# Non-commutative Iwasawa theory of abelian varieties over global function
fields

Li-Tong Deng  Li-Tong Deng, Yau Mathematical Sciences Center, Tsinghua
University, 100084, Beijing, China<EMAIL_ADDRESS>,  Yukako
Kezuka  Yukako Kezuka, Institut de mathématiques de Jussieu-Paris Rive Gauche,
4 Pl. Jussieu 75005 Paris, France<EMAIL_ADDRESS>,  Yong-Xiong Li
Yong-Xiong Li, Yau Mathematical Sciences Center, Tsinghua University, 100084,
Beijing, China<EMAIL_ADDRESS>and  Meng Fai Lim  Meng Fai Lim, School
of Mathematics and Statistics, Hubei Key Laboratory of Mathematical Sciences,
Central China Normal University, 430079, Wuhan, China<EMAIL_ADDRESS>
###### Abstract.

Let $A$ be an abelian variety defined over a global function field $F$. We
investigate the structure of the $p$-primary Selmer group
$\mathrm{Sel}(A/F_{\infty})$ for any prime number $p$ distinct from the
characteristic of $F$, over $p$-adic Lie extensions $F_{\infty}$ of $F$ which
contain the cyclotomic $\mathbb{Z}_{p}$-extension $F^{\mathrm{cyc}}$. In
particular, we prove that the Pontryagin dual of the Selmer group
$\mathrm{Sel}(A/F^{\mathrm{cyc}})$ is a torsion
$\mathbb{Z}_{p}[[\mathrm{Gal}(F^{\mathrm{cyc}}/F)]]$-module with trivial
$\mu$-invariant, and we establish the $\mathfrak{M}_{H}(G)$-conjecture of
Coates–Fukaya–Kato–Sujatha–Venjakob for $A/F_{\infty}$.

In view of the validity of the $\mathfrak{M}_{H}(G)$-conjecture, it therefore
makes sense to speak of the characteristic element (in the sense of Coates et
al.) attached to the Pontryagin dual of $\mathrm{Sel}(A/F_{\infty})$. We
relate the order of vanishing of the characteristic elements, evaluated at
Artin representations, to the corank of the Selmer group of the corresponding
twist of $A$ over the base field $F$. Combining this with the deep results of
Tate, Milne and Kato–Trihan, we show that the order of vanishing of the
characteristic elements is equal to the order of vanishing of the $L$-function
of $A/F$ at $s=1$ under appropriate assumptions.

Finally, we relate the generalised Euler characteristic of
$\mathrm{Sel}(A/F_{\infty})$ to the Euler characteristic of
$\mathrm{Sel}(A/F^{\mathrm{cyc}})$. This is a natural analogue of Zerbes’
result in the number field context and generalises previous results of Sechi
and Valentino in the function field context.

###### Key words and phrases:

Elliptic curves, Iwasawa theory, Euler characteristics

###### 2010 Mathematics Subject Classification:

11R23 (primary), 11R34 (secondary)

The second author is supported by the ANR project Coloss and the European
Union’s Horizon 2020 research and innovation programme under the Marie
Sklodowska-Curie grant agreement No. 101026826.

##  1\. Introduction

Non-commutative Iwasawa theory has emerged as a powerful framework for
understanding deep arithmetic properties over number fields contained in a
$p$-adic Lie extension and their precise relationship to special values of
complex $L$-functions. The aim of the present paper is to study non-
commutative Iwasawa theory over global function fields. To motivate our
discussion, we begin by reviewing the number field context as developed by
Coates, Fukaya, Kato, Sujatha, and Venjakob in [8] (see also [3, 4, 38]). In
the number field case, $p$ will denote an odd prime. Given a compact $p$-adic
Lie group $\mathcal{G}$, the Iwasawa algebra $\Lambda(\mathcal{G})$ of
$\mathcal{G}$ is defined to be the completed group ring

$\Lambda(\mathcal{G})=\varprojlim{\mathbb{Z}}_{p}[\mathcal{G}/\mathcal{U}],$

where $\mathcal{U}$ runs over all open normal subgroups of $\mathcal{G}$. Let
$E$ be an elliptic curve defined over a number field $F$ with good ordinary
reduction at all primes of $F$ above $p$. Denote by $F_{\infty}$ an admissible
$p$-adic Lie extension of $F$, that is, a Galois extension of $F$ such that
$F^{\mathrm{cyc}}\subset F_{\infty}$ and whose Galois group
$G={\mathrm{Gal}}(F_{\infty}/F)$ is a $p$-adic Lie group of positive
dimension. Here $F^{\mathrm{cyc}}$ denotes the cyclotomic
$\mathbb{Z}_{p}$-extension of $F$. Write
$\Gamma={\mathrm{Gal}}(F^{\mathrm{cyc}}/F)$,
$H={\mathrm{Gal}}(F_{\infty}/F^{\mathrm{cyc}})$, and set

$X(E/F_{\infty})={\mathrm{Hom}}_{\rm
cts}({\mathrm{Sel}}(E/F_{\infty}),{\mathbb{Q}}_{p}/{\mathbb{Z}}_{p})$

for the Pontryagin dual of the $p$-primary Selmer group
${\mathrm{Sel}}(E/F_{\infty})$, endowed with its natural left
$\Lambda(G)$-module structure. In the case where
$F_{\infty}=F^{\mathrm{cyc}}$, it is well-known that $X(E/F^{\mathrm{cyc}})$
is a finitely generated $\Lambda(\Gamma)$-module. It is generally expected
that $X(E/F^{\mathrm{cyc}})$ enjoys nice module-theoretical properties, as
illustrated by the following conjecture of Mazur [24]:

######  Conjecture 1.1 (Mazur).

The Iwasawa module $X(E/F^{\mathrm{cyc}})$ is a finitely generated torsion
$\Lambda(\Gamma)$-module.

This has been verified by Mazur himself in the case where
${\mathrm{Sel}}(E/F)$ is finite. The best evidence in support of this
conjecture is a deep result of Kato [17], where the torsionness assertion is
established when $E$ is defined over ${\mathbb{Q}}$ and $F$ is an abelian
extension of ${\mathbb{Q}}$. Taking the structure theory of
$\Lambda(\Gamma)$-module into account, Mazur’s conjecture is equivalent to
saying that $X(E/F^{\mathrm{cyc}})/X(E/F^{\mathrm{cyc}})(p)$ is finitely
generated over ${\mathbb{Z}}_{p}$, where $X(E/F^{\mathrm{cyc}})(p)$ denotes
the $p$-primary submodule of $X(E/F^{\mathrm{cyc}})$. Motivated by this
observation, Coates et al. introduce the category ${\mathfrak{M}}_{H}(G)$
which consists of all finitely generated $\Lambda(G)$-modules $M$ whose
quotient $M/M(p)$ by the $p$-primary submodule $M(p)$ is a finitely generated
$\Lambda(H)$-module. They then conjectured that ${\mathfrak{M}}_{H}(G)$
contains all the torsion $\Lambda(G)$-modules which are of interest in
arithmetic applications. More precisely, they formulated (see Section 5 and
Conjecture 5.1 of [8]):

######  Conjecture 1.2 (${\mathfrak{M}}_{H}(G)$-Conjecture).

The Iwasawa module $X(E/F_{\infty})$ belongs to the category
${\mathfrak{M}}_{H}(G)$.

This conjecture is crucial as it allows one to define a characteristic element
for $X(E/F_{\infty})$. This is essential for formulating the main conjectures
of non-commutative Iwasawa theory (see [8, 4]). Coates et al. further
demonstrated in their paper how these characteristic elements relate to the
Euler characteristics of $X(E/F_{\infty})$ via the theory of Akashi series
introduced in [9]. This latter aspect has been further explored by Zerbes in a
series of papers [39, 40, 41], where she related the characteristic elements
to the generalised Euler characteristics of $X(E/F_{\infty})$. Here, the
notion of the generalised Euler characteristics is an extension of the usual
Euler characteristics, whose idea emerged in [9], and was later developed and
systematically studied by Zerbes in [40].

We now shift our focus to the function field setting, the central theme of our
paper. Henceforth, $p$ denotes a fixed prime, where we allow $p=2$. Throughout
this paper, $F$ will always denote a global function field. In other words,
$F$ is a finite extension of the field $k(T)$ of rational functions in one
variable over a finite constant field $k$. We always assume that the
characteristic $\mathrm{char}(F)$ of $F$ is distinct from our fixed prime $p$.
We fix a separable closure $\bar{F}$ of $F$. The cyclotomic
${\mathbb{Z}}_{p}$-extension $F^{\rm cyc}$ of $F$ is obtained by adjoining to
$F$ the unique ${\mathbb{Z}}_{p}$-constant field extension of $k$. Thus,
writing $\mu_{p^{\infty}}$ for the group of $p$-power roots of unity in an
algebraic closure $\bar{k}$ of $k$, $F^{\rm cyc}$ is the unique
${\mathbb{Z}}_{p}$-extension of $F$ contained in $F(\mu_{p^{\infty}})$.
Following the analogy with the number field setting, we define an admissible
$p$-adic Lie extension of $F$ as an extension $F_{\infty}$ satisfying the
following conditions:

  1. (i)

$F_{\infty}$ contains the cyclotomic ${\mathbb{Z}}_{p}$-extension
$F^{\mathrm{cyc}}$ of $F$.

  2. (ii)

$G={\mathrm{Gal}}(F_{\infty}/F)$ is a compact $p$-adic group without
$p$-torsion.

  3. (iii)

$F_{\infty}/F$ is unramified outside of a finite set of primes.

It is worth noting that a global function field $F$ with $\mathrm{char}(F)\neq
p$ has only one ${\mathbb{Z}}_{p}$-extension, the cyclotomic one (see [28,
Proposition 10.3.20]). Therefore, any admissible extension of dimension at
least $2$ is necessarily non-commutative. This underscores the relevance of
non-commutative Iwasawa theory in the context of function fields. We also note
that when $\mathrm{char}(F)=p$, there are infinitely many geometric (i.e., not
containing any constant field extension) ${\mathbb{Z}}_{p}$-extensions of $F$
(see [14, Theorem 3]). This presents another intriguing difference between
number fields and function fields.

We now return to the core of the paper. Our first main result is as follows:

######  Theorem 1.3.

Let $A$ be an abelian variety defined over a global function field $F$ with
characteristic prime to $p$. Let $F_{\infty}/F$ be an admissible $p$-adic Lie
extension. Then, the $\mathfrak{M}_{H}(G)$-conjecture holds for the Pontryagin
dual $X(A/F_{\infty})$ of the Selmer group ${\mathrm{Sel}}(A/F_{\infty})$.

Furthermore, the Pontryagin dual $X(A/F^{\mathrm{cyc}})$ of the Selmer group
${\mathrm{Sel}}(A/F^{\mathrm{cyc}})$ has $\mu$-invariant equal to zero.

In particular, our result completely resolves Mazur’s conjecture and the
$\mathfrak{M}_{H}(G)$-conjecture in the function field setting. This presents
a stark contrast with the number field case. It is long established that the
Pontryagin dual of the Selmer group over the cyclotomic
${\mathbb{Z}}_{p}$-extension of a number field can have a positive
$\mu$-invariant (see [24, §10, Example 2], [13]). In contrast, our result
shows that this never occurs in function fields. We note that Sechi [31] has
given a proof of this theorem for elliptic curves $E$ without complex
multiplication defined over $F$, where $F_{\infty}$ is obtained by adjoining
the $p$-power torsion points of $E$ to $F$, under the assumption that
${\mathrm{Sel}}(E/F)$ is finite. In the case $\mathrm{char}(F)=p$, the
${\mathfrak{M}}_{H}(G)$-conjecture has been proven by Ochiai and Trihan under
a $\mu=0$ assumption (see [29, Theorem 1.9]). Our proof of this theorem
differs entirely from previous work on the ${\mathfrak{M}}_{H}(G)$-conjecture.
In particular, Theorem 1.3 provides the first example where the
${\mathfrak{M}}_{H}(G)$-conjecture holds unconditionally. A key element of our
proof is the observation that the cohomological dimension of
${\mathrm{Gal}}(\bar{F}/F^{\mathrm{cyc}})$ is equal to one. From there, our
approach is essentially cohomological in nature. The authors find it curious
that this line of approach seems to have been overlooked in literature.

In light of this result on the $\mathfrak{M}_{H}(G)$-conjecture, we can now
attach a characteristic element $\xi_{A}$ to $X(A/F_{\infty})$ in the sense of
Coates et al. This allows us to evaluate $\xi_{A}$ at a given Artin
representation $\rho:G\to{\mathrm{GL}}_{n}(\mathcal{O})$, where $\mathcal{O}$
is the ring of integers of a finite extension of ${\mathbb{Q}}_{p}$ in
$\overline{{\mathbb{Q}}}_{p}$, via the homomorphism $\Phi_{\rho}$ (for
details, see Section 7, in particular (7.2)). This yields an element in the
field of fractions of the formal power series ring $\mathcal{O}[[T]]$ in an
indeterminate $T$ with coefficients in $\mathcal{O}$. For subsequent
discussion, we shall introduce the following hypothesis to facilitate explicit
calculations:

Hypothesis ($\mathbf{G}$). $H^{i}(G,A_{p^{\infty}}(F_{\infty}))$ is finite for
all $i\geq 1$.

We are now in a position to state our next result, presented here in a
slightly simplified form. For a more precise and general statement of the
theorem below, we refer readers to Theorem 7.7, Corollary 7.8, and Theorem
7.12.

######  Theorem 1.4.

Let $A$ be an abelian variety defined over a global function field $F$ with
$\mathrm{char}(F)\neq p$, and let $F_{\infty}/F$ be an admissible $p$-adic Lie
extension. Denote by $\xi_{A}$ the characteristic element of
$X(A/F_{\infty})$, and let $\mathrm{reg}_{F}$ denote the regular
representation of $F$. Assuming Hypothesis $(\mathbf{G})$ holds, we have

${\mathrm{ord}}_{T=0}(\Phi_{\mathrm{reg}_{F}}(\xi_{A}))\geq{\mathrm{corank}}_{{\mathbb{Z}}_{p}}({\mathrm{Sel}}(A/F)).$

In addition, equality holds when Greenberg’s semi-simplicity conjecture 6.2
holds for $A/F$. Moreover, if we assume the finiteness of $\hbox{\cyr
X}(A/F)$, then

${\mathrm{ord}}_{T=0}(\Phi_{\mathrm{reg}_{F}}(\xi_{A}))={\mathrm{ord}}_{s=1}L(A/F,s).$

In fact, the finiteness assumption on the $\ell$-primary part of the Tate-
Shafarevich group $\hbox{\cyr X}(A/F)(\ell)$ for a single prime number $\ell$
suffices for the last part of the above theorem by the deep theorem of
Kato–Trihan [18].

In order to state our final result, we need to introduce the following
stronger finiteness hypothesis. Recall that we write
$H={\mathrm{Gal}}(F_{\infty}/F^{\rm cyc})$.

Hypothesis ($\mathbf{H}$). $H^{i}(H,A_{p^{\infty}}(F_{\infty}))$ is finite for
all $i\geq 0$.

Let $S$ be a nonempty finite set of primes of $F$ containing all the primes
where the abelian variety $A$ has bad reduction, and all the primes of $F$
which ramify in $F_{\infty}$, and let $S^{\prime}$ denotes the subset of
primes whose inertia group in $G$ is infinite. We write $A^{*}$ for the dual
abelian variety. The following is our final main result.

