{p}$, $\Gamma_{\mathfrak{q}}$ with $\Gamma_{p}$ and $G_{\mathfrak{q}}$ with $\Gamma_{\text{cyc}}$ via $\gamma_{\mathfrak{q}}\mapsto\gamma$. Let $\mathcal{H}(\Gamma_{\text{cyc}})$ be the set of power series

with coefficients in $\mathbf{Q}_{p}$ such that $\sum_{n\geq 0}c_{n}X^{n}$ converges on the open unit disk.

For this subsection only, let $K$ be any number field where all primes above $p$ are unramified. Note that for our purpose, $p$ will be totally split in $K$ and so we can take $K_{v}=\mathbb{Q}_{p}$ in the following discussion where $v$ is a prime of $K$ above $p$. Denote by $\mathcal{O}_{K_{v}}$ the ring of integers of $K_{v}$ and let $\chi:{\mathrm{Gal}}(K_{v}(\mu_{p^{\infty}})/K_{v})\to\mathbb{Z}_{p}^{\times}$ be the cyclotomic character. Let $\mathcal{M}_{/K}$ be a motive defined over $K$ in the sense of [20] and $\mathcal{M}_{p}$ its $p$-adic realization. Let $T$ be a $G_{K}$-stable $\mathbb{Z}_{p}$-lattice inside $\mathcal{M}_{p}$. We shall denote by $T^{\dagger}\colonequals\mathrm{Hom}(T,\mu_{p^{\infty}})$ the cartier dual of $T$ and by $T^{\ast}(1)\colonequals\mathrm{Hom}(T,\mathbb{Z}_{p}(1))$ the Tate dual of $T$. Suppose that

For simplicity, suppose that the dimension of $\mathrm{Ind}_{K/\mathbb{Q}}\mathcal{M}_{p}$ over $K_{v}$ is even. This will be the case for example when the motive $\mathcal{M}$ is the motive associated to an abelian variety. Let $2g\colonequals\mathrm{dim}_{K_{v}}(\mathrm{Ind}_{K/\mathbb{Q}}\mathcal{M}_{p})$ and let $g_{\pm}:=\mathrm{dim}_{K_{v}}(\mathrm{Ind}_{K/\mathbb{Q}}\mathcal{M}_{p})^{c=\pm 1}$. Let $v$ be a prime above $p$ in $K$ and let $\mathbb{D}_{{\mathrm{cris}},v}(\mathcal{M}_{p})$ be $(\mathbb{B}_{\mathrm{cris}}\otimes_{\mathbb{Q}_{p}}\mathcal{M}_{p})^{G_{K_{v}}}$ where $\mathbb{B}_{\mathrm{cris}}$ is the crystalline period ring defined by Fontaine. It admits the structure of a filtered $\varphi$-module. Let $\mathbb{A}_{K_{v}}^{+}\colonequals\mathcal{O}_{K_{v}}\llbracket\pi\rrbracket$ where $\pi$ is seen as a formal variable equipped with a semilinear action by a Frobenius $\varphi$ which acts as the absolute Frobenius on $\mathcal{O}_{K_{v}}$ and on $\pi$ by $\varphi(\pi)=(\pi+1)^{p}-1$, and with an action of $g\in{\mathrm{Gal}}(K_{v}(\mu_{p^{\infty}})/K_{v})$ given by $g(\pi)=(\pi+1)^{\chi(g)}-1$. Let $\mathbb{N}_{v}(T)$ be the Wach module of $T$ whose existence and properties are shown in [21, Proposition 2.1.1]. It is a free $\mathbb{A}_{K_{v}}^{+}$-module of rank $2g$. Furthermore, the quotient $\mathbb{N}_{v}(T)/\pi\mathbb{N}_{v}(T)$ is identified with a $\mathcal{O}_{K_{v}}$-lattice of $\mathbb{D}_{{\mathrm{cris}},v}(\mathcal{M}_{p})$. We denote by $\mathbb{D}_{{\mathrm{cris}},v}(T)$ this $\mathcal{O}_{K_{v}}$-lattice. It is equipped with a filtration of $\mathcal{O}_{K_{v}}$-modules $\{\mathrm{Fil}^{i}\mathbb{D}_{{\mathrm{cris}},v}(T)\}_{i\in\mathbb{Z}}$ and a Frobenius operator $\varphi$. If we suppose that

The matrix of $\varphi$ with respect to this basis is of the form

for some $C_{v}\in\mathrm{GL}_{2g}(\mathcal{O}_{K_{v}})$ and where $I_{g}$ is the identity $g\times g$ matrix.

