# Characteristic ideal of the fine Selmer group and results on
$\mu$-invariance under isogeny in the function field case.

Sohan Ghosh  Harish Chandra Research Institute, A CI of Homi Bhabha National
Institute, Chhatnag Road, Jhunsi, Prayagraj (Allahabad) 211 019 India
<EMAIL_ADDRESS>and  Jishnu Ray  Harish Chandra Research Institute, A CI
of Homi Bhabha National Institute, Chhatnag Road, Jhunsi, Prayagraj
(Allahabad) 211 019 India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Consider a function field $K$ with characteristic $p>0$. We investigate the
$\Lambda$-module structure of the Mordell-Weil group of an abelian variety
over $\mathbb{Z}_{p}$-extensions of $K$, generalizing results due to Lee.
Next, we study the algebraic structure and prove a control theorem for the
$S$-fine Mordell-Weil groups, the function field analogue for Wuthrich’s fine
Mordell–Weil groups, over a $\mathbb{Z}_{p}$-extension of $K$. In case of the
unramified $\mathbb{Z}_{p}$-extension, $K_{\infty}$, we compute the
characteristic ideal of the Pontryagin dual of the $S$-fine Mordell–Weil
group. This provides an answer to an analogue of Greenberg’s question for the
characteristic ideal of the dual fine Selmer group in the function field
setup. In the $\ell\neq p$ case, we prove the triviality of the
$\mu$-invariant for the Selmer group (same as the fine Selmer group in this
case) of an elliptic curve over a non-commutative
$GL_{2}(\mathbb{Z}_{\ell})$-extension of $K$ and thus extending Conjecture A.
In the $\ell=p$ case, we compute the change of $\mu$-invariants of the dual
Selmer groups of elliptic curves under isogeny, giving a lower bound for the
$\mu$-invariant.

###### Key words and phrases:

fine Selmer group, fine Mordell-Weil group, Iwasawa invariants, isogeny,
elliptic curves over function fields

###### 2020 Mathematics Subject Classification:

Primary: 11R23, Secondary: 11G05, 11G10, 11R58, 11S25

##  1\. Introduction

This article is divided into two parts, namely Part I and Part II dealing with
different, yet connected, topics.

###  1.1. Part I: Function Field ($\ell=p$ case)

Let $K=\mathbb{F}(t)$ be the function field over the finite field $\mathbb{F}$
of characteristic $p$, of cardinality $p^{r}$ for some $r>0$ and let
$K_{\infty}$ be the arithmetic $\mathbb{Z}_{p}$-extension of $K$ (defined in
§2.2). Let $\Lambda$ be the Iwasawa algebra of $K_{\infty}$ over $K$. In this
case, the Iwasawa main conjecture over $K_{\infty}$ for semistable abelian
varieties $A$ defined over $K$, have been settled by works of [LLTT16].

Let $S$ be a finite set of primes of $K$ that contains the set of all primes
of bad reduction of $A/K$. The $S$-fine Selmer group $R^{S}(E/K_{\infty})$
(for definition see §2.1) is a subgroup of the classical Selmer group, which
is always co-torsion as a $\Lambda$-module (cf. [OT09, Theorem 1.7]).

Therefore, a natural question is to find the characteristic ideal of
$R^{S}(E/K_{\infty})$. The first part of this article addresses this question.

The strategy of the proof is to define the $S$-fine Mordell-Weil group,
$\mathcal{M}^{S}(E/L)$, for an algebraic extension $L$ of $K$, following
[Wut07] (cf. §2.3). The fine Selmer groups fits into the following short exact
sequence:

$0\longrightarrow\mathcal{M}^{S}(A/L)\longrightarrow
R^{S}(A/L)\longrightarrow{}^{S}(A/L)\longrightarrow 0,$

where ${}^{S}(A/L)$ denotes the $S$-fine Tate-Shafarevich group (defined in
§2.3). Note that the ${}^{S}(A/L)$ is the subgroup of the
$\Sha(A/L)[p^{\infty}]$, the $p$-primary part of the Tate-Shafarevich group
(see Remark 2.7).

For each $n\geqslant 1$, let
$\Phi_{n}=\frac{(1+T)^{p^{n}}-1}{(1+T)^{p^{n-1}}-1}\in\Lambda$ be the
$p^{n}$-th cyclotomic polynomial in $1+T$. Let $K_{n}$ be the subextension of
$K_{\infty}$ such that $[K_{n}:K]=p^{n}$. One of the main result in the first
part of our article is the following.

######  Theorem 1.1.

(Theorem 4.9) Let $L$ be a finite extension of $K$. Let $L_{\infty}/L$ be the
arithmetic $\mathbb{Z}_{p}$-extension. Assume that $E/L$ has split
multiplicative reduction or good reduction at all primes of $L$ and
$\Sha(E/L_{n})[p^{\infty}]$ is finite for all $n\geqslant 0$.

Also, suppose that $E$ has split multiplicative reduction at all primes of $S$
and that ${}^{S}(E/L_{\infty})$ is finite. Then,

$\operatorname{Char}_{\Lambda}(R^{S}(E/L_{\infty})^{\vee})=\left(\underset{e_{n}\geqslant
1,n\geqslant 0}{\prod}\Phi_{n}^{e_{n}}\right)$

where
$e_{n}=\dfrac{\operatorname{rank}E(L_{n})-\operatorname{rank}E(L_{n-1})}{{\varphi}(p^{n})}$.

This theorem has the following applications.

First, under the hypothesis that ${}^{S}(E/K_{\infty})$ is finite one can see
easily that the $\mu$-invariant of the $S$-fine Selmer group is trivial
confirming the validity of Conjecture A in the function field case (cf.
[GJS22, Theorem 3.7]).

Secondly, one can deduce an algebraic functional equation of
$R^{S}(E/K_{\infty})$. More precisely, one obtains the following result.

######  Corollary 1.2.

Assume that the hypotheses in Theorem 1.1 hold. Then the characteristic ideals
of $R^{S}(E/K_{\infty})^{\vee}$ and $R^{S}(E/K_{\infty})^{\vee,\iota}$ as
$\Lambda$-modules are equal, i.e. there is a pseudo-isomorphism
$R^{S}(E/K_{\infty})^{\vee}\sim R^{S}(E/K_{\infty})^{\vee,\iota}.$

Here $\iota$ is the involution on $\Lambda$ sending a group-like element of
$\Gamma=\operatorname{Gal}(K_{\infty}/K)$ to its inverse. For any
$\Lambda$-module $M$, we write $M^{\iota}$ for the $\Lambda$-module which
coincides with $M$ as a $\mathbb{Z}_{p}$-module, with the action of $\Gamma$
given by

$\gamma\cdot_{\iota}x=\gamma^{-1}x\textrm{ for }\gamma\in\Gamma\textrm{ and
}x\in M.$

######  Remark 1.3.

We don’t know how to prove the result in Corollary 1.2 without the assumption
that ${}^{S}(E/K_{\infty})$ is finite. Even in the number field case over the
cyclotomic $\mathbb{Z}_{p}$-extension of $\mathbb{Q}$, such a result is not
known without this additional hypothesis that
$\FSha(E/\mathbb{Q}_{\mathrm{cyc}})$ is finite (cf. [Lei23, discussion after
Theorem C]). Infact, it is conjectured by Wuthrich that
$\FSha(E/\mathbb{Q}_{\mathrm{cyc}})$ should be finite [Wut07, Question 8.3 and
Conjecture 8.4]. For classical $p^{\infty}$-Selmer groups over
$\mathbb{Q}_{\infty}$, algebraic functional equations are known but the case
of fine Selmer group is considered to be much more difficult and open as of
now; partial results are proven in [HKLR23, Theorem C].

######  Remark 1.4.

Let $K_{n}$ be the finite subextension of $K_{\infty}$ such that
$[K_{n}:K]=p^{n}$. Then we know that $e_{n}=\mu\cdot p^{n}+\lambda\cdot
n+O(1)$ where $|{}^{S}(A/K_{n})|=p^{e_{n}}$. Because of Conjecture A (which is
true without the finiteness assumption of ${}^{S}(A/K_{\infty})$), we know
that $\mu$=0. Additionally, if know that the growth of the Tate-Shafarevich
group $\Sha(A/K_{n})[p^{\infty}]$ is of the kind $p^{\mu_{1}\cdot p^{n}+c}$
(where $c$ is a constant independent of $n$), then we can conclude that
$\lambda=0$ and hence the growth of ${}^{S}(A/K_{n})$ stays bounded. In this
case ${}^{S}(E/K_{\infty})$ is finite.

An example arising via this technique in the number field case was given in
[Wut07, page 11].

###  1.2. Part II: Function Field ($\ell\neq p$ case)

Let $\ell$ be a rational prime, with $\ell\neq p$. Consider $K=\mathbb{F}(t)$,
where $\mathbb{F}$ is a finite field of char $p$. The $\ell^{\infty}$-Selmer
group (defined in 2.8) of an elliptic curve $E$ coincides with the
${\ell^{\infty}}$-fine Selmer group (defined in 2.8). The main result here is
to prove an analogue of Conjecture A (i.e. the vanishing of the
$\mu$-invariant of the fine Selmer) over the non-commutative $\ell$-adic
trivializing extension $K_{\infty}$ of $K$. The Galois group of $K_{\infty}$
over $K$ is an open subgroup of $GL_{2}(\mathbb{Z}_{\ell})$ (cf. [Pal14, §4]).
In particular we show the following result.

######  Theorem 1.5.

(Theorem 6.3) Let $K_{\mathrm{cyc}}$ be the unique
$\mathbb{Z}_{\ell}$-extension of $K$ and $K_{\infty}$ be the trivialising
extension of $K$, such that $G=\operatorname{Gal}(K_{\infty}/K)$ is pro-$p$.
Assume $E$ to be a non-isotrivial elliptic curve over the function field $K$.
Then,

$\mu_{G}(S(E/K_{\infty})^{\vee})=\mu_{\Gamma}(S(E/K_{\mathrm{cyc}})^{\vee})=0.$

###  1.3. Isogeny and $\mu$-invariants ($\ell=p$ case)

We again return to the setting of §1.1. Unlike the $\ell\neq p$ case, here the
Selmer and fine Selmer groups are distinct. Our goal in this section is to
give a lower bound of $\mu$-invariant of the Selmer group over the unramified
$\mathbb{Z}_{p}$-extension $K_{\infty}$ over $K$. The strategy that we adopt
is to compare the $\mu$-invariants of isogenous elliptic curves. More
precisely, we show the following result.