######  Theorem 1.5.

Let $A$ be an abelian variety defined over a global field $F$ with
$\mathrm{char}F\neq p$ and let $F_{\infty}$ be an admissible $p$-adic Lie
extension of $F$. Assume that Hypothesis $(\mathbf{H})$ holds. Then
${\mathrm{Sel}}(A/F_{\infty})$ has finite generalised $G$-Euler characteristic
$\chi(G,{\mathrm{Sel}}(A/F_{\infty}))$ if and only if ${\mathrm{Sel}}(A/F^{\rm
cyc})$ has finite generalised $\Gamma$-Euler characteristic
$\chi(\Gamma,{\mathrm{Sel}}(A/F^{\mathrm{cyc}}))$. Moreover, if this is the
case, then

$\chi(G,{\mathrm{Sel}}(A/F_{\infty}))=\chi(\Gamma,{\mathrm{Sel}}(A/F^{\mathrm{cyc}}))\prod_{v\in
S^{\prime}}\frac{\\#A^{*}_{p^{\infty}}(F_{v})}{\\#H^{1}(\Gamma_{w},A_{p^{\infty}}(F^{\mathrm{cyc}}_{w}))},$

where, for every $v\in S^{\prime}$, $w$ is a fixed prime of $F^{\mathrm{cyc}}$
above $v$.

The above result is a natural analogue of Zerbes [40], where the result was
proved in the context of number fields. In the function field context, our
result generalises previous computations of Sechi [31] and Valentino [36]. In
the case $A$ is an elliptic curve, we may recover the formula of Sechi with
Euler factors of the complex $L$-function appearing on the right hand side.

We conclude this introduction with an outline of the paper. In Section 2, we
collect some basic notions concerning Iwasawa algebras that will be used
throughout the paper. The proof of Theorem 1.3 will be given in Section 3,
where the key observation is that the Galois group
${\mathrm{Gal}}(F_{S}/F^{\mathrm{cyc}})$, where $F_{S}$ is the maximal
extension of $F$ contained in $\bar{F}$ which is unramified outside $S$, has
$p$-cohomological dimension one (see Lemma 3.2). In Section 4, we establish
the surjectivity of the localisation map defining the Selmer group
${\mathrm{Sel}}(A/F_{\infty})$ using Jannsen’s spectral sequence and Nekovář’s
duality theorem. This will be a key input to our computation of Akashi series
in Section 5. In Section 6, we prove a control theorem, showing in particular
that the restriction map
${\mathrm{Sel}}(A/F)\to{\mathrm{Sel}}(A/F^{\mathrm{cyc}})^{\Gamma}$ has finite
kernel and cokernel. This allows us to relate the order of vanishing at $T=0$
of the characteristic power series of ${\mathrm{Sel}}(A/F^{\mathrm{cyc}})$ to
the ${\mathbb{Z}}_{p}$-corank of the Selmer group ${\mathrm{Sel}}(A/F)$. In
Section 7, we relate the order of vanishing of characteristic elements
evaluated at Artin representations to the Selmer coranks and their twists in
the intermediate subextensions of the $p$-adic Lie extension. This, in
particular, gives us Theorem 1.4. We then give the definition of the
generalised Euler characteristic in Section 8, following the work of Zerbes
[40]. Finally, the computations of the generalised Euler characteristics will
be carried out in Section 9 under Hypothesis $(\mathbf{H})$ to obtain Theorem
1.5.

##  2\. Prerequisites on Iwasawa algebras

Throughout $p$ will denote a fixed prime. Let $G$ be a compact $p$-adic Lie
group without $p$-torsion. The Iwasawa algebra of $G$ over ${\mathbb{Z}}_{p}$
is defined by

$\Lambda(G)=\varprojlim{\mathbb{Z}}_{p}[G/U],$

where $U$ runs over the open normal subgroups of $G$ and the inverse limit is
taken with respect to the usual projection maps. Suppose for now that $G$ is
pro-$p$. Then the ring $\Lambda(G)$ has a skew field of fractions
$\mathcal{Q}(G)$ which allows us to define the rank of a $\Lambda(G)$-module
$M$ via the following:

${\mathrm{rank}}_{\Lambda(G)}(M)=\dim_{\mathcal{Q}(G)}\mathcal{Q}(G)\otimes_{\Lambda(G)}M.$

The module $M$ is said to be torsion if ${\mathrm{rank}}_{\Lambda(G)}(M)=0$.

Now, suppose that $G$ is not necessarily pro-$p$. By Lazard’s theorem (see
[12, Corollary 8.34]), one can always find an open subgroup $G^{\prime}$ of
$G$ which is pro-$p$. Then a $\Lambda(G)$-module $M$ is said to be torsion
over $\Lambda(G)$ if it is torsion over $\Lambda(G^{\prime})$ in the above
sense. We shall also make use of the following equivalent formulation of a
torsion $\Lambda(G)$-module: ${\mathrm{Hom}}_{\Lambda(G)}(M,\Lambda(G))=0$.

##  3\. Selmer groups and the ${\mathfrak{M}}_{H}(G)$-Conjecture

Let $p$ be any prime number. Throughout, $F$ will always denote a global
function field of characteristic coprime to $p$. We write $k$ for its constant
field. Suppose that $A$ is an abelian variety defined over $F$. Let
$F_{\infty}$ be a $p$-adic Lie extension of $F$ satisfying the following
admissibility conditions:

  1. (i)

$F_{\infty}$ contains the cyclotomic ${\mathbb{Z}}_{p}$-extension
$F^{\mathrm{cyc}}$ of $F$.

  2. (ii)

$G={\mathrm{Gal}}(F_{\infty}/F)$ is a compact $p$-adic group without
$p$-torsion.

  3. (iii)

$F_{\infty}/F$ is unramified outside of a finite set of primes.

In particular, $F^{\mathrm{cyc}}$ itself satisfies these conditions. Let $S$
be a nonempty finite set of primes of $F$ containing all the primes where the
abelian variety $A$ has bad reduction, and all the primes of $F$ which ramify
in $F_{\infty}$.

Let $\bar{F}$ denote a separable closure of $F$. Given any extension
${\mathfrak{K}}$ inside $\bar{F}$, we can define the $p$-primary Selmer group
${\mathrm{Sel}}(A/{\mathfrak{K}})$ of $E$ over ${\mathfrak{K}}$ using the
Galois cohomology groups:

(3.1) ${\mathrm{Sel}}(A/{\mathfrak{K}})={\rm
ker}\left(H^{1}({\mathfrak{K}},A_{p^{\infty}})\rightarrow\prod_{w}H^{1}({\mathfrak{K}}_{w},A)\right),$

where $A_{p^{\infty}}$ denotes the group of all $p$-power division points in
$A(\bar{F})$. We note that when ${\mathfrak{K}}$ is an infinite extension of
$F$, then ${\mathfrak{K}}_{w}$ is given as the union of completions at $w$ of
all finite extensions over $F$ contained in ${\mathfrak{K}}$. If
${\mathfrak{K}}$ is Galois over $F$, the natural action of
${\mathrm{Gal}}({\mathfrak{K}}/F)$ on $H^{1}({\mathfrak{K}},A_{p^{\infty}})$
turns ${\mathrm{Sel}}(A/{\mathfrak{K}})$ into a
${\mathrm{Gal}}({\mathfrak{K}}/F)$-module.

Let $F_{S}$ be the maximal extension of $F$ contained in $\bar{F}$ which is
unramified outside $S$. By the choice of $S$, we know $F_{\infty}$ is
contained in $F_{S}$. For any extension $K$ of $F$ contained in $F_{S}$, we
write $G_{S}(K)$ for the Galois group ${\mathrm{Gal}}(F_{S}/K)$. When $L$ is a
finite extension of $F$, we define each prime $v$ of $F$

$J_{v}(A/L)=\bigoplus_{w\mid v}H^{1}(L_{w},A)(p),$

where $w$ runs over all primes of $L$ lying above $v$. Since $L_{w}$ has
characteristic different from $p$, we have $H^{1}(L_{w},A)(p)\simeq
H^{1}(L_{w},A_{p^{\infty}})$ by Kummer theory. We shall make this
identification without any further mention.

Let $\bar{L}_{w}$ be a separable closure of $L_{w}$, by choosing an embedding
$\bar{L}\subset\bar{L}_{w}$, we can view ${\mathrm{Gal}}(\bar{L}_{w}/L_{w})$
as a subgroup of ${\mathrm{Gal}}(\bar{L}/L)$. We then have the following
localisation map

(3.2) $\lambda_{S}(A/L):H^{1}(G_{S}(L),A_{p^{\infty}})\to\bigoplus_{v\in
S}J_{v}(A/L)$

induced by restriction.

For an infinite separable extension $K$ of $F$ contained in $F_{S}$, we define

$J_{v}(A/K)=\lim_{\longrightarrow}J_{v}(A/L),$

where $L$ runs over all finite extensions of $F$ contained in $K$ and the
limit is taken with respect to the restriction maps. We similarly define the
localisation map

$\lambda_{S}(A/K)=\lim_{\longrightarrow}\lambda_{S}(A/L).$

We will use the following result on Selmer groups in this paper, which can be
proven in the same way as in the case of elliptic curves over number fields
(see [5, Lemma 2.3]).

######  Proposition 3.1.

Let $K$ be an extension of $F$ contained in $F_{S}$. Then
${\mathrm{Sel}}(A/K)$ satisfies the exact sequence

$0\to{\mathrm{Sel}}(A/K)\to
H^{1}(G_{S}(K),A_{p^{\infty}})\stackrel{{\scriptstyle\lambda_{S}(A/K)}}{{\longrightarrow}}\bigoplus_{v\in
S}J_{v}(A/K).$

The remainder of the section will be devoted to proving the
$\mathfrak{M}_{H}(G)$-conjecture for the Pontryagin dual $X(A/F_{\infty})$ of
the $p$-primary Selmer group ${\mathrm{Sel}}(A/F_{\infty})$. In order to
achieve this, we start with the following observation which will be crucial in
our subsequent discussion.

######  Lemma 3.2.

We have

$\mathrm{cd}_{p}(G_{S}(F^{\mathrm{cyc}}))=1.$

###### Proof.

Since the set $S$ is nonempty and $\mathrm{char}(F)\neq p$, Theorem 10.1.2
(i)(c) from [28] tells us that $\mathrm{cd}_{p}(G_{S}(\bar{k}F))=1$. Note that
$G_{S}(\bar{k}F)$ is a closed subgroup of $G_{S}(F^{\mathrm{cyc}})$, and its
quotient ${\mathrm{Gal}}(\bar{k}F/F^{\mathrm{cyc}})$ is a profinite group,
where every open subgroup has an index prime to $p$. Thus, by applying
Proposition 3.3.5 (i) from [28], we find
$\mathrm{cd}_{p}(G_{S}(F^{\mathrm{cyc}}))=\mathrm{cd}_{p}(G_{S}(\bar{k}F))$,
and the lemma follows. ∎

######  Lemma 3.3.

The cohomology group $H^{1}(G_{S}(F_{\infty}),A_{p^{\infty}})$ is cofinitely
generated over $\Lambda(H)$. In particular,
$H^{1}(G_{S}(F_{\infty}),A_{p^{\infty}})^{\vee}$ is a torsion
$\Lambda(G)$-module.

###### Proof.

Without loss of generality, we may assume that both $G$ and consequently $H$,
are pro-$p$ without $p$-torsion. Therefore, by an application of Nakayama’s
lemma, we are reduced to showing that
$H^{1}(G_{S}(F_{\infty}),A_{p^{\infty}})^{H}$ is cofinitely generated over
${\mathbb{Z}}_{p}$. Since
$H^{2}(G_{S}(F^{\mathrm{cyc}}),A_{p^{\infty}})=H^{2}(G_{S}(F_{\infty}),A_{p^{\infty}})=0$
by Lemma 3.2, the spectral sequence

$H^{i}\left(H,H^{j}(G_{S}(F_{\infty}),A_{p^{\infty}})\right)\Rightarrow
H^{i+j}(G_{S}(F^{\mathrm{cyc}}),A_{p^{\infty}})$

degenerates to yield an exact sequence

(3.3) $0\to H^{1}(H,A_{p^{\infty}}(F_{\infty}))\to
H^{1}(G_{S}(F^{\mathrm{cyc}}),A_{p^{\infty}})\to
H^{1}(G_{S}(F_{\infty}),A_{p^{\infty}})^{H}\to
H^{2}(H,A_{p^{\infty}}(F_{\infty}))\to 0.$

As $H^{1}(H,A_{p^{\infty}}(F_{\infty}))$ and
$H^{2}(H,A_{p^{\infty}}(F_{\infty}))$ are cofinitely generated over
${\mathbb{Z}}_{p}$ (see the proof of Theorem 1.1 in [15]), it remains to show
that $H^{1}(G_{S}(F^{\mathrm{cyc}}),A_{p^{\infty}})$ is cofinitely generated
over ${\mathbb{Z}}_{p}$. By Nakayama’s lemma, this is equivalent to showing
that $H^{1}(G_{S}(F^{\mathrm{cyc}}),A_{p^{\infty}})[p]$ is finite. On the
other hand, from the long exact sequence of Galois cohomology arising from the
short exact sequence $0\to A_{p}\to
A_{p^{\infty}}\xrightarrow{p}A_{p^{\infty}}\to 0$, we obtain a surjection

$H^{1}(G_{S}(F_{\mathrm{cyc}}),A_{p})\twoheadrightarrow
H^{1}(G_{S}(F_{\mathrm{cyc}}),A_{p^{\infty}})[p].$

Hence, it suffices to show $H^{1}(G_{S}(F_{\mathrm{cyc}}),A_{p})$ is finite,
or equivalently,
${\mathrm{corank}}_{{\mathbb{F}}_{p}[[\Gamma]]}\left(H^{1}(G_{S}(F_{\mathrm{cyc}}),A_{p})\right)=0$.
Since
${\mathrm{corank}}_{{\mathbb{F}}_{p}[[\Gamma]]}\left(H^{0}(G_{S}(F^{\mathrm{cyc}}),A_{p})\right)=0$
and $H^{2}(G_{S}(F^{{\mathrm{cyc}}}),A_{p})=0$, we have

(3.4)
${\mathrm{corank}}_{{\mathbb{F}}_{p}[[\Gamma]]}\left(H^{1}(G_{S}(F^{\mathrm{cyc}}),A_{p})\right)=\sum_{i=0}^{2}(-1)^{i+1}{\mathrm{corank}}_{{\mathbb{F}}_{p}[[\Gamma]]}\left(H^{i}(G_{S}(F^{\mathrm{cyc}}),A_{p})\right).$

It then follows from the spectral sequence

$H^{i}\left(\Gamma,H^{j}(G_{S}(F^{\mathrm{cyc}}),A_{p})\right)\Rightarrow
H^{i+j}(G_{S}(F),A_{p})$

that the right hand side of (3.4) is

$\sum_{i=0}^{2}(-1)^{i+1}{\mathrm{corank}}_{{\mathbb{F}}_{p}}(H^{i}(G_{S}(F),A_{p})).$

The latter sum vanishes by a global Euler–Poincaré characteristic calculation
(cf. [28, (8.7.4)]). This concludes the proof of the lemma. ∎

We can now prove the main result of this section.