Let $L/\mathbb{Q}_{p}$ be a finite unramified extension. For $x\in\mathcal{O}_{L}$, define

Let $S_{L/\mathbb{Q}_{p}}$ be the image of $y_{L/\mathbb{Q}_{p}}$ in $\mathcal{O}_{L}[{\mathrm{Gal}}(L/\mathbb{Q}_{p})]$. In [22, Section 3.2], it is shown that there is an isomorphism $y_{F_{\infty}/\mathbb{Q}_{p}}$ of $\Lambda(\Gamma_{\text{ur}})$-modules

where the inverse limit is taken with respect to the trace maps on the left and the projection maps ${\mathrm{Gal}}(L^{\prime}/\mathbb{Q}_{p})\to{\mathrm{Gal}}(L/\mathbb{Q}_{p})$ for $L\subseteq L^{\prime}$ on the right. By [22, Proposition 3.2], $\varprojlim_{\mathbb{Q}_{p}\subseteq L\subseteq F_{\infty}}\mathcal{O}_{L}$ is a free $\Lambda(\Gamma_{\text{ur}})$-module of rank one, thus the Yager module $S_{F_{\infty}/\mathbb{Q}_{p}}$ is also free of rank one over $\Lambda(\Gamma_{\text{ur}})$.

Let $\{\Omega_{\mathbb{Q}_{p}}\}$ be a basis of the Yager module $S_{F_{\infty}/\mathbb{Q}_{p}}$. Recall that $T$ is a $G_{K}$-stable $\mathbb{Z}_{p}$-lattice inside $\mathcal{M}_{p}$. Let $\mathcal{L}_{T}^{\infty}$ be the two-variable big logarithm map of Loeffler–Zerbes [22]

It is a morphism of $\Lambda(\Gamma_{p})$-modules. The completed tensor product $\mathcal{H}(\Gamma_{\text{cyc}})\widehat{\otimes}\Lambda(\Gamma_{\text{ur}})$ is isomorphic to $\mathcal{H}(\Gamma_{p})$, the set of power series in $\gamma-1$ and $\sigma-1$ with coefficients in $\mathbb{Q}_{p}$ converging on the open unit disk. Thus, one may see the big logarithm map as an application sending elements of the Iwasawa cohomology to two-variables power series tensored with $\mathbb{D}_{\mathrm{cris}}(T)$. For $k_{n}$ a finite subextension of $k_{\infty}$, let $\exp_{T}^{\ast}:H^{1}(k_{n},T)\to k_{n}\otimes\mathbb{D}_{\text{cris}}(T)$ be the Bloch–Kato dual exponential map. We say that the cyclotomic part of a finite order character $\eta:\Gamma_{p}\to\overline{{\mathbb{Q}}}_{p}^{\times}$ is of conductor $p^{n+1}$ if $\eta$ sends the topological generator $\gamma$ to a primitive $p^{n}$-th root of unity. Then, the big logarithm map enjoys the following interpolation property (see [22, Theorem 4.15]):

For simplicity, let us suppose that the dimension of $\mathbb{D}_{\mathrm{cris}}(T)$ as a $\mathbb{Z}_{p}$-module is even as it will be the case for the applications we have in mind. Write $2g$ for the dimension of $\mathbb{D}_{\mathrm{cris}}(T)$ as a $\mathbb{Z}_{p}$-module where $g$ is some strictly positive integer. Choose $\{v_{i}\}_{i=1}^{2g}$ a Hodge-compatible $\mathbb{Z}_{p}$-basis of $\mathbb{D}_{\mathrm{cris}}(T)$ as in section 2.2. Let $L$ be a finite unramified extension of $\mathbb{Q}_{p}$ and let $L_{\mathrm{cyc}}$ denote the cyclotomic $\mathbb{Z}_{p}$-extension of $L$. Then, in [4], the authors show the existence of one-variable Coleman maps