######  Theorem 1.6.

(Theorem 5.1) Let $E_{1}$ and $E_{2}$ be two non-isotrivial elliptic curves
over the function field $K$ and let ${\varphi}:E_{1}\longrightarrow E_{2}$ be
an isogeny of degree $p^{r}$ for some $r>0$. Also assume that
$H^{2}_{fl}(K_{\infty},E_{i}[p^{\infty}])=0$, for $i=1,2$.

Then,

$\mu(S(E_{2}/K_{\infty})^{\vee})-\mu(S(E_{1}/K_{\infty})^{\vee})=\operatorname{ord}_{p}(\chi_{fl}(Spec(K),A)),$

where $A=\ker({\varphi})$ and $\chi_{fl}(Spec(K),A)$ is the Euler
characteristic for flat cohomology defined in §2.7.

It follows that if $\operatorname{ord}_{p}(\chi_{fl}(Spec(K),A))$ is positive,
one can obtain examples of elliptic curves with positive $\mu$-invariant over
$K_{\infty}$.

######  Remark 1.7.

  1. (i)

In [LLS+21], the authors calculate the change in $\mu$-invariant with isogeny
for Selmer groups of elliptic curves having semi-stable reduction everywhere,
for the unramified $\mathbb{Z}_{p}$-extension $K_{\infty}/K$.

In Theorem 5.1 we do not make any assumption about the nature of reduction of
the elliptic curve and use a different method to calculate the change in
$\mu$-invariant with $p^{r}$-isogeny of elliptic curves, with the additional
assumption that $H^{2}_{fl}(K_{\infty},E_{i}[p^{\infty}])=0$. Infact our
technique is much more general and can also be adapted in the $\ell\neq p$
case both for the unramified cyclotomic $\mathbb{Z}_{\ell}$-extension and for
a non-commutative $GL_{2}(\mathbb{Z}_{\ell})$ extension (see Appendix A).
However we don’t get any new result in the $\ell\neq p$ case because of the
stronger result that we showed earlier concerning the triviality of the
$\mu$-invariant (see Theorem 1.5).

  2. (ii)

The assumption $H^{2}_{fl}(K_{\infty},E_{i}[p^{\infty}])=0$, made in Theorem
5.1, is an analogue of the Weak-Leopoldt Conjecture for number fields. This
assumption is similarly adopted in various other works (see for example [BV18,
Remark 2.1]). We also rely on this assumption in our article; however, we are
unable to provide a proof for it.

In the $\ell=p$ case, the Selmer group and the $S$-fine Selmer group over the
unramified $\mathbb{Z}_{p}$-extension of $K$ are defined using flat-cohomology
and hence the number field arguments doesn’t carry over verbatim in this
setup. This is also reciprocated in the formula of the characteristic ideal of
the fine $S$-Selmer group. We note that the power of $\Phi_{n}$ appearing in
Theorem 1.1 is $e_{n}$, unlike the number field case where the power is given
by $e_{n}-1$ by Greenberg (cf. [KP07, Problem 0.7]). Furthermore, this
difficulty also appears while analyzing the local terms in proving Theorem 5.1
(cf. see (10)). We overcame this difficulty by assuming that we are working
over the unramified $\mathbb{Z}_{p}$-extension where the local terms doesn’t
contribute. The question of generalizing Theorem 5.1 for an arbitrary
$\mathbb{Z}_{p}$-extension of $K$ is much more involved and a work is
progress.

In the $\ell\neq p$, setup, our contribution is to work with the non-
commutative $GL_{2}(\mathbb{Z}_{\ell})$-extension of $K$ and to find a
strategy to give instances of trivial $\mu$-invariant of (fine) Selmer group.
The strategy is the following. Since we already know Conjecture A over the
unramified $\ell$-adic extension $K_{\mathrm{cyc}}$ over $K$, using the
analogous $\mathfrak{M}_{H}(G)$ conjecture for $E$ which is also known to hold
from [Pal14, Theorem 4.1] (the conjecture states that $S(E/K_{\infty})^{\vee}$
is a finitely generated $\mathbb{Z}_{\ell}[[G]]$-module, such that
$S(E/K_{\infty})^{\vee}/S(E/K_{\infty})^{\vee}[p^{\infty}]$ is a finitely
generated $\mathbb{Z}_{\ell}[[H]]$-module; here
$G=\operatorname{Gal}(K_{\infty}/K)$ and
$H=\operatorname{Gal}(K_{\infty}/K_{\mathrm{cyc}})$) we show that
$\mu$-invariant of the Selmer group $S(E/K_{\infty})$ is also trivial.

## Acknowledgement

We thank Aprameyo Pal and Anwesh Ray for useful insights while writing this
paper. The second author gratefully acknowledges support from the Inspire
research grant. The second author also thanks Aditya Karnataki for inviting
him to Chennai Mathematical Institute in July 2023; it was there where the
questions dealt in sections 5 and A were conceived. The first author
acknowledges the support received from the HRI postdoctoral fellowship.

##  2\. Preliminaries

Fix an odd integer prime $p$. Let $K$ be a function field in one variable over
a finite field $\mathbb{F}$ of characteristic $p$. Let $A$ be an abelian
variety defined over $K$. Consider an open dense subset $U$ of
$C_{K}=\mathbb{P}^{1}_{\mathbb{F}}$, where $\mathbb{P}^{1}_{\mathbb{F}}$
denotes the projective space of dimension $1$ over $\mathbb{F}$, such that
$A/K$ has good reductions at every place of $U$. Let $\Sigma_{K}$ be the set
of all the primes of $K$ and $S$ denote the set of primes of $K$ outside $U$
i.e., the places of $C_{K}\setminus U$. Therefore, $S$ is a finite set of
primes of $K$ that contains the set of all primes of bad reduction of $A/K$.
Let $K_{S}$ denote the maximal algebraic extension of $K$ unramified outside
$S$. Consider a finite extension of $K$, $L\subset K_{S}$.

######  Definition 2.1.

Let $B$ be an abelian group.

  1. (1)

Denote by $B[p^{n}]$ the $p^{n}$-torsion points of $B$ and let
$B[p^{\infty}]=\underset{n}{\bigcup}B[p^{n}]$.

  2. (2)

Let $T_{p}B:=\varprojlim_{k}B[p^{k}]$ the Tate module of $B$.

  3. (3)

The $p$-adic completion of $B$ is defined as
$B^{*}=\displaystyle\varprojlim_{n}B/p^{n}B$ .

  4. (4)

Also, define

$B^{\bullet}:=B^{*}\otimes\mathbb{Q}_{p}\text{ and
}V_{p}B:=T_{p}B\otimes\mathbb{Q}_{p}.$

###  2.1. The fine Selmer group

Let $v$ be any prime of $K$ and $w$ denote a prime of $L$. Define

$\displaystyle J_{v}^{1}(A/L):=\underset{w\mid
v}{\prod}\frac{H^{1}_{fl}(L_{w},A[p^{\infty}])}{im(\kappa_{w})}\ \text{and
}K_{v}^{1}(A/L):=\underset{w\mid v}{\prod}H^{1}_{fl}(L_{w},A[p^{\infty}]).$

Here $H^{i}_{\text{fl}}(-,-)$ denotes the flat cohomology [Mil86, Chapters II,
III] and
$\kappa_{w}:A(L_{w})\otimes{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}}\hookrightarrow
H^{1}_{fl}(L_{w},A[p^{\infty}])$ is induced by the Kummer map [BL09a, §2.1.2].

######  Definition 2.2.

[KT03, Prop. 2.4] Let $\Sigma_{K},S\text{ and }K\subset L\subset K_{S}$ be as
above. Then the Selmer group $S(A/L)$ is defined as:

$S(A/L):=\ker\big{(}H^{1}_{\text{fl}}(L,A[p^{\infty}])\longrightarrow\underset{v\in\Sigma_{K}}{\prod}J_{v}^{1}(A/L)\big{)}.$
(1)

We define the $S$-fine Selmer group as:

$\displaystyle
R^{S}(A/L):={}\ker\big{(}H^{1}_{\text{fl}}(L,A[p^{\infty}])\longrightarrow\underset{v\in
S}{\bigoplus}K_{v}^{1}(A/L)\underset{v\in\Sigma_{K}\setminus
S}{\prod}J_{v}^{1}(A/L)\big{)}$ (2)
$\displaystyle\cong\ker\big{(}S(A/L)\longrightarrow\underset{w\mid v,v\in
S}{\bigoplus}A(L_{w})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}\big{)}.$

######  Remark 2.3 (Dependence on $S$).

Let $A,S,K,U$ be as above and let $\mathcal{A}$ be the N$\acute{e}$ron model
of $A$ over $C_{K}$. Recall the following equivalent definition [KT03] of the
Selmer group:

$S(A/K):=\ker\big{(}H^{1}_{\text{fl}}(U,\mathcal{A}[p^{\infty}])\longrightarrow\underset{v\in
S}{\bigoplus}J^{1}_{v}(A/K)\big{)}.$ (3)

Using definitions 2 and 3, we have

$R^{S}(A/K):=\ker\big{(}H^{1}_{\text{fl}}(U,\mathcal{A}[p^{\infty}])\longrightarrow\underset{v\in
S}{\bigoplus}K^{1}_{v}(A/K)\big{)}.$ (4)

In fact, in [KT03, Proposition 2.4], the authors showed that the two
definitions (1 and 3) of $S(A/K)$ are equivalent. The key ingredient in the
proof is the following exact sequence [Mil86, Chapter 3, §7]:

$0\rightarrow H^{1}_{\text{fl}}(U,\mathcal{A}[p^{\infty}])\longrightarrow
H^{1}_{\text{fl}}(K,A[p^{\infty}])\longrightarrow\underset{v\in
U}{\oplus}H^{1}_{\text{fl}}(K_{v},A[p^{\infty}])/H^{1}_{\text{fl}}(O_{v},A[p^{\infty}]),$
(5)

where $O_{v}$ is the valuation ring of $K_{v}$ and
$H^{1}_{\text{fl}}(O_{v},A[p^{\infty}])\cong
A(K_{v})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}.$

For an infinite algebraic extension $\mathcal{L}$ of $K$, the above
definitions extend, as usual, by taking inductive limit over finite sub
extensions of $\mathcal{L}$ over $K$.