######  Theorem 3.4.

Let $A$ be an abelian variety defined over a global function field $F$ of
characteristic prime to $p$. Let $F_{\infty}/F$ be an admissible $p$-adic Lie
extension. Then the Selmer group ${\mathrm{Sel}}(A/F_{\infty})$ of $A$ over
$F_{\infty}$ is a cofinitely generated $\Lambda(H)$-module. In particular, the
$\mathfrak{M}_{H}(G)$-conjecture holds for $X(A/F_{\infty})$.

Furthermore, ${\mathrm{Sel}}(A/F^{\mathrm{cyc}})$ is a cofinitely generated
${\mathbb{Z}}_{p}$-module.

###### Proof.

Since ${\mathrm{Sel}}(A/F_{\infty})$ is contained in
$H^{1}(G_{S}(F_{\infty}),A_{p^{\infty}})$, the conclusion is now an immediate
consequence of Lemma 3.3. ∎

##  4\. Surjectivity of localisation map

In this section, we establish the surjectivity of the localisation map
$\lambda_{S}(A/F_{\infty})$ defining the Selmer group
${\mathrm{Sel}}(A/F_{\infty})$. In particular, we obtain the surjectivity of
the localisation map $\lambda_{S}(A/F^{\mathrm{cyc}})$. Before giving a proof,
we will introduce the compact Selmer group ${\mathfrak{S}}(A/F_{\infty})$. We
write $T_{p}A=\varprojlim_{n}A_{p^{n}}$ for the $p$-adic Tate module of $A$.
For every finite extension $L$ of $F$ contained in $F_{\infty}$, the compact
Selmer group ${\mathfrak{S}}(A/L)$ is defined to be

$\mathrm{ker}\Big{(}H^{1}(G_{S}(L),T_{p}A)\to\prod_{w}H^{1}(L_{w},T_{p}A)\Big{)},$

where the product runs over all primes $w$ of $L$. We then define the compact
Selmer group over $F_{\infty}$ by

${\mathfrak{S}}(A/F_{\infty})=\varprojlim_{L}{\mathfrak{S}}(A/L).$

Define the continuous Iwasawa cohomology of $T_{p}A$ by

$H^{i}_{\mathrm{Iw}}(F_{\infty},T_{p}A)=\varprojlim_{L}H^{i}(G_{S}(L),T_{p}A),$

where the inverse limit is taken over all finite Galois extensions $L$ of $F$
contained in $F_{\infty}$, via the corestriction maps. Note in particular that
${\mathfrak{S}}(A/F_{\infty})$ is a submodule of
$H^{1}_{\mathrm{Iw}}(F_{\infty},T_{p}A)$.

We now record the following result which will be a key ingredient for our
eventual proof of the surjectivity.

######  Lemma 4.1.

$H^{1}_{\mathrm{Iw}}(F_{\infty},T_{p}A)$ is a torsion $\Lambda(G)$-module.
Moreover, if $\dim(G)\geq 2$ or if $A_{p^{\infty}}(F_{\infty})$ is finite,
then $H^{1}_{\mathrm{Iw}}(F_{\infty},T_{p}A)=0$.

###### Proof.

We consider the following spectral sequence of Jannsen (see [16, Theorem 1,
Theorem 11]; see also [23, Theorem 4.5.1] and [27, Theorem 8.5.6]))

$E_{2}^{i,j}={\mathrm{Ext}}^{i}_{\Lambda(G)}\left(H^{j}\left(G_{S}(F_{\infty}),A_{p^{\infty}})\right)^{\vee},\Lambda(G)\right)\Rightarrow
H^{i+j}_{\mathrm{Iw}}(F_{\infty},T_{p}A).$

The low degree terms corresponding to this spectral sequence give us the
following exact sequence

(4.1)
$0\to{\mathrm{Ext}}^{1}_{\Lambda(G)}\left(A_{p^{\infty}}(F_{\infty})^{\vee},\Lambda(G)\right)\to
H^{1}_{\mathrm{Iw}}(F_{\infty},T_{p}A)\to{\mathrm{Ext}}^{0}_{\Lambda(G)}\left(H^{1}\left(G_{S}(F_{\infty}),A_{p^{\infty}})\right)^{\vee},\Lambda(G)\right).$

By Lemma 3.3, we have
${\mathrm{rank}}_{\Lambda(G)}\left(H^{1}\left(G_{S}(F_{\infty}),A_{p^{\infty}})\right)^{\vee}\right)=0$,
so the right-most term vanishes. Since
${\mathrm{Ext}}^{1}_{\Lambda(G)}\left(A_{p^{\infty}}(F_{\infty})^{\vee},\Lambda(G)\right)$
is a torsion $\Lambda(G)$-module (cf. [37, Proposition 3.5 (iii) (a)]), it
follows that $H^{1}_{\mathrm{Iw}}(F_{\infty},T_{p}A)$ is a torsion
$\Lambda(G)$-module. Now, $A_{p^{\infty}}(F_{\infty})^{\vee}$ is finitely
generated over ${\mathbb{Z}}_{p}$, so the assumption $\dim(G)\geq 2$ tells us
that $A_{p^{\infty}}(F_{\infty})^{\vee}$ is a pseudo-null module over
$\Lambda(G)$, and the left-most term of (4.2) also vanishes. Thus,
$H^{1}_{\mathrm{Iw}}(F_{\infty},T_{p}A)=0$, as required. ∎

######  Theorem 4.2.

Let $A$ be an abelian variety defined over a global function field $F$ of
characteristic prime to $p$. Let $F_{\infty}/F$ be an admissible $p$-adic Lie
extension. Then the localisation map $\lambda_{S}(A/F_{\infty})$ is
surjective, giving rise to the following short exact sequence:

$0\to{\mathrm{Sel}}(A/F_{\infty})\to
H^{1}(G_{S}(F_{\infty}),A_{p^{\infty}})\stackrel{{\scriptstyle\lambda_{S}(A/F_{\infty})}}{{\longrightarrow}}\bigoplus_{v\in
S}J_{v}(A/F_{\infty})\to 0.$

###### Proof.

As the conclusion of the theorem remains unchanged upon replacing the base
field field $F$ by a larger field contained in $F_{\infty}$, we may assume
that $G$ is pro-$p$. The Poitou-Tate sequence ([30, Proposition A.3.2]) gives
an exact sequence

$\displaystyle 0\to{\mathrm{Sel}}(A/F_{\infty})\to
H^{1}(G_{S}(F_{\infty}),A_{p^{\infty}})\stackrel{{\scriptstyle\lambda_{S}(A/F_{\infty})}}{{\longrightarrow}}\bigoplus_{v\in
S}J_{v}(A/F_{\infty})\to{\mathfrak{S}}(A^{*}/F_{\infty})^{\vee}\to 0,$

where the final $``0"$ term is a consequence of Lemma 3.2. It therefore
remains to show that ${\mathfrak{S}}(A^{*}/F_{\infty})=0$.

We consider the following spectral sequence of Jannsen (see [16, Theorem 1,
Theorem 11]; see also [23, Theorem 4.5.1] and [27, Theorem 8.5.6]))

$E_{2}^{i,j}={\mathrm{Ext}}^{i}_{\Lambda(G)}\left(H^{j}\left(G_{S}(F_{\infty}),A^{*}_{p^{\infty}})\right)^{\vee},\Lambda(G)\right)\Rightarrow
H^{i+j}_{\mathrm{Iw}}(F_{\infty},T_{p}A^{*}).$

The low degree terms corresponding to this spectral sequence give us the
following exact sequence

(4.2)
$0\to{\mathrm{Ext}}^{1}_{\Lambda(G)}\left(A^{*}_{p^{\infty}}(F_{\infty})^{\vee},\Lambda(G)\right)\to
H^{1}_{\mathrm{Iw}}(F_{\infty},T_{p}A^{*})\to{\mathrm{Ext}}^{0}_{\Lambda(G)}\left(H^{1}\left(G_{S}(F_{\infty}),A^{*}_{p^{\infty}})\right)^{\vee},\Lambda(G)\right).$

By Lemma 3.3, we have
${\mathrm{rank}}_{\Lambda(G)}\left(H^{1}\left(G_{S}(F_{\infty}),A_{p^{\infty}})\right)^{\vee}\right)=0$,
so the right-most term vanishes. Now, $A_{p^{\infty}}(F_{\infty})^{\vee}$ is
finitely generated over ${\mathbb{Z}}_{p}$. Therefore, if $\dim(G)\geq 2$,
then $A_{p^{\infty}}(F_{\infty})^{\vee}$ is a pseudo-null module over
$\Lambda(G)$, and so the left-most term of (4.2) also vanishes. Thus, we have
shown that ${\mathfrak{S}}(A^{*}/F_{\infty})=0$ when $\dim(G)\geq 2$.

It therefore remains to consider the case where $F_{\infty}=F^{\mathrm{cyc}}$.
This case is slightly more involved, owing to the fact that we do not know the
finiteness of $A_{p^{\infty}}(F^{\mathrm{cyc}})$ in general (but see Remark
4.3 in the case where $A$ is an elliptic curve without complex
multiplication). By base changing $F$, if necessary, we may assume that every
prime of $S$ is inert in $F^{\mathrm{cyc}}/F$. Thus, by abuse of notation,
given $v\in S$, we also denote by $v$ the unique prime of $F^{\mathrm{cyc}}$
lying above $v$, and identify the Galois groups
${\mathrm{Gal}}(F^{\mathrm{cyc}}/F)$ with
${\mathrm{Gal}}(F^{\mathrm{cyc}}_{v}/F_{v})$, denoting them both by $\Gamma$.
For each $v\in S$, we consider the following local version of Jannsen’s
spectral sequence (see [16, Theorem 11], [23, Theorem 4.2.2] or [27, Theorem
5.2.6]) for the dual abelian variety $A^{*}$:

$E_{2}^{i,j}={\mathrm{Ext}}^{i}_{\Lambda(\Gamma)}\left(H^{j}\left(F^{\mathrm{cyc}}_{v},A_{p^{\infty}}^{*})\right)^{\vee},\Lambda(\Gamma)\right)\Rightarrow
H^{i+j}_{\mathrm{Iw}}(F^{\mathrm{cyc}}_{v},T_{p}A^{*}).$

To simplify notation, we write
$\mathrm{E}^{i}(-)={\mathrm{Ext}}^{i}_{\Lambda(\Gamma)}\left(-,\Lambda(\Gamma)\right)$.
Analogously to the global case, the low degree terms of the above spectral
sequence give an exact sequence

$0\to\mathrm{E}^{1}\left(A^{*}_{p^{\infty}}(F^{\mathrm{cyc}}_{v})^{\vee}\right)\to
H^{1}_{\mathrm{Iw}}(F^{\mathrm{cyc}}_{v},T_{p}A^{*})\to\mathrm{E}^{0}\left(H^{1}(F^{\mathrm{cyc}}_{v},A^{*}_{p^{\infty}})^{\vee}\right).$

The global and local spectral sequences satisfy a functoriality property [23,
Theorem 4.5.1], yielding the following commutative diagram:

---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{E}^{1}\left(A^{*}_{p^{\infty}}(F^{\mathrm{cyc}})^{\vee}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{H^{1}_{\mathrm{Iw}}(F^{\mathrm{cyc}},T_{p}A^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus_{v\in
S}\mathrm{E}^{1}\left(A^{*}_{p^{\infty}}(F^{\mathrm{cyc}}_{v})^{\vee}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus_{v\in
S}H^{1}_{\mathrm{Iw}}(F^{\mathrm{cyc}}_{v},T_{p}A^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus_{v\in
S}\mathrm{E}^{0}\left(H^{1}(F^{\mathrm{cyc}}_{v},A^{*}_{p^{\infty}})^{\vee}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where the vertical maps are induced by the natural localisation maps. Note
that $\ker g={\mathfrak{S}}(A^{*}/F^{\mathrm{cyc}})$. Thus, it remains to show
that $f$ is injective. Since $S$ is assumed to be non-empty, writing
$f=\oplus_{v\in S}f_{v}$, it suffices to show that

$f_{v}:\mathrm{E}^{1}\left(A^{*}_{p^{\infty}}(F^{\mathrm{cyc}})^{\vee}\right)\to\mathrm{E}^{1}\left(A^{*}_{p^{\infty}}(F^{\mathrm{cyc}}_{v})^{\vee}\right)$

is injective for each $v\in S$. But since both
$A^{*}_{p^{\infty}}(F^{\mathrm{cyc}})^{\vee}$ and
$A^{*}_{p^{\infty}}(F^{\mathrm{cyc}}_{v})^{\vee}$ are finitely generated over
${\mathbb{Z}}_{p}$, it follows by [28, Corollary 5.5.7] that $f_{v}$ is given
by

${\mathrm{Hom}}_{{\mathbb{Z}}_{p}}(A^{*}_{p^{\infty}}(F^{\mathrm{cyc}})^{\vee},{\mathbb{Z}}_{p})\to{\mathrm{Hom}}_{{\mathbb{Z}}_{p}}(A^{*}_{p^{\infty}}(F^{\mathrm{cyc}}_{v})^{\vee},{\mathbb{Z}}_{p}),$

which is injective since it is induced by the inclusion
$A^{*}_{p^{\infty}}(F^{\mathrm{cyc}})\hookrightarrow
A^{*}_{p^{\infty}}(F^{\mathrm{cyc}}_{v})$. This completes the proof of the
theorem. ∎

######  Remark 4.3.

In the case where $A=E$ is an elliptic curve without complex multiplication,
the proof of Theorem 4.2 can be simplified. By a theorem of Deuring, this is
equivalent to saying that $E$ is not defined over a finite field;
equivalently, $E$ is non-isotrivial. In particular, we do not have to
separately consider the case $F_{\infty}=F^{\mathrm{cyc}}$, since Lemma 4.1
holds without the assumption $\dim(G)\geq 2$. One way to see this is the
result of Ulmer, which says that $E(\bar{k}F)$ is a finitely generated abelian
group (see [35, §6.3]). Since $F^{\rm cyc}$ is a subfield of $\bar{k}F$, it
follows that $E(F^{\rm cyc})$ is a finitely generated abelian group, and thus
$E_{p^{\infty}}(F^{\rm cyc})$ is finite. It follows that
$E_{p^{\infty}}(F^{\rm cyc})$ is pseudo-null over $\Lambda(G)$, and we always
have $H^{1}_{\mathrm{Iw}}(F_{\infty},T_{p}E)$ injecting into the
${\mathrm{Ext}}^{0}$-term.