for $1\leq i\leq 2g$. Those maps are compatible with the corestriction maps $H^{1}_{\mathrm{Iw}}(F_{m,\mathrm{cyc}},T)\to H^{1}_{\mathrm{Iw}}(F_{m-1,\mathrm{cyc}},T)$ and the trace maps $\mathcal{O}_{F_{m}}\otimes\Lambda(\Gamma_{\mathrm{cyc}})\to\mathcal{O}_{F_{m-1}}\otimes\Lambda(\Gamma_{\mathrm{cyc}})$ (see [23, Appendix A.1] where the Coleman maps $\mathrm{Col}_{T,F_{m},i}$ are denoted by $\mathrm{Col}_{\sharp/\flat,F_{m}}$ instead). We define two-variable Coleman maps by taking the inverse limit of the $\mathrm{Col}_{T,F_{m},i}$ as $F_{m}$ runs through the finite extensions between $F_{\infty}$ and $\mathbb{Q}_{p}$. In order to get a family of maps landing in $\Lambda(\Gamma_{p})$, we further compose it with $y_{F_{\infty}/\mathbb{Q}_{p}}$. More precisely, the two-variable Coleman maps are defined by

where $(z_{m})\in\varprojlim_{F_{m}}H^{1}_{\mathrm{Iw}}(F_{m,\mathrm{cyc}},T)$. By identifying $\Omega_{\mathbb{Q}_{p}}\cdot\Lambda(\Gamma_{p})$ with $\Lambda(\Gamma_{p})$, we omit $\Omega_{\mathbb{Q}_{p}}$ from the notation and see $\mathrm{Col}_{T,i}^{\infty}$ as taking value in $\Lambda(\Gamma_{p})$. By combining [4, Theorem 2.13] and [22, Theorem 4.7 (1)], we see that these Coleman maps decompose the big logarithm map

The matrix $M_{T}$ is called a logarithmic matrix and is defined in the following way: Let $\Phi_{p^{n}}(1+X)$ denote the $p^{n}$th cyclotomic polynomial. For $n\geq 1$, first define the matrices

and $M_{n}\colonequals(C_{\varphi})^{n+1}C_{n}\cdots C_{1}$ where $C$ and $C_{\varphi}$ are defined as in section 2.2 with $v=p$. In [4], it is shown that the sequence $\{M_{n}\}$ converges to a $2g\times 2g$ matrix with entries in $\mathcal{H}(\Gamma_{\text{cyc}})$ which we call $M_{T}$. Let ${\mathfrak{q}}\in\{{\mathfrak{p}},{\mathfrak{p}}^{c}\}$. If we include the primes ${\mathfrak{p}}$ and ${\mathfrak{p}}^{c}$ in our notation, then we have the Coleman maps

for $i\in\{1,...,2g\}$. They satisfy

For ${\mathfrak{q}}={\mathfrak{p}}$, $M_{T,{\mathfrak{p}}}=\lim_{r\rightarrow\infty}M_{{\mathfrak{p}},r}$. Here $M_{{\mathfrak{p}},r}\colonequals(C_{\varphi,{\mathfrak{p}}})^{r+1}C_{{\mathfrak{p}},r}\cdots C_{{\mathfrak{p}},1}$,

For ${\mathfrak{q}}={\mathfrak{p}}^{c}$ we have expressions $M_{T,{\mathfrak{p}}^{c}}=\lim_{s\rightarrow\infty}M_{{\mathfrak{p}}^{c},s}$, where $M_{{{\mathfrak{p}}^{c}},s}\colonequals(C_{\varphi,{\mathfrak{p}}^{c}})^{s+1}C_{{\mathfrak{p}}^{c},s}\cdots C_{{{\mathfrak{p}}^{c}},1}$,

For any $r\geq 1$, $r$ an integer, define $H_{{\mathfrak{p}},r}=C_{{\mathfrak{p}},r}\cdots C_{{\mathfrak{p}},1}$. Similarly, for any integer $s\geq 1$, we define $H_{{\mathfrak{p}}^{c},s}=C_{{\mathfrak{p}}^{c},s}\cdots C_{{\mathfrak{p}}^{c},1}$.
Let $H_{r,s}$ be the block diagonal matrix where the blocks on the diagonal are given by $H_{{\mathfrak{p}},r}$ and $H_{{\mathfrak{p}}^{c},s}$. Consider the tuples $\underline{I}=(I_{\mathfrak{p}},I_{{\mathfrak{p}}^{c}})$ and $\underline{J}=(J_{\mathfrak{p}},J_{{\mathfrak{p}}^{c}})$ where $I_{\mathfrak{q}}\subseteq\{1,...,2g\}$ and $J_{\mathfrak{q}}\subseteq\{1,...,2g\}$ for ${\mathfrak{q}}\in\{{\mathfrak{p}},{\mathfrak{p}}^{c}\}$. Let us suppose that $|I_{\mathfrak{p}}|+|I_{{\mathfrak{p}}^{c}}|=2g$ and $|J_{\mathfrak{p}}|+|J_{{\mathfrak{p}}^{c}}|=2g$.
We define $H_{\underline{I},\underline{J},r,s}$ to be the $(\underline{I},\underline{J})^{th}$-minor of $H_{r,s}$. We also define $\underline{I}_{0}\colonequals(I_{{\mathfrak{p}},0},I_{{\mathfrak{p}}^{c},0})$ where $I_{{\mathfrak{q}},0}=\{1,\ldots,g\}$.