######  Remark 2.4.

Consider a $p$-adic Lie extension $L_{\infty}/K$ where
$G$=$\mathrm{Gal}(L_{\infty}/K)$ is a compact $p$-adic Lie group without any
$p$-torsion. Then, one can show that $R^{S}(A/L_{\infty})$ is independent of
$S$, using the Noetherianess of $\mathbb{Z}_{p}[[G]]$. However, we cannot
determine this set $S$ explicitly (see [GJS22, Remark 3.2]).

###  2.2. $\mathbb{Z}_{p}^{d}$-extensions

Let $\mathbb{F}^{(p)}$ be the unique subfield of $\overline{\mathbb{F}}$ such
that $\mathrm{Gal}(\mathbb{F}^{(p)}/\mathbb{F})\cong\mathbb{Z}_{p}$. Set
$K_{\infty}:=K\mathbb{F}^{(p)}$. Note that $K_{\infty}/K$ is unramified
everywhere. This follows from the fact that the prime ideals of $K$ correspond
to irreducible monic polynomials of $K$ and $K_{\infty}$ is obtained by
extending the perfect field $\mathbb{F}$. This $\mathbb{Z}_{p}$-extension
$K_{\infty}$ is referred to as in the literature as the "arithmetic"
$\mathbb{Z}_{p}$-extension.

The second type of $\mathbb{Z}_{p}$-extension that bears a close analogy with
the cyclotomic $\mathbb{Z}_{p}$-extension of a number field, is the
“cyclotomic extension at the prime ideal $\mathfrak{P}$". We briefly outline
its construction below:

Let $P(t)=a_{n}t^{n}+\cdots a_{0}\in\mathbb{F}[t]$. We define the Carlitz
polynomial $[P(t)](X)$ with coefficients in $\mathbb{F}[t]$ recursively as
follows:

$[1](X)=X$,

$[t](X)=X^{p}+tX$,

$[t^{n}](X)=[t]([t^{n-1}](X))$ and

$[a_{n}t^{n}+\cdots+a_{1}t+a_{0}](X)=a_{n}[t^{n}](X)+\cdots+a_{1}[t](X)+a_{0}(X).$

Consider a field extension $F$ of $K$. Then $F$ can be thought of as a
$\mathbb{F}[t]$-module, where the action of $\mathbb{F}[t]$ is given by the
Carlitz polynomials.

Choose a prime $\mathfrak{P}$ of $\mathbb{F}[t]$. For $n>0$, let

$\Lambda_{\mathfrak{P}^{n}}:=\\{\lambda\in\overline{\mathbb{F}(t)}|[\mathfrak{P}^{n}](\lambda)=0\\}.$

Here $K(\Lambda_{\mathfrak{P}^{n}})/K$ is Galois with
$\mathrm{Gal}(K(\Lambda_{\mathfrak{P}^{n}})/K)\cong(\mathbb{F}[t]/\mathfrak{P}^{n})^{\times}$.
Put $\widetilde{K}:=\underset{n\geqslant
1}{\bigcup}K(\Lambda_{\mathfrak{P}^{n}})$, then
$\mathrm{Gal}(\widetilde{K}/K)\cong\mathbb{Z}_{p}^{\mathbb{N}}\times(\mathbb{F}[t]/\mathfrak{P})^{\times}$.

The $\mathbb{Z}_{p}^{d}$-extension obtained from $\widetilde{K}$, for
$d\geqslant 1$ is ramified only at the prime $\mathfrak{P}$ and it is totally
ramified at that prime [Ros02, Proposition 12.7]. The
$\mathbb{Z}_{p}$-extension, thus obtained, is referred to as the "geometric"
$\mathbb{Z}_{p}$-extension.

###  2.3. Fine Mordell-Weil groups

We define the $S$-fine Mordell-Weil group similar to that of [Wut07] as
follows:

######  Definition 2.5.

Let $k\geqslant 1$ and an extension $L/K$, the $p^{k}$-$S$-fine Mordell-Weil
group of $A/L$, $M^{S}_{p^{k}}(A/L)$ is defined as:

$M^{S}_{p^{k}}(A/L)=\ker(A(L)/p^{k}\longrightarrow\underset{v\mid
S}{\bigoplus}{A(L_{v})/p^{k}A(L_{v})})$

The $p$-primary $S$-fine Mordell-Weil group of $A/L$ is given by:

$\mathcal{M}^{S}(A/L)=\varinjlim_{k}M^{S}_{p^{k}}(A/L)=\ker(A(L)\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}\longrightarrow\underset{v\mid
S}{\bigoplus}A(L_{v})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}).$

Again following [Wut07], we define the $S$-fine Tate-Shafarevich as follows:

######  Definition 2.6.

For an extension $L/K$, $S$-fine Tate-Shafarevich is defined as:

${}^{S}(A/L):=\dfrac{R^{S}(A/L)}{\mathcal{M}^{S}(A/L)}.$

Hence, we have the following short exact sequence:

$0\longrightarrow\mathcal{M}^{S}(A/L)\longrightarrow
R^{S}(A/L)\longrightarrow{}^{S}(A/L)\longrightarrow 0.$

######  Remark 2.7.

Let $k\geqslant 1$ and $L$ be a finite extension of $K$. Note that
$\mathcal{M}^{S}(A/L)$ can also be defined as the intersection of $R^{S}(A/L)$
with $A(L)\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}$ inside $S(A/L)$ [Wut07, §2].
Now, by using the exact sequence

$0\longrightarrow A(L)\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}\longrightarrow
S(A/L)\longrightarrow\Sha(A/L)[p^{\infty}]\longrightarrow 0,$

it is easy to see that ${}^{S}(A/L)\subset\Sha(A/L)[p^{\infty}]$.

######  Remark 2.8.

By Remark 2.7, we observe that for any separable extension $\mathcal{L}$ of
$K$, the finiteness of ${}^{S}(A/\mathcal{L})$ follows from finiteness of
$\Sha(A/\mathcal{L})[p^{\infty}]$.

The finiteness of $\Sha(A/\mathcal{L})$ (i.e., the BSD conjecture) has been
establised for a class of elliptic curves over function fields (see for
example [HYZ23]).

###  2.4. The functor $\mathfrak{G}$.

Let $K$ be a function field of char $p$. Let $K_{\infty}$ be a
$\mathbb{Z}_{p}$-extension of $K$, with Galois group $\Gamma$. Denote by
$\Lambda$ the Iwasawa algebra $\mathbb{Z}_{p}[[\Gamma]]$. We identify
$\mathbb{Z}_{p}[[\Gamma]]$ with the power series ring $\mathbb{Z}_{p}[[T]]$ by
identifying $T$ with $\gamma-1$, where $\gamma$ is a topological generator of
$\Gamma$. For $n\geqslant 0$, define $\omega_{n}=(1+T)^{p^{n}}-1$ and
$\omega_{0,-1}:=T$.

For a finitely generated $\Lambda$-module $X$, we define

$\mathfrak{G}:=\displaystyle\varprojlim_{n}\left(\frac{X}{\omega_{n}X}[p^{\infty}]\right).$

The properties of the functor $\mathfrak{G}$ are discussed in [Lee20].

Let $\Phi_{n}:=\frac{\omega_{n}}{\omega_{n-1}}$ for $n\geqslant 1$ and
$\Phi_{0}=X$. The functor $\mathfrak{G}$ has the following properties [Lee20,
Lemma A.2.9].

  1. (1)

$\mathfrak{G}(\Lambda)=0$. Hence, $\mathfrak{G}(X)$ is a torsion
$\Lambda$-module.

  2. (2)

$\mathfrak{G}(\Lambda/g^{e})=\Lambda/g^{e}$ if $g$ is coprime to $\omega_{n}$
for all $n$.

  3. (3)

For $m\geqslant 0$,

$\mathfrak{G}(\Lambda/\Phi_{m}^{e})=\begin{cases}\Lambda/\Phi_{m}^{e-1},&e\geqslant
2\\\ 0,&e=1\end{cases}$

  4. (4)

$\mathfrak{G}$ is a covariant functor and preserves the pseudo-isomorphism.

Consider a finitely generated $\mathbb{Z}_{p}$-module $M$. Denote
$M_{\text{div}}$ as the maximal divisible subgroup of $M$. For an integral
domain $R$ and an $R$-module $A$, let $A_{R-\text{tor}}$ represent the
elements of $A$ that are $R$-torsion. Let us recall the following lemma from
[Lee20]:

######  Lemma 2.9.

[Lee20, Lemma 2.1.4]

  1. (1)

Let $R$ be an integral domain and let $Q(R)$ be the quotient field of $R$.
Consider an exact sequence of $R$-modules $0\longrightarrow A\longrightarrow
B\longrightarrow C\longrightarrow 0$, where $A$ is an $R$-torsion module. Then
we have a short exact sequence $0\longrightarrow
A_{R-\text{tor}}\longrightarrow B_{R-\text{tor}}\longrightarrow
C_{R-\text{tor}}\longrightarrow 0$.

  2. (2)

Let $0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0$ be
a short exact sequence of finitely generated $\mathbb{Z}_{p}$-modules. If $A$
has finite cardinality, then we have a short exact sequence

$0\longrightarrow A=A[p^{\infty}]\longrightarrow B[p^{\infty}]\longrightarrow
C[p^{\infty}]\longrightarrow 0.$

  3. (3)

If $0\longrightarrow X\longrightarrow Y\longrightarrow Z\longrightarrow
W\longrightarrow 0$ is an exact sequence of cofinitely generated
$\mathbb{Z}_{p}$-modules with finite $W$,then the sequence

$X_{div}\longrightarrow Y_{div}\longrightarrow Z_{div}\longrightarrow
W\longrightarrow 0$

is exact.