##  5\. Akashi series

Denote by $\mathcal{Q}(\Gamma)$ the field of fractions of $\Lambda(\Gamma)$.
To each $M$ in ${\mathfrak{M}}_{H}(G)$, we now attach a non-zero element
$\mathrm{Ak}(M)$ of $\mathcal{Q}(\Gamma)$, which is uniquely determined up to
multiplication by a unit in $\Lambda(\Gamma)$. It can be shown (see [8, Lemma
3.1]) that the homology groups

$H_{i}(H,M)\qquad\text{for }i\geq 0$

are finitely generated torsion $\Lambda(\Gamma)$-modules. We fix from now on a
topological generator of $\Gamma$, and identify $\Lambda(\Gamma)$ with the
formal power series ring ${\mathbb{Z}}_{p}[[T]]$ by mapping this topological
generator to $1+T$. Let $f_{M,i}\in\Lambda(\Gamma)$ be a characteristic power
series for $H_{i}(H,M)$, and define

(5.1) $\mathrm{Ak}(M)=\prod_{i\geq 0}f_{M,i}^{(-1)^{i}}.$

This product is finite because $H_{i}(H,M)=0$ for $i\geq\dim(H)$, and it is
well defined up to multiplication by a unit in $\Lambda(\Gamma)$ because each
$f_{M,i}$ is. We call $\mathrm{Ak}(M)$ the Akashi series of $M$. If $Y$ is a
discrete $p$-primary $G$-module such that
$M=Y^{\vee}\in{\mathfrak{M}}_{H}(G)$, then we call $\mathrm{Ak}(M)$ also the
Akashi series of $Y$, and by abuse of notation, we also denote it by
$\mathrm{Ak}(Y)$. Via the duality

$H_{i}(H,M)=H^{i}(H,Y)^{\vee},$

we will work with $H$-cohomology groups in our computation of Akashi series
without further mention. It is indeed a generalisation of the notion of
characteristic power series, because if $M$ is a finitely generated torsion
$\Lambda(\Gamma)$-module (we can view $H=\\{1\\}$), then the Akashi series of
$M$ is exactly the $\Lambda(\Gamma)$-characteristic power series of $M$. In
this case, we simply denote $\mathrm{Ak}(M)$ by $f_{M}$, or $f_{Y}$, again by
abuse of notation.

######  Lemma 5.1.

Given an exact sequence

(5.2) $0\to M_{1}\to M_{2}\to M_{3}\to 0$

of modules in the category ${\mathfrak{M}}_{H}(G)$, their Akashi series
satisfy $\mathrm{Ak}(M_{2})=\mathrm{Ak}(M_{1})\cdot\mathrm{Ak}(M_{3})$.

###### Proof.

This follows from considering the long exact sequence of $H$-homology derived
from (5.2) and applying the multiplicativity of
$\Lambda(\Gamma)$-characteristic series in short exact sequences. ∎

######  Remark 5.2.

We note that the original definition of ${\mathfrak{M}}_{H}(G)$ in [9] differs
from the definition in the subsequent paper [8] or in this paper. One
consequence of this is that it is no longer true that if $M$ is a module in
${\mathfrak{M}}_{H}(G)$ then $\mathrm{Ak}(M)=\mathrm{Ak}(M/M(p))$, as shown in
[9, Lemma 4.1]. This is because $M(p)$ need not be finitely generated over
$\Lambda(H)$, and so its homology group may not be finitely generated over
${\mathbb{Z}}_{p}$. Therefore, even though its homology group is annihilated
by some power of $p$, we cannot conclude that they are finite. In general,
$\mathrm{Ak}(M(p))$ can be non-trivial. In case $G$ is pro-$p$ without
$p$-torsion, it is given by $\mathrm{Ak}(M(p))=p^{\mu(M)}$ (see [1, 1.6]),
where $\mu(M)$ denotes the $\mu$-invariant of $M$ defined in [37, 4].

By virtue of Theorem 4.2, we have a short exact sequence

$0\to{\mathrm{Sel}}(A/F_{\infty})\to
H^{1}(G_{S}(F_{\infty}),A_{p^{\infty}})\to\bigoplus_{v\in
S}J_{v}(A/F_{\infty})\to 0.$

Since ${\mathrm{Sel}}(A/F_{\infty})$ and
$H^{1}(G_{S}(F_{\infty}),A_{p^{\infty}})$ are cofinitely generated
$\Lambda(H)$-modules by Lemma 3.3 and Theorem 3.4, so is $\bigoplus_{v\in
S}J_{v}(A/F_{\infty})$. Therefore, we may apply Lemma 5.1 to conclude that

(5.3)
$\mathrm{Ak}\Big{(}H^{1}(G_{S}(F_{\infty}),A_{p^{\infty}})\Big{)}=\mathrm{Ak}\Big{(}{\mathrm{Sel}}(A/F_{\infty})\Big{)}\mathrm{Ak}\left(\bigoplus_{v\in
S}J_{v}(A/F_{\infty})\right).$

Therefore, the determination of
$\mathrm{Ak}\Big{(}{\mathrm{Sel}}(A/F_{\infty})\Big{)}$ is reduced to
computing $\mathrm{Ak}\Big{(}H^{1}(G_{S}(F_{\infty}),A_{p^{\infty}})\Big{)}$
and $\mathrm{Ak}\left(\bigoplus_{v\in S}J_{v}(A/F_{\infty})\right)$. These
will be the content of the next two lemmas.

######  Lemma 5.3.

$\mathrm{Ak}\Big{(}H^{1}(G_{S}(F_{\infty}),A_{p^{\infty}})\Big{)}=f_{H^{1}(G_{S}(F^{\mathrm{cyc}}),A_{p^{\infty}})}\prod_{i\geq
1}\left(f_{H^{i}\left(H,A_{p^{\infty}}(F_{\infty})\right)}\right)^{(-1)^{i}}$.

###### Proof.

Since
$H^{2}(G_{S}(F^{\mathrm{cyc}}),A_{p^{\infty}})=H^{2}(G_{S}(F_{\infty}),A_{p^{\infty}})=0$
by Lemma 3.2, the spectral sequence

$H^{i}\left(H,H^{j}(G_{S}(F_{\infty}),A_{p^{\infty}})\right)\Rightarrow
H^{i+j}(G_{S}(F_{\infty}),A_{p^{\infty}})$

degenerates to yield an exact sequence

(5.4) $0\to H^{1}(H,A_{p^{\infty}}(F_{\infty}))\to
H^{1}(G_{S}(F^{\mathrm{cyc}}),A_{p^{\infty}})\to
H^{1}(G_{S}(F_{\infty}),A_{p^{\infty}})^{H}\to
H^{2}(H,A_{p^{\infty}}(F_{\infty}))\to 0$

and isomorphisms

(5.5) $H^{i}\left(H,H^{1}(G_{S}(F_{\infty}),A_{p^{\infty}})\right)\simeq
H^{i+2}(H,A_{p^{\infty}}(F_{\infty}))$

for $i\geq 1$. The conclusion now follows from a direct calculation using
(5.4) and (5.5). ∎

Since $\mathrm{char}(F_{v})\neq p$ for each $v\in S$, it follows from a
theorem of Iwasawa [28, Theorem 7.5.3] that the decomposition subgroup $G_{w}$
of $w$ in $G$ has dimension at most $2$. Let $S^{\prime}$ denote the subset of
$S$ consisting of primes whose inertia group in $G$ is infinite. We note that
this is equivalent to requiring that the decomposition subgroup $G_{w}$ has
dimension exactly $2$. Indeed, since the residue characteristic is prime to
$p$ and $G$ is a $p$-adic Lie group, the wild inertia subgroup is finite.
Moreover, the maximal unramified $p$-extension of $F_{v}$ is
$F_{w}^{\mathrm{cyc}}$, and the maximal tamely ramified extension of $F_{v}$
is a topologically cyclic extension over the maximal unramified extension.

######  Lemma 5.4.

We have $\mathrm{Ak}\left(\bigoplus_{v\in
S}J_{v}(A/F_{\infty})\right)=\prod_{v\in S\setminus
S^{\prime}}f_{J_{v}(A/F^{\mathrm{cyc}})}$

###### Proof.

For each $v\in S$, we have

$\displaystyle J_{v}(A/F_{\infty})$ $\displaystyle=\bigoplus_{w\mid
v}\mathrm{Ind}_{H_{w}}^{H}H^{1}(F_{\infty,w},A_{p^{\infty}}),$

where the sum runs over places $w$ of $F^{\mathrm{cyc}}$ lying above $v$, and
we also write $w$ for a fixed prime in $F_{\infty}$ lying above $w$. We claim
that $J_{v}(A/F_{\infty})=0$ for $v\in S^{\prime}$. Indeed, if we let $w$
denote a prime of $F_{\infty}$ above $v$, then $\dim G_{w}=2$ by our
assumption that $v\in S^{\prime}$. The theorem of Iwasawa [28, Theorem 7.5.3]
then tells us that $F_{\infty,w}$ has no nontrivial $p$-extension. Hence
$H^{1}(F_{\infty,w},A_{p^{\infty}})=0$ and this establishes our claim.

Taking this claim into account, it remains to compute
$\mathrm{Ak}\left(J_{v}(A/F_{\infty})\right)$ for $v\in S\setminus
S^{\prime}$. For a prime $v$ of $F$, we write $w$ for a prime of
$F^{\mathrm{cyc}}$ lying above $v$, and by abuse of notation, we also write
$w$ for a fixed prime in $F_{\infty}$ lying above $w$. We then denote by
$H_{w}$ the decomposition group in $H$. Shapiro’s lemma gives

(5.6) $H^{i}(H,J_{v}(A/F_{\infty}))\simeq\bigoplus_{w\mid
v}H^{i}(H_{w},H^{1}(F_{\infty,w},A_{p^{\infty}})).$

If $v\in S\>\backslash\>S^{\prime}$, then $\mathrm{cd}_{p}(H_{w})=0$. Thus,
the right hand side of (5.6) vanishes for all $i\geq 1$. For the
$J_{v}(A/F_{\infty})^{H}$ term, one can check easily that the restriction map

$J_{v}(A/F^{\mathrm{cyc}})\rightarrow J_{v}(A/F_{\infty})^{H}$

is an isomorphism using the fact that $\mathrm{cd}_{p}(H_{w})=0$. In
conclusion, we have

$\mathrm{Ak}\Big{(}J_{v}(A/F_{\infty})\Big{)}=\left\\{\begin{array}[]{ll}f_{J_{v}(A/F^{\mathrm{cyc}})}&\hbox{if
$v\in S\setminus S^{\prime}$;}\\\ 0,&\hbox{if $v\in
S^{\prime}$.}\end{array}\right.$

∎

######  Theorem 5.5.

Let $A$ be an abelian variety defined over a global function field $F$ with
$\mathrm{char}(F)\neq p$. Let $F_{\infty}/F$ be an admissible $p$-adic Lie
extension. Then we have

$\mathrm{Ak}\left({\mathrm{Sel}}(A/F_{\infty})\right)=f_{{\mathrm{Sel}}(A/F^{\mathrm{cyc}})}\prod_{i\geq
1}\left(f_{H^{i}\left(H,A_{p^{\infty}}(F_{\infty})\right)}\right)^{(-1)^{i}}\prod_{v\in
S^{\prime}}f_{J_{v}(A/F^{\mathrm{cyc}})}.$

###### Proof.

Putting the calculations of Lemma 5.3 and 5.4 into (5.3), we obtain

$f_{H^{1}(G_{S}(F^{\mathrm{cyc}}),A_{p^{\infty}})}\prod_{i\geq
1}\left(f_{H^{i}\left(H,A_{p^{\infty}}(F_{\infty})\right)}\right)^{(-1)^{i}}=\mathrm{Ak}\left({\mathrm{Sel}}(A/F_{\infty})\right)\prod_{v\in
S\setminus S^{\prime}}f_{J_{v}(A/F^{\mathrm{cyc}})}.$

Thanks to Theorem 4.2, there is a short exact sequence

$0\to{\mathrm{Sel}}(A/F^{\mathrm{cyc}})\to
H^{1}(G_{S}(F^{\mathrm{cyc}}),A_{p^{\infty}})\to\bigoplus_{v\in
S}J_{v}(A/F^{\mathrm{cyc}})\to 0,$

where all the terms are cofinitely generated over ${\mathbb{Z}}_{p}$ by Lemma
3.3 and Theorem 3.4. Consequently, we have

$f_{H^{1}(G_{S}(F^{\mathrm{cyc}}),A_{p^{\infty}})}=f_{{\mathrm{Sel}}(A/F^{\mathrm{cyc}})}\prod_{v\in
S}f_{J_{v}(A/F^{\mathrm{cyc}})}.$

Combining all of the above, we obtain the required conclusion of the theorem.
∎

We record an interesting corollary of our Theorem 5.5 which is an analogue of
a result of Howson [15] (also see [6]).

######  Corollary 5.6.

Retain the settings of Theorem 5.5. Suppose further that $G$ is pro-$p$ with
dimension $\geq 2$. We then have the following formula

$\begin{split}\mathrm{rank}_{{\mathbb{Z}}_{p}[[H]]}X(A/F_{\infty})=\mathrm{rank}_{{\mathbb{Z}}_{p}}X(A/F^{\mathrm{cyc}})-\mathrm{rank}_{{\mathbb{Z}}_{p}}A_{p^{\infty}}(F^{\mathrm{cyc}})^{\vee}\\\
+\sum_{w\in
S^{\prime}(F^{\mathrm{cyc}})}\mathrm{rank}_{{\mathbb{Z}}_{p}}A_{p^{\infty}}(F^{{\mathrm{cyc}}}_{w})^{\vee}.\end{split}$

###### Proof.

In the discussion of the proof, whenever $N$ is a finitely torsion
${\mathbb{Z}}_{p}[[\Gamma]]$-module with trivial $\mu$-invariant, we shall
represent the characteristic series $f_{N}$ by a distinguished polynomial, and
so it makes sense to speak of $\deg f_{N}$. In particular, we have
${\mathrm{rank}}_{{\mathbb{Z}}_{p}}(N)=\deg f_{N}$. Now, if $M$ is a
${\mathbb{Z}}_{p}[[G]]$-module which is finitely generated over
${\mathbb{Z}}_{p}[[H]]$, then $H_{i}(H,M)$ is finitely generated over
${\mathbb{Z}}_{p}$ for every $i$. Hence, we have

${\mathrm{rank}}_{{\mathbb{Z}}_{p}[[H]]}(M)=\sum_{i\geq
0}(-1)^{i}{\mathrm{rank}}_{{\mathbb{Z}}_{p}}H_{i}(H,M)$

(cf. [15, Theorem 1.1]). Combining this with the above, we see that
${\mathrm{rank}}_{{\mathbb{Z}}_{p}[[H]]}(M)$ can be read off from
$\mathrm{Ak}(M)$. More precisely, we have

${\mathrm{rank}}_{{\mathbb{Z}}_{p}[[H]]}(M)=\sum_{i\geq 0}(-1)^{i}\deg
f_{H_{i}(H,M)}.$

Now, rewriting the conclusion of Theorem 5.5, we have

$\mathrm{Ak}\left({\mathrm{Sel}}(A/F_{\infty})\right)=\mathrm{Ak}\left(A_{p^{\infty}}(F_{\infty})\right)f_{{\mathrm{Sel}}(A/F^{\mathrm{cyc}})}f_{A_{p^{\infty}}(F^{\mathrm{cyc}})}^{-1}\prod_{v\in
S^{\prime}}f_{J_{v}(A/F^{\mathrm{cyc}})}.$

Since $\dim H\geq 1$ by hypothesis, we see that
${\mathrm{rank}}_{{\mathbb{Z}}_{p}[[H]]}(A_{p^{\infty}}(F_{\infty})^{\vee})=0$.
It remains to show that

$\deg
f_{J_{v}(A/F^{\mathrm{cyc}})}=\sum_{w|v}{\mathrm{rank}}_{{\mathbb{Z}}_{p}}A_{p^{\infty}}(F^{{\mathrm{cyc}}}_{w})^{\vee}$

for $v\in S^{\prime}$, where $w$ runs through the primes of $F^{\mathrm{cyc}}$
above $v$. For this, we note that
$J_{v}(A/F^{\mathrm{cyc}})=\oplus_{w|v}H^{1}(F^{\mathrm{cyc}}_{w},A_{p^{\infty}})$.
Since $v\in S^{\prime}$, it follows from the proof of Lemma 5.4 that
$H^{1}(F^{\mathrm{cyc}}_{w},A_{p^{\infty}})=H^{1}(H_{w},A_{p^{\infty}}(F_{\infty,w}))$.
Since $\dim H_{w}\geq 1$ and
${\mathrm{rank}}_{{\mathbb{Z}}_{p}[[H]]}(A_{p^{\infty}}(F_{\infty,w})^{\vee})=0$,
we have

${\mathrm{rank}}_{{\mathbb{Z}}_{p}}H^{1}(H_{w},A_{p^{\infty}}(F_{\infty,w}))^{\vee}={\mathrm{rank}}_{{\mathbb{Z}}_{p}}H^{0}(H_{w},A_{p^{\infty}}(F_{\infty,w}))^{\vee}={\mathrm{rank}}_{{\mathbb{Z}}_{p}}A_{p^{\infty}}(F^{{\mathrm{cyc}}}_{w})^{\vee}.$

This completes the proof of the corollary. ∎

##  6\. Control Theorem

In this section, we will prove the following control theorem which will allow
us to relate the order of vanishing at $T=0$ of the characteristic power
series of ${\mathrm{Sel}}(A/F^{\mathrm{cyc}})$ to the
${\mathbb{Z}}_{p}$-corank of the Selmer group over ${\mathrm{Sel}}(A/F)$ over
the base field $F$.