The multi-signed Selmer group can be defined without the assumption that $K(1)\cap K_{\infty}=K$. For this subsection only, we work in more generality and drop this assumption. Suppose that there are $p^{t}$ primes above ${\mathfrak{p}}$ and ${\mathfrak{p}}^{c}$ in $K_{\infty}$. Fix a choice of coset representatives $\gamma_{1},\ldots,\gamma_{p^{t}}$ and $\delta_{1},\ldots,\delta_{p^{t}}$ for $\Gamma/\Gamma_{\mathfrak{p}}$ and $\Gamma/\Gamma_{{\mathfrak{p}}^{c}}$ respectively. Since $p$ splits in $K$, we can identify $K_{\mathfrak{q}}$ with $\mathbb{Q}_{p}$ and $\Gamma_{\mathfrak{q}}$ with $\Gamma_{p}$. Consider the ``semi-local" decomposition coming from Shapiro's lemma

where $v$ runs through the primes above ${\mathfrak{p}}$ in $K_{\infty}$. By choosing a Hodge-compatible basis of $\mathbb{D}_{\mathrm{cris},{\mathfrak{p}}}(T)$, define the Coleman map for $T$ at ${\mathfrak{p}}$ by

for all $1\leq i\leq 2g$.

Let $\mathcal{L}_{T,{\mathfrak{p}}}^{k_{\infty}}=\oplus_{j=1}^{p^{t}}\mathcal{L}_{T,{\mathfrak{p}}}^{\infty}\cdot\gamma_{j}$. Define $\mathrm{Col}_{T,{\mathfrak{p}}^{c},i}^{k_{\infty}}$ and $\mathcal{L}_{T,{\mathfrak{p}}^{c}}^{k_{\infty}}$ in an analogous way. Let $J_{\mathfrak{q}}$ denote a subset of $\{1,\ldots,2g\}$ and let

Tate's local pairing induces a pairing

for all places $v$ of $K_{\infty}$ above ${\mathfrak{q}}$. We define $H^{1}_{J_{\mathfrak{q}}}(K_{\infty,{\mathfrak{q}}},T^{\dagger})\subseteq\bigoplus_{v|{\mathfrak{q}}}H^{1}(K_{\infty,v},T^{\dagger})$ as the orthogonal complement of $\ker\mathrm{Col}_{T,J_{\mathfrak{q}}}$ under the previous pairing. If $L$ is a finite extension of $K_{\mathfrak{q}}$, define

where $L_{\mathrm{ur}}$ is the maximal unramified extension of $L$. Let $H^{1}_{\mathrm{ur}}(L,T^{\dagger})$ be the image of $H^{1}_{\mathrm{ur}}(L,\mathcal{M}_{p}^{\ast}(1))$ under the natural map $H^{1}(L,\mathcal{M}_{p}^{\ast}(1))\to H^{1}(L,T^{\dagger})$. Let

where $w$ is a place in $K_{\infty}$ and $L$ runs through finite extensions of $\mathbb{Q}_{p}$ contained in $K_{\infty,w}$. Let $\underline{J}\colonequals(J_{\mathfrak{p}},J_{{\mathfrak{p}}^{c}})$ where as before $J_{\mathfrak{p}}$ and $J_{{\mathfrak{p}}^{c}}$ are subsets of $\{1,\ldots,2g\}$ such that $|J_{\mathfrak{p}}|+|J_{{\mathfrak{p}}^{c}}|=2g$. Fix $\Sigma$ a finite set of prime of $K$ containing the primes above $p$, the archimedean primes and the primes of ramification of $T^{\dagger}$. Let $K_{\Sigma}$ be the maximal extension of $K$ unramified outside $\Sigma$. Let $\Sigma^{\prime}$ be the set of primes of $K_{\infty}$ lying above the primes in $\Sigma$. If $M$ is any ${\mathrm{Gal}}(K_{\Sigma}/K_{\infty})$-module, we denote $H^{1}(K_{\Sigma}/K_{\infty},M)$ by $H^{1}_{\Sigma}(K_{\infty},M)$.