We will use Lemma 2.9 and the properties of functor $\mathfrak{G}$ later in
the proof of Theorem 3.5.

###  2.5. Elliptic curves over function fields

Let $E$ be an elliptic curve defined over the function field $K$ of char $p$.
Let $E[p^{\infty}]:=E(\overline{K})[p^{\infty}]$, where $\overline{K}$ is the
separable closure of $K$.

We recall the Mordell-Weil-Lang-Néron theorem:

######  Theorem 2.10.

[Ulm11, Lecture 1, Theorem 5.1] Assume that $K=\mathbb{F}(C)$ is the function
field of a curve $C$ over a finite field $\mathbb{F}$ and let $E$ be an
elliptic curve over $K$. Then $E(K)$ is a finitely generated abelian group.

For any $v\in\Sigma_{K}$, choose a minimal Weierstrass equation for $E$. Let
$E_{v}$ be the reduction of $E$ modulo $v$. For any point $P\in E$, let
$P_{v}$ be its image in $E_{v}$. Let $E_{v,ns}(\mathbb{F}_{v})$ be the set of
non-singular points of $E_{v}(\mathbb{F}_{v})$ and define,

$E_{0}(K_{v}):=\\{P\in E(K_{v})\,|\,P_{v}\in E_{v,ns}(\mathbb{F}_{v})\\}$

######  Remark 2.11.

  1. (i)

Let $v$ be a prime of $K$, where the elliptic curve $E$ has split
multiplicative reduction. Then according to the theory of Tate curves,
$E_{0}(K_{v})$ is isomorphic to $\mathcal{O}_{v}^{*}$, where $\mathcal{O}_{v}$
represents the ring of integers of $K_{v}$ [BL09a, §2.1.2]. Since
$E_{0}(K_{v})$ has a finite index within $E(K_{v})$, it follows that
$E(K_{v})[p^{\infty}]$ is also finite.

  2. (ii)

Consider $v$, a prime of $K$, where $E$ has good reduction. According to
[Tan10, Lemma 2.5.1], the formal group linked to $E$ at $v$, also a subgroup
of finite index of $E(K_{v})$, is a torsion-free $\mathbb{Z}_{p}$-module.
Consequently, this implies that $E(K_{v})[p^{\infty}]$ is finite.

Now, let us recall the lemma stated in [BL09a]:

######  Lemma 2.12.

[BL09a, Lemma 4.1] Let $G\cong\mathbb{Z}_{p}^{d}$, for some $d\geqslant$ and
$B$ be a finite $p$-primary $G$-module. Then,

$|H^{1}(G,B)|\leqslant|B|^{d}\text{and
}|H^{1}(G,B)|\leqslant|B|^{\frac{d(d-1)}{2}}.$

######  Remark 2.13.

Let $\mathcal{K}_{d}$ be a $\mathbb{Z}_{p}^{d}$-extension of $K$ for some
$d\geqslant 1$. Then the group $E[p^{\infty}](\mathcal{K}_{d})$ is finite
[BL09a, Lemma 4.3]. Therefore, by Lemma 2.12, we get that
$H^{1}(G,E[p^{\infty}](\mathcal{K}_{d}))$ is bounded by
$|E[p^{\infty}](\mathcal{K}_{d})|^{d}$.

###  2.6. Euler Characteristic and the $\mu$-invariant

Let $G$ be pro-$p$, $p$-adic Lie group without any elements of order $p$. Let
$\Lambda(G)$ represent the Iwasawa algebra of $G$. The completed group
algebra, denoted as $\Omega(G)$, is defined as follows:

$\Omega(G)=\varprojlim_{U}\mathbb{F}_{p}[G/U],$

where $U$ varies across the open normal subgroups of $G$.

######  Definition 2.14.

[How02] Let $M$ be a finitely generated $\Lambda(G)$-module then we define

$\mu(M):=\sum_{i\geqslant
0}\operatorname{rank}_{\Omega(G)}(p^{i}(M[p^{\infty}])/p^{i+1}).$

######  Remark 2.15.

Note that the definition of $\mu(M)$ does not require $M$ to be a torsion
$\Lambda(G)$-module.

######  Definition 2.16.

Consider a compact $p$-adic Lie group $G$ with no element of order $p$. Let
$M$ be a finitely generated $\Lambda(G)$-module. If $H^{i}(G,M)$ is finite for
every $i\geqslant 0$, then the $G$-Euler characteristic of $M$ is defined and
it is given by:

$\chi(G,M):=\underset{i\geqslant 0}{\prod}(\\#H^{i}(G,M))^{(-1)^{i}}\ .$

We recall the following results from [How02]:

######  Corollary 2.17.

[How02, Corollary 1.7] Assume that $G$ contains no non-trivial element of
finite order. If $M$ is a finitely generated $\Lambda(G)$-module then

$\mu(M)=\operatorname{ord}_{p}(\chi(G,M[p^{\infty}]),$

where $ord_{p}(\alpha)$ denotes the maximum power of $p$ which divides
$\alpha$.

######  Proposition 2.18.

[How02, Proposition 1.8] Assume that $G$ contains no non-trivial element of
finite order. In a short exact sequence of finitely generated
$\Lambda(G)$-modules, one has

$0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0.$

we have $\mu(B)\leqslant\mu(A)+\mu(C)$ with equality holding if $B$, and hence
also $A$ and $C$, is $\Lambda(G)$-torsion.

###  2.7. Euler characteristic for flat cohomology

######  Definition 2.19.

[TW11, §5] Let $S$ be a scheme of characteristic $p>0$ and a finite flat group
scheme $N/S$ we define the Euler characteristic of $N/S$ as

$\chi_{fl}(S,N):=\underset{i}{\prod}(\\#H^{i}_{fl}(S,N))^{(-1)^{i}}.$

whenever the groups $H^{i}_{fl}(S,N)$ are finite.

####  2.7.1. Relation with Hochschild-Serre spectral sequence

Let $K$ be a function field of char $p$ and $K_{\infty}$ be an unramified
$\mathbb{Z}_{p}$-extension with $\Gamma=\mathrm{Gal}(K_{\infty}/K)$. Let $A$
be a sheaf for the flat topology on $Spec(K)$. Now, recall the Hochschild-
Serre spectral sequence [Mil80, Chapter 3, Remark 2.21]

$E_{2}^{a,b}:=H^{a}(\Gamma,H^{b}_{fl}(K_{\infty},A))\implies
H_{fl}^{a+b}(K,A),$ (6)

where $A$ is a sheaf for flat topology on $Spec(K)$.

######  Remark 2.20.

If $A$ is a finite flat group scheme over $Spec(K)$, then by using (6), we
obtain that

$\chi_{fl}(Spec(K),A)=\dfrac{\chi(\Gamma,H^{2}_{fl}(K_{\infty},A))\times\chi(\Gamma,H^{0}_{fl}(K_{\infty},A))}{\chi(\Gamma,H^{1}_{fl}(K_{\infty},A)}$

###  2.8. The $\ell\neq p$ case

Choose and fix a rational prime $\ell\neq p$. Consider $K=\mathbb{F}(t)$,
where $\mathbb{F}$ is a finite field of char $p$ of cardinality $p^{r}$, for
some $r>0$. In this subsection, we will recall the properties of the
$\ell^{\infty}$-Selmer groups of an elliptic curve $E$ over certain compact
$\ell$-adic Lie extensions over the function field $K$ of characteristic $p$.
Note that the image of the Kummer map, in this set up, is identically 0
[BL09b, Proposition 3.1], thus the ${\ell^{\infty}}$-fine Selmer group
coincides with the ${\ell^{\infty}}$-Selmer group.

Let $E/K$ be an elliptic curve. Let $S$ be a finite set of places of $K$ that
contains the places which ramify over $K_{\infty}$ and the places where $E$
has a bad reduction. Let $K_{S}$ be the maximal seperable extension of $K$,
which is unramified outside $S$. For each $v\in S$, put
$J^{1}_{v}(E/L):=\underset{w\mid v}{\prod}H^{1}(L_{w},E[\ell^{\infty}])$.

######  Definition 2.21.

Let $L\subset K_{S}$ be a finite extension of $K$. We define the
${\ell}^{\infty}$-Selmer group $S(E/L)$ of $E/L$ as:

$S(E/L):=\ker(H^{1}(G_{S}(L),E[{\ell}^{\infty}])\longrightarrow\underset{v\in
S}{\bigoplus}J^{1}_{v}(E/L)),$ (7)

where $G_{S}(L):=\mathrm{Gal}(K_{S}/L)$.

The definition of $S(E/\mathcal{L})$ extends to an infinite extension
$\mathcal{L}/K$ by taking direct limits over intermediate finite
subextensions.

Denote by $\mathbb{F}_{p}^{(\ell)}$ the unique $\mathbb{Z}_{\ell}$-extension
of $\mathbb{F}_{p}$ contained in $\overline{\mathbb{F}}_{p}$. Then
$K_{\mathrm{cyc}}:=\mathbb{F}_{p}^{(\ell)}K$ is the unique
$\mathbb{Z}_{\ell}$-extension of $K$ [BL09b, Proposition 4.3]. This is known
in the literature as the “arithmetic" $\mathbb{Z}_{\ell}$-extension. The
extension $K_{\mathrm{cyc}}$ of $K$ is unramified everywhere.

Another $p$-adic Lie extension relevant to our discussion is the trivializing
extension of $K$. It is defined as follows:

Let $E/K$ be an elliptic curve such that $j(E)\notin\mathbb{F}$. Put
$K_{\infty}:=K(E[\ell^{\infty}])$. This is an $\ell$-adic Lie extension of the
global field $K$. We know $G_{\infty}:=\mathrm{Gal}(K_{\infty}/K)$ is an open
subgroup of $GL_{2}(\mathbb{Z}_{\ell})$ [Pal14, §4].

######  Remark 2.22.

The condition $j(E)\notin\mathbb{F}$ ensures that the elliptic curve $E$ has
no complex multiplication [Ulm11, Lecture 1, §4]. Elliptic curves satisfying
this property are called non-iostrivial elliptic curves.