######  Theorem 6.1.

Let $A$ be an abelian variety defined over a global function field $F$ of
characteristic different from $p$. Let $F^{\mathrm{cyc}}$ denote the
cyclotomic ${\mathbb{Z}}_{p}$-extension of $F$. Write
$\Gamma_{n}={\mathrm{Gal}}(F_{n}/F)$, where $F_{n}$ for the $n$-th layer with
of the ${\mathbb{Z}}_{p}$-extension $F^{\mathrm{cyc}}/F$ satisfying
$[F_{n}:F]=p^{n}$. Then the restriction map

${\mathrm{Sel}}(A/F_{n})\to{\mathrm{Sel}}(A/F^{\mathrm{cyc}})^{\Gamma_{n}}$

has finite kernel and cokernel which are bounded independently of $n$.

###### Proof.

We will closely follow the method of the proof of the control theorem for fine
Selmer groups over a number field as presented in [20]. We consider the
following commutative diagram with exact rows

(6.1)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{Sel}}(A/F_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{n}}$$\textstyle{H^{1}(G_{S}(F_{n}),A_{p^{\infty}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{n}}$$\textstyle{\bigoplus_{v\in
S}J_{v}(A/F_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{n}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{Sel}}(A/F^{\rm
cyc})^{\Gamma_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(G_{S}(F^{\rm
cyc}),A_{p^{\infty}})^{\Gamma_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus_{v\in
S}J_{v}(A/F^{\rm cyc})^{\Gamma_{n}}.}$

Since $\Gamma_{n}$ has $p$-cohomological dimension $1$, the inflation-
restriction sequence tells us that
$\ker\beta_{n}=H^{1}(\Gamma_{n},A_{p^{\infty}}(F^{\mathrm{cyc}}))$ and
$\mathrm{coker}~{}\\!\beta_{n}=0$. Write
$M=\left(A_{p^{\infty}}(F^{\mathrm{cyc}})\right)^{\vee}$. Then $M$ is a
finitely generated ${\mathbb{Z}}_{p}$-module with a continuous action by
$\Gamma$, so that it is a finitely generated torsion
$\Lambda(\Gamma_{n})$-module for every $n$. It follows by [28, Proposition
5.3.20] that

${\mathrm{rank}}_{{\mathbb{Z}}_{p}}M_{\Gamma_{n}}={\mathrm{rank}}_{{\mathbb{Z}}_{p}}M^{\Gamma_{n}}<\infty.$

But we have ${\mathrm{rank}}_{{\mathbb{Z}}_{p}}M_{\Gamma_{n}}=0$ since
$M_{\Gamma_{n}}=\left(A_{p^{\infty}}(F_{n})\right)^{\vee}$ is finite by the
Mordell–Weil theorem. It follows that $M^{\Gamma_{n}}$ is also finite. Since
$\Gamma_{n}$ has $p$-cohomological dimension $1$, we have

$H^{1}(\Gamma_{n},A_{p^{\infty}}(F^{\mathrm{cyc}}))=\left(A_{p^{\infty}}(F^{\mathrm{cyc}})\right)_{\Gamma_{n}},$

which is the Pontryagin dual of $M^{\Gamma_{n}}$. Since $M$ is a finitely
generated ${\mathbb{Z}}_{p}$-module, the order of $M^{\Gamma_{n}}$ is bounded
independently of $n$ by $M(p)$. It remains to show that $\ker\gamma_{n}$ is
finite and bounded independently of $n$ in order to conclude that the same
holds for $\ker\alpha_{n}$ and $\mathrm{coker}~{}\\!\alpha_{n}$, by the snake
lemma. For this, since $v\in S$ does not lie above $p$, it suffices to show
that for each prime $w$ of $F^{\mathrm{cyc}}$ lying above $v\in S$,
$H^{1}(\Gamma_{w},A_{p^{\infty}}(F^{\mathrm{cyc}}_{w}))$ is finite and bounded
independently of $n$. This can be shown by replacing $\Gamma_{n}$ with
$\Gamma_{w}$ and $M$ with
$M_{w}=\left(A_{p^{\infty}}(F^{\mathrm{cyc}}_{w})\right)^{\vee}$, which is
still a finitely generated ${\mathbb{Z}}_{p}$-module, in the above argument
showing that $H^{1}(\Gamma_{n},A_{p^{\infty}}(F^{\mathrm{cyc}}))$ is finite
and bounded independently of $n$. ∎

We have shown in Theorem 3.4 that $X(A/F^{\mathrm{cyc}})$ is a finitely
generated torsion $\Lambda(\Gamma)$-module with $\mu$-invariant zero. Thus,
the structure theorem for finitely generated $\Lambda(\Gamma)$-modules tells
us that there is a pseudo-isomophism

(6.2)
$X(A/F^{\mathrm{cyc}})\sim\bigoplus_{i=1}^{k}\Lambda(\Gamma)/(f_{i}^{a_{i}}),$

where we have identified $\Lambda(\Gamma)\simeq{\mathbb{Z}}_{p}[[T]]$ using
our fixed topological generator of $\Gamma$, and the $f_{i}$ for
$i=1,\ldots,k$ are polynomials of degree $\lambda_{i}$, say, which are
distinguished and irreducible. Thus, $\Lambda(\Gamma)/(f_{i}^{a_{i}})$ is
isomorphic to ${\mathbb{Z}}_{p}^{a_{i}\lambda_{i}}$ as abelian groups, and

${\mathrm{rank}}_{{\mathbb{Z}}_{p}}\left(X(A/F^{\mathrm{cyc}})\right)=\lambda=\sum_{i=1}^{k}a_{i}\lambda_{i}.$

Furthermore, we have
$\prod_{i=1}^{k}f_{i}^{a_{i}}=f_{{\mathrm{Sel}}(A/F^{\mathrm{cyc}})}$ up to a
unit in $\Lambda(\Gamma)$.

We will make use of the following analogue of the semi-simplicity conjecture
due to Greenberg, which was posed for elliptic curves over a number field, for
abelian varieties over a global function field.

######  Conjecture 6.2 (Greenberg’s semi-simplicity conjecture for $A/F$).

Let $A$ be an abelian variety defined over a global function field $F$. Then
the action of $\Gamma$ on $X(A/F^{\mathrm{cyc}})$ is completely reducible,
that is, $a_{i}=1$ for all $i$ in (6.2).

We then have the following.

######  Proposition 6.3.

Let $A$ be an abelian variety defined over a global function field $F$. Then
we have

${\mathrm{ord}}_{T=0}(f_{{\mathrm{Sel}}(A/F^{\mathrm{cyc}})})\geq{\mathrm{corank}}_{{\mathbb{Z}}_{p}}({\mathrm{Sel}}(A/F)).$

In addition, equality holds when Conjecture 6.2 holds.

###### Proof.

We have shown in Theorem 6.1 that
${\mathrm{Sel}}(A/F)\to{\mathrm{Sel}}(A/F^{\mathrm{cyc}})^{\Gamma}$ has finite
kernel and cokernel. Taking the Pontryagin dual, we see that the map
$X(A/F^{\mathrm{cyc}})_{\Gamma}\to X(A/F)$ has finite kernel and cokernel. It
follows that

${\mathrm{rank}}_{{\mathbb{Z}}_{p}}(X(A/F^{\mathrm{cyc}})_{\Gamma})={\mathrm{corank}}_{{\mathbb{Z}}_{p}}({\mathrm{Sel}}(A/F)).$

Now, we have
${\mathrm{rank}}_{{\mathbb{Z}}_{p}}(X(A/F^{\mathrm{cyc}})_{\Gamma})={\mathrm{rank}}_{{\mathbb{Z}}_{p}}\left(X(A/F^{\mathrm{cyc}})/TX(A/F^{\mathrm{cyc}})\right)$,
and since
${\mathrm{rank}}_{{\mathbb{Z}}_{p}}\left(\Lambda(\Gamma)/(T)\right)=1$ and
$\Lambda(\Gamma)/(f,T)\simeq{\mathbb{Z}}_{p}/f(0){\mathbb{Z}}_{p}$ for a
distinguished polynomial $f$, the ${\mathbb{Z}}_{p}$-rank of the latter is $1$
if and only if $T\mid f$. It thus follows from (6.2) that
${\mathrm{rank}}_{{\mathbb{Z}}_{p}}\left(X(A/F^{\mathrm{cyc}})/TX(A/F^{\mathrm{cyc}})\right)$
is equal to the number of $f_{i}$ equal to $T$. Assuming Conjecture 6.2, this
is precisely to the power of $T$ dividing
$f_{{\mathrm{Sel}}(A/F^{\mathrm{cyc}})}$, and the result follows. ∎

##  7\. The order of vanishing of the characteristic element

In this section, we prove our main result on the order of vanishing of the
characteristic element attached to the Selmer group over a $p$-adic Lie
extension. We shall begin recalling the definition of these characteristic
elements. As before, $G$ will denote a compact $p$-adic Lie group without
$p$-torsion and $H$ is a closed normal subgroup of $G$ such that
$\Gamma:=G/H\simeq{\mathbb{Z}}_{p}$. First, we introduce the characteristic
element $\xi_{M}$ attached to $M\in\mathfrak{M}_{H}(G)$. In order to do this,
we recall from [8] an equivalent definition of the category
$\mathfrak{M}_{H}(G)$. A multiplicatively closed subset $\Sigma$ of
$\Lambda(G)$ is said to be a left and right Ore set if, for each $f\in\Sigma$
and $x\in\Lambda(G)$, there exist $g_{1},g_{2}\in\Sigma$ and
$y_{1},y_{2}\in\Lambda(G)$ such that

$fy_{1}=xg_{1}\quad\text{and}\quad y_{2}f=g_{2}x.$

We define

$\Sigma=\\{f\in\Lambda(G):\Lambda(G)/\Lambda(G)f\text{ is a finitely generated
$\Lambda(H)$-module}\\}.$

It is shown in [8, Theorem 2.4] that the set $\Sigma$ is a multiplicatively
closed left and right Ore set in $\Lambda(G)$. Since $p\notin\Sigma$, we also
need to consider

$\Sigma^{*}=\cup_{n\geq 0}p^{n}\Sigma,$

which is again a multiplicatively closed left and right Ore set in
$\Lambda(G)$. We may therefore localise $\Lambda(G)$ at $\Sigma$ and at
$\Sigma^{*}$ to obtain $\Lambda(G)_{\Sigma}$ and
$\Lambda(G)_{\Sigma^{*}}=\Lambda(G)_{\Sigma}\left[\frac{1}{p}\right]$,
respectively. It follows from [8, Proposition 2.3] that, given a finitely
generated $\Lambda(G)$-module $M$, we have $M\in\mathfrak{M}_{H}(G)$ if and
only if $M$ is $\Sigma^{*}$-torsion. Since the Iwasawa algebra $\Lambda(G)$ is
noetherian and $G$ has no element of order $p$, $\Lambda(G)$ has finite global
dimension equal to $\dim(G)+1$. Then $K_{0}(\Lambda(G))$ can be identified
with the Grothendieck group of the category of all finitely generated
$\Lambda(G)$-modules. It follows from the localisation sequence of $K$-theory
for the Ore set $\Sigma^{*}$ that we have an exact sequence

$\cdots\to K_{1}(\Lambda(G))\to
K_{1}(\Lambda(G)_{\Sigma^{*}})\stackrel{{\scriptstyle\partial_{G}}}{{\longrightarrow}}K_{0}(\mathfrak{M}_{H}(G))\to
K_{0}(\Lambda(G))\to K_{0}(\Lambda(G)_{\Sigma^{*}})\to 0.$

By [8, Proposition 3.4], the map $\partial_{G}$ is surjective. In view of this
result, we can define the notion of a characteristic element for each $M$ in
$\mathfrak{M}_{H}(G)$.

######  Definition 7.1.

Given $M$ in $\mathfrak{M}_{H}(G)$, a characteristic element for $M$ is any
element $\xi_{M}\in K_{1}(\Lambda(G)_{\Sigma^{*}})$ satisfying

$\partial_{G}(\xi_{M})=[M],$

where $[M]$ denotes the class of $M$ in $K_{0}(\Lambda(G))$.

We fix an algebraic closure $\overline{{\mathbb{Q}}}_{p}$ of
${\mathbb{Q}}_{p}$. Let $\rho:G\to{\mathrm{GL}}_{n}(\mathcal{O})$ be an Artin
representation, where $\mathcal{O}$ is the ring of integers of a finite
extension of ${\mathbb{Q}}_{p}$ in $\overline{{\mathbb{Q}}}_{p}$. We write
$\Lambda_{\mathcal{O}}(\Gamma)$ for the Iwasawa algebra of $\Gamma$ with
coefficients in $\mathcal{O}$, which we identify with the formal power series
ring $\mathcal{O}[[T]]$ in an indeterminate $T$ by sending our fixed
topological generator of $\Gamma$ to $1+T$. Let
$\mathcal{Q}_{\mathcal{O}}(\Gamma)$ denote the field of fractions of
$\Lambda_{\mathcal{O}}(\Gamma)$. If $M$ is a finitely generated
$\Lambda(G)$-module, we put
$M_{\mathcal{O}}=M\otimes_{{\mathbb{Z}}_{p}}\mathcal{O}$, and define

(7.1)
$\mathrm{tw}_{\rho}(M)=M_{\mathcal{O}}\otimes_{\mathcal{O}}\mathcal{O}^{n}$

endowed with the diagonal action of $G$, where $G$ acts on $\mathcal{O}^{n}$
via $\rho$.