##  3\. Structure of the Mordell-Weil group

Let $K$ be a function field of char $p$ and $A/K$ be an abelian variety
defined over $K$. Let $K_{\infty}$ be a $\mathbb{Z}_{p}$-extension of $K$,
with Galois group $\Gamma$ (either $K_{\infty}$ is the arithmetic
$\mathbb{Z}_{p}$-extension or the geometric $\mathbb{Z}_{p}$-extension). From
now on, $S$ will be the finite set of primes of $K$, containing the primes of
bad reduction of $A$ and the primes of $K$ ramified in $K_{\infty}$.

Let $K_{n}$ be the unique sub-extension of $K_{\infty}/K$ such that
$[K_{n}:K]=p^{n}$. Denote by
$\Gamma_{n}:=\operatorname{Gal}(K_{\infty}/K_{n})$. Now consider the natural
restriction map:

$S^{A}_{n}:S(A/K_{n})\longrightarrow S(A/K_{\infty})^{\Gamma_{n}}.$

######  Remark 3.1.

The following theorem of [Tan10] shows that $\ker(S^{A}_{n})$ and
$\operatorname{coker}(S^{A}_{n})$ are both finite and bounded independently of
$n$:

######  Theorem 3.2.

[Tan10, Theorem 4] Let $K_{d}$ be a $\mathbb{Z}_{p}^{d}$-extension of a global
field $K$ of characteristic $p$ with Galois group
$\Gamma_{d}:=\mathrm{Gal}(K_{d}/K)$. Assume that $K_{d}/K$ is unramified
outside a finite set $S$ of places of $K$. Let $A$ be an abelian variety over
$K$ with good ordinary reduction at every place in $S$. Then for every finite
intermediate extension $F$ of $K_{d}/K$, the kernel and the cokernel of the
restriction map $\text{res}\ K_{d}/F:S(A/F)\longrightarrow
S(A/K_{d})^{\Gamma_{F}}$, where $\Gamma_{F}=\mathrm{Gal}(K_{d}/F)$ are finite.
Furthermore, if $d=1$, then the orders of the kernel and the cokernel of
$\text{res}\ K_{d}/F$ are bounded as $F$ varies.

The Tate-Shafarevich group of an abelian variety $A/K$ is defined as:

$\Sha(A/K):=\ker(H^{1}(K,A)\longrightarrow\underset{v\in\Sigma_{K}}{\prod}H^{1}(K_{v},A))$

######  Proposition 3.3.

Let $A/K$ be an abelian variety and $S$ be as above. Assume that
$\ker(S^{A}_{n})$ and $\operatorname{coker}(S^{A}_{n})$ are finite and bounded
independently of $n$.

Further, if the groups $A(K_{n})$ and $\Sha(A/K_{n})[p^{\infty}]$ are finite
for each $n>0$, then there exists $\mu,\lambda,\nu\geqslant 0$, such that

$\\#S(A/K_{n})=\\#\Sha(A/K_{n})[p^{\infty}]=p^{e_{n}}$

for $n\gg 0$, where $e_{n}=p^{n}\mu+n\lambda+\nu$.

###### Proof.

The proof follows easily from the structure of finitely generated
$\Lambda$-modules (see [Gre01, Corollary 4.11]). ∎

The subsequent lemma extends the result of [Lee20, Lemma 2.0.1] to function
fields of characteristic $p$. We provide the proof below:

######  Lemma 3.4.

Let $L$ be a finite extension of $K$ or $K_{v}$ for some prime $v$ and let
$A/L$ be an abelian variety. Let $L_{\infty}/L$ be a
$\mathbb{Z}_{p}$-extension and $X=(A(L_{\infty})[p^{\infty}])^{\vee}$. Then,

  1. (i)

$X$ is a finitely generated torsion $\Lambda$-module with $\mu=0$ and
$char_{\Lambda}(X)$ is coprime to $\omega_{n}$ for all $n$.

  2. (ii)

The modules $\frac{A(L_{\infty})[p^{\infty}]}{p^{n}A(L_{\infty})[p^{\infty}]}$
and $\frac{A(L_{\infty})[p^{\infty}]}{\omega_{n}A(L_{\infty})[p^{\infty}]}$
are finite and bounded independent of $n$.

  3. (iii)

For the natural maps

$MW^{A}_{n}:A(L_{n})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}\longrightarrow(A(L_{\infty})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p})^{\Gamma_{n}}$

and

$S^{A}_{n}:S(A/L_{n})\longrightarrow S(A/L_{\infty})^{\Gamma_{n}}$

the groups $\ker(MW^{A}_{n})$ and $\ker(S^{A}_{n})$ are finite and bounded
independently of $n$

###### Proof.

Note that (ii) follows from (i), via the structure theorem of finitely
generated $\Lambda$-modules. For (i), observe that

$\dfrac{X}{pX}\cong(A(L_{\infty})[p])^{\vee},\dfrac{X}{\omega_{n}X}\cong(A(L_{n})[p^{\infty}])^{\vee}$

$A(L_{\infty})[p]$ and $A(L_{n})[p^{\infty}]$ are finite (see proof of [GJS22,
Proposition 3.3]). This proves (i).

Now, notice that by the definition of Selmer groups, we have injections:

$\ker(MW^{A}_{n})\hookrightarrow\ker(S^{A}_{n})\hookrightarrow\dfrac{A(L_{\infty})[p^{\infty}]}{\omega_{n}A(L_{\infty})[p^{\infty}]}$

The second injectivity follows from the fact that $\ker(S^{A}_{n})\subset
H^{1}(\Gamma_{n},A(F_{\infty})[p^{\infty}])$. Now, (iii) follows from (ii). ∎

The next proposition is a generalisation of [Lee20, Theorem 3.3] in the
setting of function fields.

######  Proposition 3.5.

Let $L$ be a finite extension of $K$ or $K_{v}$ for some prime $v$ and let
$A/L$ be an abelian variety. Let $L_{\infty}/L$ be a
$\mathbb{Z}_{p}$-extension. Then, we have:

  1. (1)

$\mathfrak{G}((A(L_{\infty})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p})^{\vee})=0$.

  2. (2)

There is a $\Lambda$-linear injection

$(A(L_{\infty})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p})^{\vee}\hookrightarrow\Lambda^{r}\oplus\left(\bigoplus_{i=1}^{u}\dfrac{\Lambda}{\Phi_{c_{i}}}\right)$
(8)

with finite cokernel for some $r,u,c_{i}\geqslant 0$.

The proof essentially follows from the arguments of [Lee20, Theorem 3.3]. We
present a brief sketch below:

###### Proof.

We first identify $\mathbb{Z}_{p}[[\mathrm{Gal}(L_{\infty}/L)]]$ with
$\mathbb{Z}_{p}[[X]]$. Let $C_{n}$ be the cokernel of the natural map

$MW^{A}_{n}:A(L_{n})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}\longrightarrow(A(L_{\infty})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p})^{\Gamma_{n}}$

As $\varinjlim_{n}A(L_{n})\otimes\
\mathbb{Q}_{p}/\mathbb{Z}_{p}=A(L_{\infty})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}$
by definition, it follows that $\varinjlim_{n}C_{n}=0$ and
$\varinjlim_{n}(C_{n})_{div}=0$.

Now, from the short exact sequence

$A(L_{n})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}\longrightarrow(A(L_{\infty})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p})^{\Gamma_{n}}\longrightarrow
C_{n}\longrightarrow 0$

we obtain the following short exact sequence

$0=(A(L_{n})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p})_{div}\longrightarrow((A(L_{\infty})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p})^{\Gamma_{n}})_{div}\longrightarrow(C_{n})_{div}\longrightarrow
0.$

by applying Lemma 2.9(iii). Taking direct limits to the above sequence, we get
that

$\varinjlim_{n}((A(L_{\infty})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p})^{\Gamma_{n}})_{div}=0.$

Taking Pontryagin dual, we obtain that

$\mathfrak{G}((A(L_{\infty})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p})^{\vee})=\displaystyle\varprojlim_{n}\frac{(A(L_{\infty})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p})^{\vee}}{\omega_{n}(A(L_{\infty})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p})^{\vee}}[p^{\infty}]=0.$

Recall from the description of functor $\mathfrak{G}$, given before the
statement of Lemma 2.9, that $\mathfrak{G}(\Lambda)=0$ and
$\mathfrak{(}\frac{\Lambda}{\omega_{m+1,m}^{e}})=0,$ if $e=1$ and
$=\frac{\Lambda}{\omega_{m+1,m}^{e-1}}$, if $e\geqslant 2$. Using the fact
that $(A(L_{\infty})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p})^{\vee}$ is a
finitely generated $\Lambda$ module and
$\mathfrak{G}((A(L_{\infty})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p})^{\vee})=0$,
we get the following $\Lambda$-linear pseudo-isomorphism:

$(A(L_{\infty})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p})^{\vee}\longrightarrow\Lambda^{r}\oplus\left(\bigoplus_{i=1}^{u}\dfrac{\Lambda}{\Phi_{c_{i}}}\right)$

for some $r,u,c_{i}\geqslant 0$.

The injectivity follows from the fact that
$(A(L_{\infty})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p})^{\vee}$ has no non-
trivial finite $\Lambda$-module since it is $\mathbb{Z}_{p}$-torsion free. ∎

######  Remark 3.6.

Let $L$ be a finite extension of $K$. Let $L_{\infty}$ be a
$\mathbb{Z}_{p}$-extension of the function field $L$. Assuming that
$\Sha(E/L_{n})[p^{\infty}]$ is finite for all $n$, we can use the finiteness
of the kernel of the map $S^{A}_{n}$ and apply the snake lemma to deduce that
the natural map

$E(L_{n})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}\longrightarrow(E(L_{\infty})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p})^{\Gamma_{n}}$

has both finite kernel and cokernel. Moreover, by comparing the
$\mathbb{Z}_{p}$-ranks of the two modules as in [Lee20, Page-2409], we get
that

$r=\lim_{n\to\infty}\frac{\operatorname{rank}_{\mathbb{Z}}A(L_{n})}{p^{n}}\text{
and
}e_{n}=\frac{\operatorname{rank}_{\mathbb{Z}}E(L_{n})-\operatorname{rank}_{\mathbb{Z}}E(L_{n-1})}{{\varphi}(p^{n})}-r;$

for $n\geqslant 1$ and $e_{n}$ is defined as the number between $1\leqslant
i\leqslant t$ satisfying $i=n$ in (8).

##  4\. Structure of Fine Mordell Weil Groups

######  Corollary 4.1.

Assume that $r=0$ in equation (8). Then there is a pseudo-isomorphism of
$\Lambda$-modules

$\mathcal{M}^{S}(A/L_{\infty})^{\vee}\sim\bigoplus_{i=1}^{u}\dfrac{\Lambda}{\Phi_{c_{i}}},$

for some $u,c_{i}\geqslant 0$.

We now prove a control theorem for the Fine Mordell-Weil group:

######  Proposition 4.2.

Let $L$ be a finite extension of $K$. Let $E/L$ be an elliptic curve and let
$m_{n}$ denote the natural morphism

$m_{n}:\mathcal{M}^{S}(E/L_{n})\longrightarrow\mathcal{M}^{S}(E/L_{\infty})^{\Gamma_{n}}$

induced by the inclusion $E(L_{n})\hookrightarrow E(L_{\infty})$.

  1. (a)

The kernel of $m_{n}$ is finite, with order bounded independently of $n$.

  2. (b)

Suppose that ${}^{S}(E/L_{n})$ is finite, then the cokernel of $m_{n}$ is
finite

###### Proof.

Consider the following commutative diagram:

${0}$${R^{S}(E/L_{\infty})^{\Gamma_{n}}}$${H^{1}(L_{\infty},E[p^{\infty}])^{\Gamma_{n}}}$${(\underset{v\in
S}{\prod}K^{1}_{v}(E/L_{\infty})\underset{v\in\Sigma_{K}\setminus
S}{\prod}J^{1}_{v}(E/L_{\infty}))^{\Gamma_{n}}}$${0}$${R^{S}(E/L_{n})}$${H^{1}(L_{n},E[p^{\infty}])}$${\underset{v\in
S}{\prod}K^{1}_{v}(E/L_{n})\underset{v\in\Sigma_{K}\setminus
S}{\prod}J^{1}_{v}(E/L_{n})}$$\scriptstyle{\lambda_{L_{\infty}}}$$\scriptstyle{f_{n}}$$\scriptstyle{\lambda_{L_{n}}}$$\scriptstyle{g_{n}}$$\scriptstyle{h_{n}}$

First, we show that the kernel of $f_{n}$ is finite and bounded independently
of $n$ and that the cokernel of $f_{n}$ is also finite.

Now, using snake lemma, we get the following exact sequence:

$0\longrightarrow\ker(f_{n})\longrightarrow\ker(g_{n})\longrightarrow\ker(h_{n})\cap\text{image}(\lambda_{L_{n}})\longrightarrow\operatorname{coker}(f_{n})\longrightarrow
0.$

Therefore, it is enough to show that $\ker(g_{n})$ is finite and bounded
independently of $n$ and $\ker(h_{n})$ is finite. Notice, that
$\ker(g_{n})=H^{1}(\Gamma_{n},E(L_{\infty})[p^{\infty}])$, which is finite and
bounded independently of $n$ by Remark 2.13.

For $v\notin S$, the kernel of the map from $J^{1}_{v}(E/L_{n})\longrightarrow
J^{1}_{v}(E/L_{\infty})^{\Gamma_{n,v}}$, where $\Gamma_{n,v}$ is the
decomposition subgroup of $\Gamma_{n}$ at the prime $v$, is 0 [BL09a, §4.2.4].
Therefore, by Shapiro’s lemma, we get that $\ker(h_{n})\cong\underset{v\in
S}{\bigoplus}H^{1}(\Gamma_{n,v},E(L_{\infty,v})[p^{\infty}])$. Now, the
finiteness of $\ker(h_{n})$ follows from the fact that
$H^{0}(\Gamma_{n,v},E(L_{\infty,v})[p^{\infty}])=E(L_{n,v})[p^{\infty}]$ is
finite (see Remark 2.11).

To prove (a) and (b), we consider the commutative diagram:

${0}$${\mathcal{M}^{S}(E/L_{\infty})^{\Gamma_{n}}}$${R^{S}(E/L_{\infty})^{\Gamma_{n}}}$${{}^{S}(E/L_{\infty})^{\Gamma_{n}}}$${0}$${\mathcal{M}^{S}(E/L_{n})}$${R^{S}(E/L_{n})}$${{}^{S}(E/L_{n})}$${0}$$\scriptstyle{m_{n}}$$\scriptstyle{f_{n}}$$\scriptstyle{\gamma_{n}}$

Applying snake lemma, we get that $\ker(m_{n})$ is finite and bounded
independently of $n$ as $\ker(f_{n})$ is finite and bounded independently of
$n$. This proves (a).

Now, if we assume that ${}^{S}(E/L_{n})$ is finite. Then, (b) follows from the
fact that $\operatorname{coker}(f_{n})$ is finite.

∎

The proof of the next Corollary is similar to that of [Lei23, Corollary 3.8].
We briefly sketch it below:

######  Corollary 4.3.

Assume that $r=0$ in equation (8) and ${}^{S}(E/L_{n})$ is finite for each
$n$. Then there is a $\Lambda$-isomorphism

$T_{p}\mathcal{M}^{S}(E/L_{n})\longrightarrow\bigoplus_{c_{i}\leqslant
n}\dfrac{\Lambda}{\Phi_{c_{i}}},$

where the integers $c_{i}$ are as in Corollary 4.1.

###### Proof.

First, note that $(\Lambda/\Phi_{c_{i}})_{\Gamma_{n}}$ has finite cardinality
for all $c_{i}>n$. Now, from Proposition 4.2, it follows that
$\mathcal{M}^{S}(E/L_{n})^{\vee}$ is given by, up to finite modules,
$\underset{c_{i}\leqslant n}{\bigoplus}{\Lambda/\Phi_{c_{i}}}$. The proof now
follows from [Lei23, Remark 2.4]. ∎

######  Remark 4.4.

If $L_{\infty}$ is the arithmetic $\mathbb{Z}_{p}$-extension of $L$, then
$S(E/L_{\infty})^{\vee}$ is a torsion $\Lambda$-module [OT09, Theorem 1.7].
Now, recall the following short exact sequence:

$0\longrightarrow
E(L_{\infty})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}\longrightarrow
S(E/L_{\infty})\longrightarrow\Sha(E/L_{\infty})[p^{\infty}]\longrightarrow
0.$

By additivity of co-rank of $\Lambda$-modules, we get that $r=0$ in equation
(8).

######  Example 4.5.

Let $p>3$ and $K=\mathbb{F}_{p}(t)$. Consider the family of elliptic curves
$E_{q}$ defined over $K$ given by the Weierstrass equation of the form

$y^{2}=x^{3}+t^{q}-t,$

where $q=p^{f}$ for some $f>0$. From, [GU20, Theorem 8.2] it follows that
$\Sha(E_{q}/L_{n})[p^{\infty}]$ is finite.

Now, considering the observations mentioned in Remark 4.4, we can conclude
that all the conditions specified in Corollary 4.3 are satisfied.

######  Proposition 4.6.

Let $L$ be a finite extension of $K$. Let $L_{\infty}/L$ be the arithmetic
$\mathbb{Z}_{p}$-extension. Assume that $E/L$ has split multiplicative
reduction or good reduction at all primes of $L$ and
$\Sha(E/L_{n})[p^{\infty}]$ is finite for all $n\geqslant 0$.

Also, suppose that $E$ has split multiplicative reduction at all primes of
$S$. Then,

$\operatorname{Char}_{\Lambda}(\mathcal{M}^{S}(E/L_{\infty}))^{\vee}=\left(\underset{e_{n}\geqslant
1,n\geqslant 0}{\prod}\Phi_{n}^{e_{n}}\right),$

where
$e_{n}=\dfrac{\operatorname{rank}E(L_{n})-\operatorname{rank}E(L_{n-1})}{{\varphi}(p^{n})}$.

###### Proof.

By using Remark 3.6 and the definition of $E(L_{n})^{\bullet}$ as in [Lei23,
§4], we have the following isomorphism of $\Lambda$-modules:

$E(L_{n})^{\bullet}\cong\bigoplus_{m=0}^{n}(\mathbb{Q}_{p}[X]/\Phi_{m})^{e_{m}}.$

Let $v\in S$ and $f_{n}:E(L_{n})^{\bullet}\longrightarrow
E(L_{n,v})^{\bullet}$ denote the natural map induced by inclusion
$E(L_{n})\hookrightarrow E(L_{n,v})$. Now, note that $E_{0}(L_{n,v})$, the
points of $E(L_{n,v})$ that do not reduce to a singular point, have finite
index in $E(L_{n,v})$. Hence, $E_{0}(L_{n,v})^{\bullet}\cong
E(L_{n,v})^{\bullet}$. Note that since $v\in S$,
$E_{0}(L_{n,v})\cong\mathcal{O}_{n,v}^{\times}$, the units in the ring of
integers $\mathcal{O}_{n,v}$ (Remark 2.11). By the structure of
$\mathcal{O}_{n,v}$, we get that $E_{0}(L_{n,v})^{\bullet}$ is isomorphic to
countable copies of $\mathbb{Q}_{p}$. Hence, we conclude that
$(\text{Image}f_{n})[\Phi_{n}]=0$.

Now, consider the following short exact sequences,

$0\longrightarrow M^{S}_{p^{k}}(E/L_{n})\longrightarrow
E(L_{n})/p^{k}\longrightarrow\underset{v\in S}{\bigoplus}E(L_{n,v})/p^{k}$

Taking inverse limit as $k$ varies and tensoring with $\mathbb{Q}_{p}$, we get

$0\longrightarrow V_{p}\mathcal{M}^{S}_{p^{k}}(E/L_{n})\longrightarrow
E(L_{n})^{\bullet}\overset{f_{n}}{\longrightarrow}\underset{v\in
S}{\bigoplus}E(L_{n,v})^{\bullet}$

Using Corollary 4.3, we get that

$V_{p}\mathcal{M}^{S}_{p^{k}}(E/L_{n})\cong\bigoplus_{m=0}^{n}(\mathbb{Q}_{p}[X]/\Phi_{m})^{s_{m}},$

where $s_{m}$ is the number of times $\Lambda/\Phi_{n}$ appears on the right-
hand side of the pseudo-isomorphism given by Corollary 4.1.