The representation $\rho$ defines a continuous group homomorphism

$G\to(M_{n}(\mathcal{O})\otimes_{{\mathbb{Z}}_{p}}\Lambda(\Gamma))^{\times}$

which maps $g$ to $\rho(g)\otimes\bar{g}$, where $\bar{g}$ denotes the image
of $g$ in $\Gamma$. By [8, Lemma 3.3], this extends to a ring homomorphism

$\Lambda(G)_{\Sigma^{*}}\to M_{n}(\mathcal{Q}_{\mathcal{O}}(\Gamma)).$

From the functoriality of $K_{1}$-groups, this in turns induces a group
homomorphism

$K_{1}(\Lambda(G)_{\Sigma^{*}})\to
K_{1}(M_{n}(\mathcal{Q}_{\mathcal{O}}(\Gamma))).$

We then set $\Phi_{\rho}$ to be the composite map

(7.2) $\Phi_{\rho}:K_{1}(\Lambda(G)_{\Sigma^{*}})\to
K_{1}(M_{n}(\mathcal{Q}_{\mathcal{O}}(\Gamma)))\stackrel{{\scriptstyle\text{Morita}}}{{\simeq}}K_{1}(\mathcal{Q}_{\mathcal{O}}(\Gamma))\simeq\mathcal{Q}_{\mathcal{O}}(\Gamma)^{\times}.$

Here, the first isomorphism is given by Morita’s equivalence. We also identify
$\mathcal{Q}_{\mathcal{O}}(\Gamma)^{\times}$ with
$\mathcal{Q}_{\mathcal{O}}(T)^{\times}$, where $\mathcal{Q}_{\mathcal{O}}(T)$
denotes the field of fractions of the power series ring $\mathcal{O}[[T]]$.

######  Definition 7.2 (Burns [3]).

_Let $\xi\in K_{1}(\Lambda(G)_{\Sigma^{*}})$ and let $\rho$ be an Artin
representation of $G$. Then there exists an integer $r_{\rho}(\xi)$ and
$g(T)\in\mathcal{Q}_{\mathcal{O}}(T)^{\times}$ such that $g(0)\neq 0$ and_

$\Phi_{\rho}(\xi)=T^{r_{\rho}(\xi)}g(T).$

_We call $r_{\rho}(\xi)$ the_ order of vanishing of $\xi$ and denote by
${\mathrm{ord}}_{T=0}(\Phi_{\rho}(\xi))$.

We now relate the evaluation of the characteristic element of a module $M$ in
the category $\mathfrak{M}_{H}(G)$ at Artin representations of $G$ to the
Akashi series of the twists of $M$ by such representations defined by (7.1).

######  Lemma 7.3.

Let $M$ be a module in the category $\mathfrak{M}_{H}(G)$, and let $\xi_{M}\in
K_{1}(\Lambda(G)_{\Sigma^{*}})$ denote its characteristic element. Then we
have

$\Phi_{\rho}(\xi_{M})=\mathrm{Ak}(\mathrm{tw}_{\hat{\rho}}(M))\bmod\Lambda_{\mathcal{O}}(\Gamma)^{\times},$

where $\hat{\rho}$ denotes the contragradient representation of $\rho$, i.e.
$\hat{\rho}(g)=\rho(g^{-1})^{t}$ for $g\in G$, where $t$ denotes the transpose
matrix.

###### Proof.

By [8, Lemma 3.2], the module $\mathrm{tw}_{\hat{\rho}}(M)$ also lies in the
category $\mathfrak{M}_{H}(G)$. Thus,
$\mathrm{Ak}(\mathrm{tw}_{\hat{\rho}}(M))$ is well-defined, and the equality
in the lemma is [8, (57)]. ∎

Given an Artin representation $\rho:G\to GL_{n}(\mathcal{O})$ of $G$ with
coefficients in $\mathcal{O}$, let $F_{\rho}$ be the Galois extension of $F$
contained in $F_{\infty}$ such that
${\mathrm{Gal}}(F_{\infty}/F_{\rho})=\ker\rho$. We shall then write
$G_{\rho}={\mathrm{Gal}}(F_{\infty}/F_{\rho})$. In the case the regular
representation $\rho=\mathrm{reg}_{L}$ for a finite Galois extension $L$ of
$F$ contained in $F_{\infty}$, we simply write
$G_{L}={\mathrm{Gal}}(F_{\infty}/L)$ for $G_{\mathrm{reg}_{L}}$. We now
introduce two hypotheses which will be useful for the computations in this
section and in Section 9.

Hypothesis ($\mathbf{G}$). $H^{i}(G,A_{p^{\infty}}(F_{\infty}))$ is finite for
all $i\geq 1$.

Hypothesis ($\mathbf{H}$). $H^{i}(H,A_{p^{\infty}}(F_{\infty}))$ is finite for
all $i\geq 0$.

######  Remark 7.4.

Hypothesis $(\mathbf{G})$ is clearly satisfied if $A_{p^{\infty}}(F_{\infty})$
is finite. It also holds in the case when $F_{\infty}=F(A_{p^{\infty}})$ (see
[2, Section 3]). Note that Hypothesis $(\mathbf{H})$ implies Hypothesis
$(\mathbf{G})$.

Similarly, we write Hypothesis ($\mathbf{G_{\rho}}$) in case we replace $G$
with $G_{\rho}$, and Hypothesis $(\mathbf{G_{L}})$ in the case
$\rho=\mathrm{reg}_{L}$ for a finite Galois extension $L$ of $F$ contained in
$F_{\infty}$ with $G_{L}={\mathrm{Gal}}(F_{\infty}/L)$.

The next proposition provides a criterion for verifying Hypothesis
$(\mathbf{H})$.

######  Proposition 7.5.

Let $A$ be an abelian variety defined over a global function field $F$ with
$\mathrm{char}(F)\neq p$. Suppose that $A_{p^{\infty}}(F^{\mathrm{cyc}})$ is
finite and $H\simeq{\mathbb{Z}}_{p}^{r}$. Then Hypothesis $(\mathbf{H})$
holds, i.e., $H^{i}(H,A_{p^{\infty}}(F_{\infty}))$ is finite for all $i\geq
0$.

###### Proof.

The conclusion of the proposition is a consequence of the following algebraic
observation: If $H\cong{\mathbb{Z}}_{p}^{r}$, then for a given
$\Lambda(H)$-module $M$, one has $H_{i}(H,M)$ being finite for every $i\geq 1$
whenever $H_{0}(H,M)$ is finite. This latter assertion essentially follows
from [33, P. 56-57], although the discussion given there is rather terse. We
refer the reader to [22, Proof of Theorem 2.3] for a detailed proof of this
fact. ∎

We recall that, by Remark 4.3, the finiteness condition on
$A_{p^{\infty}}(F^{\mathrm{cyc}})$ automatically holds when $A$ is an elliptic
curve without complex multiplication.

We now prove the following proposition which illustrates the significance of
Hypothesis $(\mathbf{G})$.

######  Proposition 7.6.

Let $A$ be an abelian variety defined over a global field of characteristic
different from $p$. Suppose that Hypothesis $(\mathbf{G})$ holds. Then we have
${\mathrm{ord}}_{T=0}\left(f_{H^{i}(H,A_{p^{\infty}}(F_{\infty}))}\right)=0$
for all $i\geq 1$.

###### Proof.

To see that the characteristic series of $H^{i}(H,A_{p^{\infty}}(F_{\infty}))$
does not contribute a factor of $T$, it suffices to show that
$H^{i}(H,A_{p^{\infty}}(F_{\infty}))^{\Gamma}$ is finite. Since
$\mathrm{cd}_{p}\Gamma=1$, the spectral sequence

$H^{r}(\Gamma,H^{s}(H,A_{p^{\infty}}(F_{\infty})))\Rightarrow
H^{r+s}(G,A_{p^{\infty}}(F_{\infty}))$

degenerates yielding a short exact sequence

$0\to H^{1}(\Gamma,H^{i-1}(H,A_{p^{\infty}}(F_{\infty})))\to
H^{i}(G,A_{p^{\infty}}(F_{\infty}))\to
H^{i}(H,A_{p^{\infty}}(F_{\infty}))^{\Gamma}\to 0$

for every $i\geq 1$. Since $H^{i}(G,A_{p^{\infty}}(F_{\infty}))$ is finite for
every $i\geq 1$ by Hypothesis $(\mathbf{G})$, it follows that
$H^{i}(H,A_{p^{\infty}}(F_{\infty}))^{\Gamma}$ is also finite for every $i\geq
1$, as required. ∎

Now, if $L$ is a finite Galois extension of $F$ contained in $F_{\infty}$, we
write $\mathrm{reg}_{L}$ for the regular representation of
${\mathrm{Gal}}(L/F)$ and let $G_{L}$ denote the Galois group
${\mathrm{Gal}}(F_{\infty}/L)$.

######  Theorem 7.7.

Let $A$ be an abelian variety defined over a global function field $F$ with
$\mathrm{char}(F)\neq p$, and let $F_{\infty}/F$ be an admissible $p$-adic Lie
extension. Denote by $\xi_{A}=\xi_{X(A/F_{\infty})}$ the characteristic
element of $X(A/F_{\infty})$. Suppose that Hypothesis $(\mathbf{G_{L}})$
holds. Then we have

${\mathrm{ord}}_{T=0}(\Phi_{\mathrm{reg}_{L}}(\xi_{A}))\geq{\mathrm{corank}}_{{\mathbb{Z}}_{p}}({\mathrm{Sel}}(A/L)).$

In addition, equality holds when Greenberg’s semi-simplicity conjecture 6.2
holds for $A/L$.

###### Proof.

Set $H_{L}=G_{L}\cap H$ and $\Gamma_{L}={\mathrm{Gal}}(L^{\mathrm{cyc}}/L)$.
Fix an appropriate power of the generator of $\Gamma$ so that it is a
generator of $\Gamma_{L}$. With these choices of generators, we have

${\mathbb{Z}}_{p}[[T_{L}]]\simeq\Lambda(\Gamma_{L})\subseteq\Lambda(\Gamma)\simeq{\mathbb{Z}}_{p}[[T]].$

By the discussion in [21, Subsection 2.2 and Lemma 2.5], there is a
restriction map

$\mathrm{res}:K_{1}(\Lambda(G)_{\Sigma^{*}})\longrightarrow
K_{1}(\Lambda(G_{L})_{\Sigma_{G_{L}}^{*}})$

such that $\mathrm{res}(\xi_{A})$ is a characteristic element of
$X(A/F_{\infty})$ viewed in $\mathfrak{M}_{H_{L}}(G_{L})$. On the other hand,
[21, Proposition 2.10] tells us that

${\mathrm{ord}}_{T=0}(\Phi_{\mathrm{reg}_{L}}(\xi_{A}))={\mathrm{ord}}_{T_{L}=0}(\Phi_{\mathrm{reg}_{L},G_{L}}(\xi_{A})).$

Hence, in view of the above observations, it suffices to consider the case
$F=L$. It follows from Lemma 7.3 that

$\Phi_{\mathrm{reg}_{F}}(\xi_{A})=\mathrm{Ak}({\mathrm{Sel}}(A/F_{\infty}))u(T)$

for some $u(T)\in\Lambda(T)^{\times}$. Taking the hypothesis of the theorem
into account, we may apply Proposition 7.6 to conclude that
${\mathrm{ord}}_{T=0}\left(f_{H^{j}(H,A_{p^{\infty}}(F_{\infty}))}\right)=0$
for all $j\geq 1$. Combining this latter observation into Theorem 5.5, we
obtain

${\mathrm{ord}}_{T=0}\left(\mathrm{Ak}\left({\mathrm{Sel}}(A/F_{\infty})\right)\right)={\mathrm{ord}}_{T=0}\left(f_{{\mathrm{Sel}}(A/F^{\mathrm{cyc}})}\prod_{v\in
S^{\prime}}f_{J_{v}(A/F^{\mathrm{cyc}})}\right).$

By Proposition 6.3, the order of vanishing of
$f_{{\mathrm{Sel}}(A/F^{\mathrm{cyc}})}$ at $T=0$ is greater or equal to
${\mathrm{corank}}_{{\mathbb{Z}}_{p}}({\mathrm{Sel}}(A/F))$, where equality
holds if Conjecture 6.2 is valid. Hence it remains to show that the terms
$f_{J_{v}(A/F^{\mathrm{cyc}})}$ at $v\in S^{\prime}$ does not contribute to
the order of vanishing. For this, it suffices to show that
$J_{v}(A/F^{\mathrm{cyc}})^{\Gamma}$ is finite. Indeed, noting that
$\Gamma_{w}\simeq\Gamma\simeq{\mathbb{Z}}_{p}$ has $p$-cohomological dimension
one, the inflation–restriction sequence gives us a surjection

$H^{1}(F_{v},A_{p^{\infty}})\twoheadrightarrow
J_{v}(A/F^{\mathrm{cyc}})^{\Gamma}.$

By Tate duality (cf. [26, Chapter I, Remark 3.6]), we see that
$H^{1}(F_{v},A_{p^{\infty}})$ is dual to $A^{*}_{p^{\infty}}(F_{v})$, and the
latter is a finite group by Mattuck’s theorem for local fields. This completes
the proof of the theorem. ∎

We now present a corollary which relates the characteristic element to the
$L$-function of $A$ in the intermediate subfields of $F_{\infty}/F$. For a
finite extension $L$ of $F$ contained in $F_{\infty}$, we write $L_{S}(A/L,s)$
for the $L$-function of $A$ with Euler factors removed at the primes of $L$
lying above $S$. The following is our corollary.

######  Corollary 7.8.

Let $A$ be an abelian variety defined over a global function field $F$ with
$\mathrm{char}(F)\neq p$, and let $F_{\infty}/F$ be an admissible $p$-adic Lie
extension. Suppose that for a finite Galois extension $L$ of $F$ contained in
$F_{\infty}$ the following statements are valid.

  1. (1)

The Tate–Shafarevich group $\hbox{\cyr X}(A/L)$ is finite.

  2. (2)

Greenberg’s semi-simplicity conjecture 6.2 holds for $A/L$.

  3. (3)

Hypothesis $(\mathbf{G_{L}})$ holds.

Then we have

${\mathrm{ord}}_{T=0}(\Phi_{\mathrm{reg}_{L}}(\xi_{A}))={\mathrm{ord}}_{s=1}L_{S}(A/L,s).$

###### Proof.

Under the finiteness of $\hbox{\cyr X}(A/L)$, we have
${\mathrm{corank}}_{{\mathbb{Z}}_{p}}({\mathrm{Sel}}(A/L))={\mathrm{rank}}_{\mathbb{Z}}A(L)$.
On the other hand, under the finiteness of $\hbox{\cyr X}(A/L)$, the deep
works of Artin–Tate [34], Milne [25] and Kato–Trihan [18] show that
${\mathrm{rank}}_{\mathbb{Z}}A(L)={\mathrm{ord}}_{s=1}L_{S}(A/L,s)$. The
conclusion of the corollary then follows from combining these equalities with
Theorem 7.7. ∎

It is now natural to ask whether similar results can be obtained for Artin
representations which are not the regular representation. In order to do this,
we introduce the following notation. Let $\rho:G\to GL_{n}(\mathcal{O})$ be an
Artin representation of $G$ with coefficients in $\mathcal{O}$. Denote by
$W_{\rho}$ a free $\mathcal{O}$-module of rank $n$ realizing $\rho$. Let
$F_{\rho}$ be the Galois extension of $F$ contained in $F_{\infty}$ such that
${\mathrm{Gal}}(F_{\infty}/F_{\rho})=\ker\rho$. We shall then write
$G_{\rho}={\mathrm{Gal}}(F_{\infty}/F_{\rho})$.