As $(\text{Image}f_{n})[\Phi_{n}]=0$, we get that $e_{m}=s_{m}$. This
completes the proof.

∎

######  Remark 4.7.

Note that the characteristic ideal of $\mathcal{M}^{S}(E/L_{\infty})$ depends
on the choice of $S$. In particular, as seen in the proof of Proposition 4.6,
we obtained $s_{m}=e_{m}$ by showing that $(\text{Image }f_{n})[\Phi_{n}]=0$.
This is obtained by observing the behavior of the elliptic curve $E$ at the
primes in $S$.

######  Remark 4.8.

The condition that $E/K$ has split multiplicative reduction or good reduction
at all primes of $K$ is trivially satisfied by constant elliptic curves (i.e.,
elliptic curves defined over a finite field).

Also note that for any elliptic curve $E$ over $K$, there exists a finite
extension $L/K$ such that $E$ has good or split multiplicative reduction at
all places of $L$ [Sil09, Proposition 5.4].

The following result is an obvious consequence of Proposition 4.6. Therefore,
we state it without proof:

######  Theorem 4.9.

Let $L$ be a finite extension of $K$. Let $L_{\infty}/L$ be the arithmetic
$\mathbb{Z}_{p}$-extension. Assume that $E/L$ has split multiplicative
reduction or good reduction at all primes of $L$ and
$\Sha(E/L_{n})[p^{\infty}]$ is finite for all $n\geqslant 0$.

Also, suppose that $E$ has split multiplicative reduction at all primes of $S$
and that ${}^{S}(E/L_{\infty})$ is finite. Then,

$\operatorname{Char}_{\Lambda}(R^{S}(E/L_{\infty})^{\vee})=\left(\underset{e_{n}\geqslant
1,n\geqslant 0}{\prod}\Phi_{n}^{e_{n}}\right)$

##  5\. Behaviour of $\mu$-invariants under isogeny ($\ell=p$ case).

Let $K=\mathbb{F}(t)$ be the function field over the finite field of char $p$.
Assume that $E_{1}/K$ and $E_{2}/K$ are two non-isotrivial elliptic curves
(i.e., $j(E_{i})\notin\mathbb{F}$ for $i=1,2$). Consider
${\varphi}:E_{1}\longrightarrow E_{2}$ as an isogeny of degree $p^{r}$ for
some $r>0$.

In this section, our goal is to consider the change in the $\mu$-invariant of
$p^{\infty}$-Selmer groups for $E_{1}$ and $E_{2}$ over $K_{\infty}$, the
unramified $\mathbb{Z}_{p}$-extension.

######  Theorem 5.1.

Let $E_{1}$ and $E_{2}$ be two non-isotrivial elliptic curves over the
function field $K$ and let ${\varphi}:E_{1}\longrightarrow E_{2}$ be an
isogeny of degree $p^{r}$ for some $r>0$. Also assume that
$H^{2}_{fl}(K_{\infty},E_{i}[p^{\infty}])=0$, for $i=1,2$.

Then,

$\mu(S(E_{2}/K_{\infty})^{\vee})-\mu(S(E_{1}/K_{\infty})^{\vee})=\operatorname{ord}_{p}(\chi_{fl}(Spec(K),A)),$

where $A=\ker({\varphi})$.

###### Proof.

Consider the following commutative diagrams:

${0}$${S(E_{2}/K_{\infty})}$${H^{1}(K_{\infty},E_{2}[p^{\infty}])}$${\operatorname{Im}(\lambda_{2})}$${0}$${0}$${S(E_{1}/K_{\infty})}$${H^{1}(K_{\infty},E_{1}[p^{\infty}])}$${\operatorname{Im}(\lambda_{1})}$${0}$$\scriptstyle{\lambda_{2}}$$\scriptstyle{f_{1}}$$\scriptstyle{\lambda_{1}}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{3}}$
(9)

Here, $\lambda_{i}$ denotes the maps:

$\lambda_{i}:H^{1}(K_{\infty},E_{i}[p^{\infty}])\longrightarrow\underset{v\in\Sigma_{K}}{\prod}J^{1}_{v}(E_{i}/K_{\infty}).$
${0}$${\operatorname{Im}(\lambda_{2})}$${\underset{v\in\Sigma_{K}}{\prod}J^{1}_{v}(E_{2}/K_{\infty})}$${\operatorname{coker}(\lambda_{2})}$${0}$${0}$${\operatorname{Im}(\lambda_{1})}$${\underset{v\in\Sigma_{K}}{\prod}J^{1}_{v}(E_{1}/K_{\infty})}$${\operatorname{coker}(\lambda_{1})}$${0}$$\scriptstyle{f_{3}}$$\scriptstyle{f_{4}}$$\scriptstyle{f_{5}}$
(10)

Let $\Gamma=\mathrm{Gal}(K_{\infty}/K)$. As $S(E_{i}/K_{\infty})^{\vee}$ is
torsion $\Lambda$-module for $i=1,2$, we get by Proposition 2.18 that

$\mu(\ker(f_{1})^{\vee})-\mu(\operatorname{coker}(f_{1})^{\vee})=\mu(S(E_{1}/K_{\infty})^{\vee})-\mu(S(E_{2}/K_{\infty})^{\vee})$

We start by showing that $\ker(f_{1})$ and $\operatorname{coker}(f_{1})$ are
both finite and therefore annihilated by a power of $p$. By Corollary 2.17, we
know that for a finitely generated $\Lambda$ module $M$,
$\mu(M)=\operatorname{ord}_{p}(\chi(\Gamma,M[p^{\infty}]^{\vee}))$. Therefore
it suffices to calculate the logarithm to the $p$-base of $\frac{\chi(\Gamma,\
\operatorname{coker}{f_{1}})}{\chi(\Gamma,\ \ker{f_{1}})}$, once we establish
the finiteness of $\ker(f_{1})$ and $\operatorname{coker}(f_{1})$.

By applying snake lemma to equation (9), we obtain the following exact
sequence:

$0\longrightarrow\ker(f_{1})\longrightarrow\ker(f_{2})\longrightarrow\ker(f_{3})\longrightarrow\operatorname{coker}(f_{1})\longrightarrow\operatorname{coker}(f_{2})\longrightarrow\operatorname{coker}(f_{3})\longrightarrow
0.$

Now, we claim that for $i=2,3$, $\ker(f_{i})$ and
$\operatorname{coker}(f_{i})$ are finite. This in turn will show that
$\ker(f_{1})$ and $\operatorname{coker}(f_{1})$ are finite. Following this, we
will conclude our calculation of $\frac{\chi(\Gamma,\
\operatorname{coker}{f_{1}})}{\chi(\Gamma,\ \ker{f_{1}})}$, by using the
relation

$\dfrac{\chi(\Gamma,\ \operatorname{coker}{f_{1}})}{\chi(\Gamma,\
\ker{f_{1}})}=\dfrac{\chi(\Gamma,\ \operatorname{coker}{f_{2}})}{\chi(\Gamma,\
\ker{f_{2}})}\times\dfrac{\chi(\Gamma,\ \ker{f_{3}})}{\chi(\Gamma,\
\operatorname{coker}{f_{3}})},$

which can be easily derived from equation (5).

Recall that the kernel of an isogeny is finite and its order divides the
degree of the isogeny. Hence, $A=\ker({\varphi}(p))$, where ${\varphi}(p)$ is
the map between $E_{1}[p^{\infty}]$ and $E_{2}[p^{\infty}]$, which is induced
by ${\varphi}$.

We claim that
$\chi(\Gamma,\ker(f_{3}))=\chi(\Gamma,\operatorname{coker}(f_{3}))=1$.
Consider the long-exact sequence:

$\displaystyle 0$ $\displaystyle\longrightarrow
S(E_{1}/K_{\infty})^{\Gamma}\longrightarrow
H^{1}(K_{\infty},E_{1}[p^{\infty}])^{\Gamma}\longrightarrow(\operatorname{im}(\lambda_{1}))^{\Gamma}\longrightarrow
H^{1}(\Gamma,S(E_{1}/K_{\infty}))$ $\displaystyle\longrightarrow
H^{1}(\Gamma,H^{1}(K_{\infty},E_{1}[p^{\infty}]))\longrightarrow
H^{1}(\Gamma,\operatorname{im}(\lambda_{1}))\longrightarrow 0.$

The assumption $H^{2}_{fl}(K_{\infty},E_{1}[p^{\infty}])=0$ implies that
$H^{1}(\Gamma,H^{1}(K_{\infty},E_{1}[p^{\infty}]))=0$ ([BV18, Proposition
2.2]). Hence, $H^{1}(\Gamma,\operatorname{Im}(\lambda_{1}))=0$.

Again, consider the exact sequence

$\displaystyle 0$
$\displaystyle\longrightarrow(\operatorname{Im}(\lambda_{1}))^{\Gamma}\longrightarrow(\underset{v\in\Sigma_{K}}{\prod}J^{1}_{v}(E_{1}/K_{\infty}))^{\Gamma}\longrightarrow(\operatorname{coker}(\lambda_{1}))^{\Gamma}\longrightarrow
H^{1}(\Gamma,\operatorname{Im}(\lambda_{1}))$ $\displaystyle\longrightarrow
H^{1}(\Gamma,\underset{v\in\Sigma_{K}}{\prod}J^{1}_{v}(E_{1}/K_{\infty}))\longrightarrow
H^{1}(\Gamma,\operatorname{coker}(\lambda_{1}))\longrightarrow 0$

By [BV18, Remark 2.7(3)], we know that

$\operatorname{coker}\big{(}(\operatorname{Im}(\lambda_{1}))^{\Gamma}\longrightarrow(\underset{v\in\Sigma_{K}}{\prod}J^{1}_{v}(E_{1}/K_{\infty}))^{\Gamma}\big{)}=0.$

This along with the fact that $H^{1}(\Gamma,\operatorname{Im}(\lambda_{1}))=0$
implies that $(\operatorname{coker}(\lambda_{1}))^{\Gamma}=0$. Hence,
$(\ker(f_{5}))^{\Gamma}=0$.