######  Proposition 7.9.

Let $A$ be an abelian variety defined over a global field of characteristic
different from $p$. Suppose that Hypothesis $(\mathbf{G_{\rho}})$ holds. Then
we have
${\mathrm{ord}}_{T=0}\left(f_{H^{i}(H,\mathrm{tw}_{\rho}(A_{p^{\infty}})(F_{\infty}))}\right)=0$
for all $i\geq 1$.

###### Proof.

Let $H_{\rho}=G_{\rho}\cap H$. Consider the following spectral sequence

$H^{r}(H/H_{\rho},H^{s}(H_{\rho},\mathrm{tw}_{\rho}(A_{p^{\infty}})(F_{\infty}))\Rightarrow
H^{r+s}(H,\mathrm{tw}_{\rho}(A_{p^{\infty}})(F_{\infty})).$

Since $H/H_{\rho}$ is finite, it remains to show that the characteristic
series of $H^{i}(H_{\rho},\mathrm{tw}_{\rho}(A_{p^{\infty}})(F_{\infty}))$
does not contribute a factor of $T$. By the definition of $H_{\rho}$, the
group $H_{\rho}$ acts trivially on $W_{\rho}$, and so we have

(7.3)
$H^{i}(H_{\rho},\mathrm{tw}_{\rho}(A_{p^{\infty}})(F_{\infty}))=H^{i}(H_{\rho},A_{p^{\infty}}(F_{\infty}))\otimes_{\mathcal{O}}W_{\rho}.$

Since Proposition 7.6 tells us that the characteristic series of
$H^{i}(H_{\rho},A_{p^{\infty}}(F_{\infty}))$ does not contribute a factor of
$T$, we see from (7.3) that the same conclusion holds for
$H^{i}(H_{\rho},\mathrm{tw}_{\rho}(A_{p^{\infty}})(F_{\infty}))$. ∎

Given any infinite separable extension $K$ of $F$ contained in $F_{S}$, we can
define the twisted Selmer group ${\mathrm{Sel}}(\mathrm{tw}_{\rho}(A)/K)$ to
be

${\mathrm{Sel}}(\mathrm{tw}_{\rho}(A)/K)=\ker\left(H^{1}(G_{S}(K),\mathrm{tw}_{\rho}(A_{p^{\infty}}))\longrightarrow\bigoplus_{v\in
S}J_{v}(\mathrm{tw}_{\rho}(A)/K)\right).$

Here, for each $v\in S$,

$J_{v}(\mathrm{tw}_{\rho}(A)/K)=\varinjlim_{L}\bigoplus_{w\mid
v}H^{1}(L_{w},\mathrm{tw}_{\rho}(A_{p^{\infty}})),$

where $L$ runs over all finite extensions of $F$ contained in $K$, $w$ runs
over all primes of $L$ lying above $v$ and the limit is taken with respect to
the restriction maps. We shall write $X(\mathrm{tw}_{\rho}(A)/K)$ for the
Pontryagin dual of ${\mathrm{Sel}}(\mathrm{tw}_{\rho}(A)/K)$.

######  Lemma 7.10.

We have the following short exact sequence

$0\longrightarrow{\mathrm{Sel}}(\mathrm{tw}_{\rho}(A)/F_{\infty})\longrightarrow
H^{1}(G_{S}(F_{\infty}),\mathrm{tw}_{\rho}(A_{p^{\infty}}))\longrightarrow\bigoplus_{v\in
S}J_{v}(\mathrm{tw}_{\rho}(A)/F_{\infty})\longrightarrow 0$

of cofinitely generated $\Lambda_{\mathcal{O}}(H)$-modules.

###### Proof.

By the definition of $\rho$, we see that $G_{S}(F_{\infty})$ acts trivially on
$W_{\rho}$, and so we have
${\mathrm{Sel}}(\mathrm{tw}_{\rho}(A)/F_{\infty})={\mathrm{Sel}}(A/F_{\infty})\otimes_{\mathcal{O}}W_{\rho}$.
Since ${\mathrm{Sel}}(A/F_{\infty})$ is cofinitely generated over $\Lambda(H)$
by Lemma 3.3, it follows that
${\mathrm{Sel}}(\mathrm{tw}_{\rho}(A)/F_{\infty})$ is cofinitely generated
over $\Lambda_{\mathcal{O}}(H)$. Via similar arguments, we see that
$H^{1}(G_{S}(F_{\infty}),\mathrm{tw}_{\rho}(A_{p^{\infty}}))$ and
$\bigoplus_{v\in S}J_{v}(\mathrm{tw}_{\rho}(A)/F_{\infty})$ are cofinitely
generated over $\Lambda_{\mathcal{O}}(H)$. Since $W_{\rho}$ is free over
$\mathcal{O}$, the short exact sequence of the lemma can be obtained from
Theorem 4.2 by tensoring the short exact sequence in the said theorem with
$W_{\rho}$. ∎

By a similar argument to that in [10, Proposition 2.5], the restriction map

${\mathrm{Sel}}(\mathrm{tw}_{\rho}(A)/F^{\mathrm{cyc}})\rightarrow{\mathrm{Sel}}(\mathrm{tw}_{\rho}(A)/F_{\infty})^{H}$

has kernel which is cofinitely generated over $\mathcal{O}$. It then follows
from this and the preceding lemma that
${\mathrm{Sel}}(\mathrm{tw}_{\rho}(A)/F^{\mathrm{cyc}})$ is cofinitely
generated over $\mathcal{O}$. Therefore, we may apply the structure theory for
$\Lambda_{\mathcal{O}}(\Gamma)$-modules to obtain the following pseudo-
isomorphism

(7.4)
$X(\mathrm{tw}_{\rho}(A)/F^{\mathrm{cyc}})\sim\bigoplus_{j=1}^{t}\Lambda_{\mathcal{O}}(\Gamma)/(f_{j}^{\beta_{j}})$

where the $f_{j}$’s are irreducible distinguished polynomials.

We can now state the semi-simplicity conjecture of Greenberg for Artin twists
of abelian varieties over a global function field.

######  Conjecture 7.11 (Greenberg’s semi-simplicity conjecture for
$\mathrm{tw}_{\rho}(A)/F$).

Let $A$ be an abelian variety defined over a global function field $F$. Then
the action of $\Gamma$ on $X(\mathrm{tw}_{\rho}(A)/F^{\mathrm{cyc}})$ is
completely reducible, that is, $\beta_{j}=1$ for all $j$ in (7.4).

We are now in position to present the following result on the order of
vanishing of characteristic elements evaluated at Artin representations.

######  Theorem 7.12.

Let $A$ be an abelian variety defined over a global function field $F$ with
$\mathrm{char}(F)\neq p$, and let $F_{\infty}/F$ be an admissible $p$-adic Lie
extension. Let $\rho$ be an irreducible Artin representation of
$G={\mathrm{Gal}}(F_{\infty}/F)$ and let $\xi_{A}$ denote a characteristic
element of $X(A/F_{\infty})$. Suppose that Hypothesis $(\mathbf{G_{\rho}})$
holds. Then we have

${\mathrm{ord}}_{T=0}(\Phi_{\rho}(\xi_{A}))\geq{\mathrm{corank}}_{\mathcal{O}}({\mathrm{Sel}}(\mathrm{tw}_{\rho}(A)/F)).$

In addition, equality holds when Greenberg’s semi-simplicity conjecture 7.11
holds for $\mathrm{tw}_{\rho}(A)/F$.

###### Proof.

The arguments used to prove Theorem 7.7 carry over for the Artin twists,
taking Proposition 7.9 and Lemma 7.10 into account, and noting that
$X(\mathrm{tw}_{\rho}(A)/F_{\infty})=\mathrm{tw}_{\hat{\rho}}X(A/F_{\infty})$
(cf. [7, Lemma 3.4]). ∎

##  8\. The Generalised Euler characteristics

In this section, we give the definition of the generalised Euler
characteristic. The idea of defining a finite Euler characteristic $\chi(G,Y)$
for a discrete $p$-primary $G$-module $Y$ when the cohomology groups
$H^{i}(G,Y)$ are infinite was first conceived by Coates–Schneider–Sujatha in
the form of the truncated Euler characteristic [9]. Our presentation here will
follow that of Zerbes [40].

Let $W$ be a discrete $p$-primary $\Gamma$-module. Then we have

$H^{0}(\Gamma,W)=W^{\Gamma}\quad H^{1}(\Gamma,W)=W_{\Gamma},$

where $W_{\Gamma}$ is the maximal quotient of $W$ on which $\Gamma$ acts
trivially. Hence, we have a map

$\phi_{W}:H^{0}(\Gamma,W)\to H^{1}(\Gamma,W),\quad f\mapsto\textrm{ residue
class of }f.$

For a discrete $p$-primary $G$-module $Y$, we define

$d^{0}:H^{0}(G,Y)=H^{0}(\Gamma,Y^{H})\longrightarrow
H^{1}(\Gamma,Y^{H})\hookrightarrow H^{1}(G,Y),$

where the middle map is $\phi_{Y^{H}}$ and the last map is given by inflation.
Similarly, for $i\geq 1$, define

$d^{i}:H^{i}(G,Y)\twoheadrightarrow H^{0}(\Gamma,H^{i}(H,Y))\longrightarrow
H^{1}(\Gamma,H^{i}(H,Y))\hookrightarrow H^{i+1}(G,Y),$

where the first map is given by restriction and the middle map is equal to
$\phi_{H^{i}(H,Y)}$. Note that the first map is surjective since ${\rm
cd}_{p}(\Gamma)=1$. We define $d^{-1}$ to be the zero map.

All the above properties for the morphisms involving in the definition of the
complex $(H^{i}(G,Y),d^{i})$ will become clear in the proof of the following
proposition.

######  Proposition 8.1.

The sequence $(H^{\bullet}(G,Y),d^{\bullet})$ forms a cochain complex.

###### Proof.

We have the Hochschild–Serre spectral sequence is given by

$H^{r}(\Gamma,H^{s}(H,Y))\Rightarrow H^{r+s}(G,Y).$

Since ${\rm cd}_{p}(\Gamma)=1$, we have $H^{r}(\Gamma,H^{s}(H,Y))=0$ unless
$r=0$ or $1$. Thus, there exists a short exact sequence

(8.1) $0\to H^{1}(\Gamma,H^{s-1}(H,Y))\to H^{s}(G,Y)\to
H^{0}(\Gamma,H^{s}(H,Y))\to 0$

for all $s\geq 1$. Now, we see that $d^{s}:H^{s}(G,Y)\to H^{s+1}(G,Y)$ is
equal to the composition

$H^{s}(G,Y)\twoheadrightarrow
H^{0}(\Gamma,H^{s}(H,Y))\stackrel{{\scriptstyle\phi_{H^{s}(H,X)}}}{{\longrightarrow}}H^{1}(\Gamma,H^{s}(H,Y))\hookrightarrow
H^{s+1}(G,Y),$

where the surjective map and injective map are exactly given by the short
exact sequence (8.1). Therefore, $d^{s+1}\circ d^{s}$ must factor through the
following composition

$H^{1}(\Gamma,H^{s}(H,Y))\to H^{s+1}(G,Y)\to H^{0}(\Gamma,H^{s+1}(H,Y)),$

which is the exact sequence (8.1), and so must be the zero map. Hence, we have
$d^{s+1}\circ d^{s}=0$, as desired. ∎

######  Definition 8.2.

Denote by ${\mathfrak{H}}^{\bullet}$ the cohomology of the complex in
Proposition 8.1. If ${\mathfrak{H}}^{i}$ is finite for all $i\geq 0$, we say
that the $G$-module $Y$ has finite generalised $G$-Euler characteristic
$\chi(G,Y)$, defined by

$\chi(G,Y)=\prod_{i\geq 0}\\#({\mathfrak{H}}^{i})^{(-1)^{i}}.$

######  Lemma 8.3.

The generalised $G$-Euler characteristic satisfies the following properties:

  1. (1)

If the groups $H^{i}(G,Y)$ are finite for all $i$, the generalised $G$-Euler
characteristic for $Y$ agrees with the usual $G$-Euler characteristic for $Y$.

  2. (2)

If $H^{0}(G,Y)$ and $H^{1}(G,Y)$ are infinite and $H^{i}(G,Y)$ is finite for
all $i\geq 2$. The module $Y$ has finite generalised $G$-Euler characteristic
if and only if the map $d^{0}$ has finite kernel and cokernel, in which case

(8.2)
$\chi(G,Y)=\frac{\\#(\ker(d^{0}))}{\\#(\mathrm{coker}~{}\\!(d^{0}))}\times\prod_{i\geq
2}\\#\left(H^{i}(G,Y)\right)^{(-1)^{i}}.$

###### Proof.

This is essentially established in the work of Zerbes [40] but not explicitly
stated in the form here. For the convenience of the readers, we shall supply a
proof of the lemma. For the first assertion, it suffices to show that for a
cochain complex $({\mathfrak{X}}^{\bullet},\delta^{\bullet})$ with
${\mathfrak{X}}^{i}$ finite for all $i\geq 0$, we have

$\prod^{\infty}_{i=0}\\#(H^{i}({\mathfrak{X}}^{\cdot}))^{(-1)^{i}}=\prod^{\infty}_{i=0}\\#({\mathfrak{X}}^{i})^{(-1)^{i}}.$

To see this, consider the following two short exact sequences

$0\to Z^{i}\to{\mathfrak{X}}^{i}\to B^{i+1}\to 0,$ $0\to B^{i}\to Z^{i}\to
H^{i}({\mathfrak{X}}^{\cdot})\to 0.$

As all modules in the sequences are finite, we have

$\\#({\mathfrak{X}}^{i})=\\#(Z^{i})\cdot\\#(B^{i+1}).$

Therefore, noting that $\\#(B^{0})=1$, we have

$\prod_{i\geq 0}\\#({\mathfrak{X}}^{i})^{(-1)^{i}}=\prod_{i\geq
0}(\\#(Z^{i})\cdot\\#(B^{i+1}))^{(-1)^{i}}=\prod_{i\geq
0}\\#(H^{i}({\mathfrak{X}}^{\cdot}))^{(-1)^{i}},$

and (1) follows. To prove the second assertion, we first note that
$\ker(d^{0})={\mathfrak{H}}^{0}$. Thus, if $d^{0}$ has finite kernel and
cokernel, then ${\mathfrak{H}}^{0}$ is finite. Now, from the exact sequence

$0\to{\mathfrak{H}}^{1}\to\mathrm{coker}~{}\\!d^{0}\to H^{2}(G,Y),$

where the third map is induced by $d^{1}$, we see that ${\mathfrak{H}}^{1}$ is
also finite. As ${\mathfrak{H}}^{i}$ are finite for $i\geq 2$ by assumption,
$Y$ has finite generalised $G$-Euler characteristic. Conversely, suppose that
$Y$ has finite generalised $G$-Euler characteristic. Since
$\ker(d^{0})={\mathfrak{H}}^{0}$, we see that $d^{0}$ has finite kernel. Since
$H^{2}(G,Y)$ is finite, $\ker(d^{2})$ is finite. Now, considering the exact
sequence

$0\to{\mathfrak{H}}^{1}\to\mathrm{coker}~{}\\!d^{0}\to\ker
d^{2}\to{\mathfrak{H}}^{2}\to 0,$

we see that $\mathrm{coker}~{}\\!(d^{0})$ is finite. It remains to show the
formula (8.2). The complex $(H^{\bullet}(G,Y),d^{\bullet})$ induces the
following cochain complex

$0\to\mathrm{coker}~{}\\!(d^{0})\stackrel{{\scriptstyle\bar{d}^{1}}}{{\longrightarrow}}H^{2}(G,Y)\stackrel{{\scriptstyle
d^{2}}}{{\longrightarrow}}\cdots.$

Since all terms in the cochain complex are finite, noting that
$\ker(\bar{d}^{1})\simeq{\mathfrak{H}}_{1}$, we see from (1) that

(8.3)
$\prod^{\infty}_{i=1}\\#({\mathfrak{H}}^{i})^{(-1)^{i+1}}=\\#(\mathrm{coker}~{}\\!(d^{0}))\times\prod^{\infty}_{i=2}\\#\left(H^{i}(G,Y)\right)^{(-1)^{i+1}}.$

Then noting that $\\#({\mathfrak{H}}^{0})=\\#(\ker(d^{0}))$ and dividing both
sides by the equality (8.3), we arrive at the following equality

$\chi(G,Y)=\frac{\\#(\ker(d^{0}))}{\\#(\mathrm{coker}~{}\\!(d^{0}))}\times\prod_{i\geq
2}\\#\left(H^{i}(G,Y)\right)^{(-1)^{i}},$

which is (8.2), as required. ∎

We now record the following important result which relates the generalised
Euler characteristic to Akashi series.

######  Proposition 8.4.

Let $Y$ be a discrete $p$-primary $G$-module such that
$Y^{\vee}\in{\mathfrak{M}}_{H}(G)$. If $Y$ has finite generalized $G$-Euler
characteristic, then the leading term of its Akashi series $\mathrm{Ak}(Y)$ is
$\alpha_{Y}T^{k}$, where $k$ is equal to the alternating sum

$k=\sum_{i\geq 0}(-1)^{i}{\rm corank}_{{\mathbb{Z}}_{p}}H^{i}(H,Y)^{\Gamma}$

and

$\chi(G,Y)=|\alpha_{Y}|^{-1}_{p}.$

###### Proof.

See [40, Proposition 2.10]. ∎

We note that it is in general not true that the generalised $G$-Euler
characteristic is multiplicative in short exact sequences (see [40, Remark
after Lemma 2.13]). However, the following lemma suffices for most of our
purposes.

######  Lemma 8.5.

Let $0\to B_{1}\to B_{2}\to B_{3}\to B_{4}\to 0$ be an exact sequence of
discrete $G$-modules with the property that $B_{1}$ and $B_{4}$ have finite
$G$-Euler characteristics. Then $B_{2}$ has finite generalised $G$-Euler
characteristic if and only if $B_{3}$ does.

###### Proof.

Set $C=\ker(B_{3}\to B_{4})$. By separately considering the following two
short exact sequences

$0\to B_{1}\to B_{2}\to C\to 0$ $0\to C\to B_{3}\to B_{4}\to 0,$

we may assume that either $B_{1}=0$ or $B_{4}=0$. We shall prove the case when
$B_{4}=0$, the case of $B_{1}=0$ being similar. Thus, we may suppose that we
have a short exact sequence of $G$-modules

$0\to B_{1}\to B_{2}\to B_{3}\to 0$

with $B_{1}$ having finite $G$-Euler characteristics. In particular, we have
that $H^{i}(G,B_{1})$ is finite for every $i\geq 0$. Taking the long exact
sequence of $G$-cohomology of the short exact sequence above, we see that

$\psi_{i}:H^{i}(G,B_{2})\to H^{i}(G,B_{3})$

have finite kernel and cokernel for all $i\geq 0$. By the naturalness of the
Hochschild–Serre spectral sequence and our definition of $d^{i}$, we see that
$\psi_{i}$ are cochain maps between the complexes
$(H^{\bullet}(G,B_{2}),d^{\bullet}_{B_{2}})$ and
$(H^{\bullet}(G,B_{3}),d^{\bullet}_{B_{3}})$. Since the kernels and cokernels
of these cochain map are finite, a direct homological algebraic argument shows
that the induced maps on their cohomology also have finite kernel and
cokernel. Hence it follows that $B_{2}$ has finite generalised Euler
characteristic if and only if $B_{3}$ does. ∎

######  Remark 8.6.

One can even show that

$\chi(G,B_{2})\chi(G,B_{4})=\chi(G,B_{1})\chi(G,B_{3})$

in the preceding lemma. As this latter calculation is not required for the
paper, we do not provide a detailed proof, leaving it to the interested
readers.

##  9\. Computations of the Generalised Euler Characteristics

In order to compute the Generalised Euler characteristics, we impose in this
section Hypothesis $(\mathbf{H})$ on the abelian variety $A$ defined over a
function field $F$, which we now recall:

Hypothesis $(\mathbf{H})$. $H^{i}(H,A_{p^{\infty}}(F_{\infty}))$ is finite for
all $i\geq 0$.

######  Theorem 9.1.

Let $A$ be an abelian variety defined over a global field $F$ with
$\mathrm{char}F\neq p$ and let $F_{\infty}$ be an admissible $p$-adic Lie
extension of $F$. Suppose that Hypothesis $(\mathbf{H})$ holds. Then
${\mathrm{Sel}}(A/F_{\infty})$ has finite generalised $G$-Euler characteristic
if and only if ${\mathrm{Sel}}(A/F^{\rm cyc})$ has finite generalised
$\Gamma$-Euler characteristics. In the event of such, we have the equality

$\chi(G,{\mathrm{Sel}}(A/F_{\infty}))=\chi(\Gamma,{\mathrm{Sel}}(A/F^{\mathrm{cyc}}))\prod_{v\in
S^{\prime}}\frac{\\#A^{*}_{p^{\infty}}(F_{v})}{\\#H^{1}(\Gamma_{w},A_{p^{\infty}}(F^{\mathrm{cyc}}_{w}))},$

where, for every $v\in S^{\prime}$, $w$ is a fixed prime of $F^{\mathrm{cyc}}$
above $v$.

###### Proof.

Consider the following fundamental commutative diagram

(9.1)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{Sel}}(A/F^{\rm
cyc})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{H^{1}(G_{S}(F^{\rm
cyc}),A_{p^{\infty}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{\bigoplus_{v\in
S}J_{v}(A/F^{\rm
cyc})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{Sel}}(A/F_{\infty})^{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(G_{S}(F_{\infty}),A_{p^{\infty}})^{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\bigoplus_{v\in
S}J_{v}(A/F_{\infty})^{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(H,{\mathrm{Sel}}(A/F_{\infty}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

From the proof of Lemma 5.4, we see that the $p$-cohomological dimension of
the decomposition group of $H$ at every prime of $F_{\infty}$ is at most one.
It follows from this that the map $\gamma$ is surjective which in turn implies
the same for $\phi$. The proof of the same lemma also gives
$H^{i}(H,J_{v}(A/F_{\infty}))=0$ for every $i\geq 1$. Applying the snake lemma
to the diagram (9.1), we obtain the following exact sequence

(9.2) $0\to\ker\alpha\to
H^{1}(H,A_{p^{\infty}}(F_{\infty}))\to\ker\gamma\to\mathrm{coker}~{}\\!\alpha\to
H^{2}(H,A_{p^{\infty}}(F_{\infty}))\to 0$

and

$H^{i}(H,{\mathrm{Sel}}(A/F_{\infty}))\simeq
H^{i+2}(H,A_{p^{\infty}}(F_{\infty}))$

for every $i\geq 1$. In view of this latter identification and Hypothesis
$(\mathbf{H})$, we see that $H^{i}(H,{\mathrm{Sel}}(A/F_{\infty}))$ is finite
for ech $i\geq 1$. It then follows from Lemma 8.3 that
${\mathrm{Sel}}(A/F_{\infty})$ has finite generalised $G$-Euler characteristic
if and only if the map

$d^{0}:H^{0}(G,{\mathrm{Sel}}(A/F_{\infty}))=H^{0}(\Gamma,{\mathrm{Sel}}(A/F_{\infty})^{H})\stackrel{{\scriptstyle
g}}{{\to}}H^{1}(\Gamma,{\mathrm{Sel}}(A/F_{\infty})^{H})\stackrel{{\scriptstyle
h}}{{\hookrightarrow}}H^{1}(G,{\mathrm{Sel}}(A/F_{\infty}))$

has finite kernel and cokernel. Since
$\mathrm{coker}~{}\\!h=H^{1}(H,{\mathrm{Sel}}(A/F_{\infty}))^{\Gamma}\simeq
H^{3}(H,A_{p^{\infty}}(F_{\infty}))^{\Gamma}$ is also finite, we see that the
map $d_{0}$ has finite kernel and cokernel if and only if $g$ does. The latter
is precisely equivalent to saying that ${\mathrm{Sel}}(A/F_{\infty})^{H}$ has
finite generalised $\Gamma$-Euler characteristics.

On the other hand, from diagram (9.1), we have the following exact sequence

(9.3) $0\to\ker\alpha\to{\mathrm{Sel}}(A/F^{\rm
cyc})\to{\mathrm{Sel}}(A/F_{\infty})^{H}\to\mathrm{coker}~{}\\!\alpha\to 0.$

Since $\ker\alpha\subseteq H^{1}(H,A_{p^{\infty}}(F_{\infty}))$, we see that
$\ker\alpha$ is finite. We claim that $\mathrm{coker}~{}\\!\alpha$ has finite
$\Gamma$-Euler characteristic. Suppose for now the claim holds. Then it
follows from Lemma 8.5 and (9.3) that ${\mathrm{Sel}}(A/F_{\infty})^{H}$ has
finite generalised $\Gamma$-Euler characteristics if and only if
${\mathrm{Sel}}(A/F^{\mathrm{cyc}})$ has finite generalised $\Gamma$-Euler
characteristics, establishing the first assertion of the theorem. It therefore
remains to prove our claim. Taking Hypothesis $(\mathbf{H})$ into account, the
exact sequence (9.2) reduces our task to showing that $\ker\gamma$ has finite
$\Gamma$-Euler characteristic. For this, we note that as observed in the proof
of Theorem 7.7, there is a surjection

$H^{1}(F_{v},A_{p^{\infty}})\twoheadrightarrow
J_{v}(A/F^{\mathrm{cyc}})^{\Gamma}.$

with $H^{1}(F_{v},A_{p^{\infty}})$ being finite. In particular,
$J_{v}(A/F^{\mathrm{cyc}})^{\Gamma}$ is finite, and this in turn implies that
$J_{v}(A/F^{\mathrm{cyc}})$ has a finite $\Gamma$-Euler characteristic.

From now on, we suppose that both generalised Euler characteristics in
question are well-defined. Then we may apply Theorem 5.5 and Proposition 8.4
to conclude that

(9.4)
$\chi(G,{\mathrm{Sel}}(A/F_{\infty}))=\chi(\Gamma,{\mathrm{Sel}}(A/F^{\mathrm{cyc}}))\prod_{v\in
S^{\prime}}\chi(\Gamma,J_{v}(A/F^{\mathrm{cyc}})).$

Therefore, the final assertion of the theorem will follow once we can show
that

(9.5)
$\chi(\Gamma,J_{v}(A/F^{\mathrm{cyc}}))=\frac{\\#A^{*}_{p^{\infty}}(F_{v})}{\\#H^{1}(\Gamma_{w},A_{p^{\infty}}(F^{{\mathrm{cyc}}}_{w}))}$

for each $v$. The restriction map yields a short exact sequence

$0\to H^{1}(\Gamma_{w},A_{p^{\infty}}(F^{{\mathrm{cyc}}}_{w}))\to
H^{1}(F_{v},A_{p^{\infty}})\to J_{v}(A/F^{\mathrm{cyc}})^{\Gamma}\to 0,$

where Tate duality tells us that $H^{1}(F_{v},A_{p^{\infty}})$ is isomorphic
to the Pontryagin dual of $A^{*}_{p^{\infty}}(F_{v})$. Thus, we have

(9.6)
$\\#J_{v}(A/F^{\mathrm{cyc}})^{\Gamma}=\frac{\\#A^{*}_{p^{\infty}}(F_{v})}{\\#H^{1}(\Gamma_{w},A_{p^{\infty}}(F^{{\mathrm{cyc}}}_{w}))}.$

On the other hand, we have

(9.7) $H^{1}\left(\Gamma,J_{v}(A/F^{\mathrm{cyc}})\right)\simeq
H^{1}(\Gamma_{w},H^{1}(F^{\mathrm{cyc}}_{w},A_{p^{\infty}}))$

by Shapiro’s lemma. Now, consider the following spectral sequence

$H^{r}(\Gamma_{w},H^{s}(F^{\mathrm{cyc}}_{w},A_{p^{\infty}}))\Rightarrow
H^{r+s}(F_{v},A_{p^{\infty}}).$

Since $\mathrm{cd}_{p}~{}\Gamma=1$,
$H^{2}(F^{\mathrm{cyc}}_{w},A_{p^{\infty}})=0$ (cf. [28, Theorem 7.1.8(i)])
and $H^{2}(F_{v},A_{p^{\infty}})=0$ (see [11, Lemma 1.12]), the spectral
sequence degenerates yielding

(9.8) $H^{1}(\Gamma_{w},H^{1}(F^{\mathrm{cyc}}_{w},A_{p^{\infty}}))=0.$

Combining (9.6), (9.7) and (9.8), we obtain (9.5), as required. This completes
the proof of our theorem. ∎

######  Remark 9.2.

When $A=E$ is an elliptic curve, it follows from well-known properties of
elliptic curves that

$\chi(\Gamma,J_{v}(E/F^{\rm cyc}))=|L_{v}(E,1)|_{p}$

where $L_{v}(E,1)$ denotes the Euler factor at $v$ of the complex $L$-function
of $E$ over $F$, and $|\cdot|_{p}$ is the $p$-adic valuation of
${\mathbb{Q}}$, normalised so that $|p|_{p}=p^{-1}$. This recovers the formula
of Zerbes in the number field setting and a formula of Sechi using the
standard Euler characteristics in the function field setting.

######  Corollary 9.3.

Retain the settings of Theorem 9.1. Suppose further that Greenberg’s
conjecture semi-simplicity conjecture 6.2 holds for $A/F$. Set
$s_{A}=\mathrm{corank}_{{\mathbb{Z}}_{p}}{\mathrm{Sel}}(A/F)$. We then have

$\frac{1}{T^{s_{A}}}\Phi_{\mathrm{reg}_{F}}(\xi_{A})\Big{|}_{T=0}=\chi(\Gamma,{\mathrm{Sel}}(A/F^{\mathrm{cyc}}))\prod_{v\in
S^{\prime}}\frac{\\#A^{*}_{p^{\infty}}(F_{v})}{\\#H^{1}(\Gamma_{w},A_{p^{\infty}}(F^{\mathrm{cyc}}_{w}))}.$

where, for every $v\in S^{\prime}$, $w$ is a fixed prime of $F^{\mathrm{cyc}}$
above $v$.

###### Proof.

This follows from a combination of Theorems 7.7 and 9.1. ∎

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