As $K_{\infty}$ is unramified at every prime $v$, we obtain that for any prime
$v$ and $w\mid v$,
$\mathrm{Gal}(\overline{K_{\infty,w}}/K_{\infty,w})\cong\underset{\ell\neq
p}{\prod}\mathbb{Z}_{\ell}$. Therefore, $\ker
f_{4}=\operatorname{coker}f_{4}=0$.

By applying a snake lemma on equation (10), we obtain that $\ker(f_{3})=0$ and
$\ker(f_{5})\cong\operatorname{coker}(f_{3})$. Since,
$(\operatorname{coker}(\lambda_{1}))^{\Gamma}=0$, hence
$(\ker(f_{5}))^{\Gamma}=(\operatorname{coker}(f_{3})^{\Gamma})=0$.

By imitating the arguments above, the assumption
$H^{2}_{fl}(K_{\infty},E_{2}[p^{\infty}])$ implies that
$H^{1}(\Gamma,\operatorname{Im}(\lambda_{2}))=0$ as well. Now, from the long
exact sequence corresponding to the short exact sequence

$0\longrightarrow\ker(f_{3})\longrightarrow\operatorname{Im}(\lambda_{1})\longrightarrow\operatorname{Im}(\lambda_{2})\longrightarrow\operatorname{coker}(f_{3})\longrightarrow
0,$

we obtain that $H^{1}(\Gamma,\operatorname{coker}(f_{3}))=0$. This proves over
claim that
$\chi(\Gamma,\ker(f_{3}))=\chi(\Gamma,\operatorname{coker}(f_{3}))=1$.

Now, considering the long exact sequence of flat cohomology corresponding to
the short exact sequence $0\longrightarrow A\longrightarrow
E_{1}[p^{\infty}]\longrightarrow E_{2}[p^{\infty}]\longrightarrow 0$ and using
the fact $H^{2}_{fl}(K_{\infty},E_{2}[p^{\infty}])=0$, we get

$\dfrac{\chi(\Gamma,\operatorname{coker}(f_{1})}{\chi(\Gamma,\ker(f_{1}))}=\dfrac{\chi(\Gamma,\operatorname{coker}(f_{2})}{\chi(\Gamma,\ker(f_{2}))}=\dfrac{\chi(\Gamma,H^{2}_{fl}(K_{\infty},A))\times\chi(\Gamma,E_{1}[p^{\infty}](K_{\infty}))\times\chi(\Gamma,H^{0}_{fl}(K_{\infty},A))}{\chi(\Gamma,H^{1}_{fl}(K_{\infty},A)\times\chi(\Gamma,E_{2}[p^{\infty}])}$
(11)

Since, $E_{1}$ and $E_{2}$ are non-iostrivial, hence we get that
$E_{1}(K_{\infty})[p^{\infty}]$ and $E_{2}(K_{\infty})[p^{\infty}]$ are finite
(Remark 2.13). Also note that $A$ is finite. Therefore,
$\chi(\Gamma,E_{1}[p^{\infty}](K_{\infty}))=\chi(\Gamma,E_{2}[p^{\infty}](K_{\infty}))=\chi(\Gamma,H^{0}_{fl}(K_{\infty},A))=1$.

Using the Hochschild-Serre spectral sequence ( [Mil80]) and its connection
with the Euler characteristic, as mentioned in Remark 2.20, in equation (11),
we derive our required result. ∎

######  Remark 5.2.

Let $E/K$ be a non-isotrivial elliptic curve and $q=p^{r}$, for some $r>0$.
Consider the elliptic curve $E^{(q)}$, which is obtained by raising the
coefficients in Weierstrass equation of $E$ by a power of $q$. Clearly,
$E^{(q)}$ is also non-isotrivial.

Let us call $E$ as $E_{1}$ and $E^{(q)}$ as $E_{2}$. The obvious isogeny map
${\varphi}$ between $E_{1}$ and $E_{2}$ is the Frobenius isogeny, $Fr_{q}$.
This is a purely inseparable isogeny. As we know that the order of the kernel
of isogeny equals its separable degree, we get that
$\mu(S(E_{2}/K_{\infty})^{\vee})=\mu(S(E_{1}/K_{\infty})^{\vee})$.

##  6\. Vanishing of $\mu_{G}(S(E/K_{\infty})^{\vee})$ in the $\ell\neq p$
case

Let $p$ be an odd prime and $\ell\geqslant 5$ be a prime distinct from $p$.
Set $K=\mathbb{F}(t)$, where $\mathbb{F}$ is a field of char $p$ as defined in
§2.8. Let $E$ be a non-isotrivial elliptic curve over the function field $K$.
Let $K_{\infty}$ be the trivialising extension of $K$ (defined in §2.8). As a
consequence of Weil pairing, we get that $K_{\mathrm{cyc}}\subset K_{\infty}$.
Denote by $G=\operatorname{Gal}(K_{\infty}/K)$ and
$H=\operatorname{Gal}(K_{\infty}/K_{\mathrm{cyc}})$.

The proof of the following result is similar to that of [CSS03a, Lemma 2.5].
We recall the key steps below:

######  Proposition 6.1.

Let $G,H$ be as above, and $E/K$ be a non-isotrivial elliptic curve. Then
$H^{i}(H,S(E/K_{\infty}))=0,$ for $i\geqslant 1$.

###### Proof.

Recall that the maps

$\lambda(\mathcal{K}):H^{1}(G_{S}(\mathcal{K}),E[\ell^{\infty}])\longrightarrow\underset{v\in
S}{\prod}J^{1}_{v}(E/\mathcal{K})$

are surjective for $\mathcal{K}=K_{\mathrm{cyc}}\text{ or
}\mathcal{K}=K_{\infty}$ (see [Pal14, proof of Theorem 4.4]). Now, by
arguments similar to the proof of [CSS03a, Lemma 2.3], we obtain the following
exact sequence:

$0\longrightarrow S(E/K_{\infty})^{H}\longrightarrow
H^{1}(G_{S}(K_{\infty}),E[\ell^{\infty}])^{H}\longrightarrow\underset{v\in
S}{\prod}J^{1}_{v}(E/K_{\infty})^{H}\longrightarrow 0$ (12)

Then, by using Hochschild-Serre spectral sequence and the finiteness of
$H^{3}(H,E[\ell^{\infty}])$, which follows from [Sec07, Chapter-2, Remark
3.14], we obtain as in [CSS03a, Lemma 2.4] that
$H^{i}(H,H^{1}(G_{S}(F_{\infty}),E[\ell^{\infty}]))=0$ for all $i\geqslant 1$.
By [Sec07, Chapter-1, Lemma III.7], we obtain that
$H^{i}(H,J^{1}_{v}(E/K_{\infty}))=0$. The result now easily follows by taking
the $H$-cohomology of the short exact sequence

$0\longrightarrow S(E/K_{\infty})\longrightarrow
H^{1}(G_{S}(K_{\infty}),E[\ell^{\infty}])\longrightarrow\underset{v\in
S}{\prod}J^{1}_{v}(E/K_{\infty})\longrightarrow 0,$

and considering the corresponding long exact sequence. ∎

######  Proposition 6.2.

Let $E/K$ be a non-isotrivial elliptic curve and
$G=\operatorname{Gal}(K_{\infty}/K)$ a pro-$p$ group. Then,
$\mu_{G}(S(E/K_{\infty})^{\vee})=\mu_{\Gamma}(S(E/K_{\mathrm{cyc}})^{\vee})$.

###### Proof.

By [Pal14, Theorem 4.4], we obtain that the kernel of the map
$S(E/K_{\mathrm{cyc}})\overset{f}{\rightarrow}S(E/K_{\infty})^{H}$ is finite
and the cokernel of $f$ is a co-finitely generated $\mathbb{Z}_{\ell}$-module.
Furthermore, we know from [Wit20, Corollary 4.38] that
$S(E/K_{\mathrm{cyc}})^{\vee}$ is a finitely generated
$\mathbb{Z}_{\ell}$-module. Then, it follows from Nakayama lemma that
$S(E/K_{\infty})^{\vee}$ is finitely generated over $\Lambda(H)$. The proof
now follows by using arguments similar to the proof of [CSS03b, Proposition
2.13]. ∎

As discussed in [GJS22, §2.1], we know that $S(E/K_{\mathrm{cyc}})^{\vee}$ is
a finitely generated $\mathbb{Z}_{\ell}$-module. Therefore, as an immediate
consequence of Proposition 6.2, we obtain the following theorem:

######  Theorem 6.3.

Let $K_{\mathrm{cyc}}$ be the unique $\mathbb{Z}_{\ell}$-extension of $K$ and
$K_{\infty}$ be the trivialising extension of $K$, such that
$G=\operatorname{Gal}(K_{\infty}/K)$ is pro-$p$. Assume $E$ to be a non-
isotrivial elliptic curve over the function field $K$. Then,

$\mu_{G}(S(E/K_{\infty})^{\vee})=\mu_{\Gamma}(S(E/K_{\mathrm{cyc}})^{\vee})=0.$

##  Appendix A Behaviour of $\mu$-invariants under isogeny ($\ell\neq p$ case)

We choose and fix a rational prime $\ell$ distinct from $p$. Set
$K=\mathbb{F}(t)$, where $\mathbb{F}$ is a field of char $p$ as defined in
§2.8. Now, let $E_{1}$ and $E_{2}$ are two non-isotrivial elliptic curves over
the function field $K$. Let ${\varphi}:E_{1}\longrightarrow E_{2}$ be an
isogeny of any degree. Let $\mathcal{K}$ be either the unramified
$\mathbb{Z}_{\ell}$-extension $K_{\mathrm{cyc}}$ or the trivialising extension
$K_{\infty}$.

In this appendix, although we don’t get any new result, we show that the
arguments given in the proof of Theorem 5.1 in the $\ell=p$ case also carries
over in the $\ell\neq p$ case. Infact in this case we don’t get any "extra"
factor of the Euler-characteristic and obtain an equality of the
$\mu$-invariants of the (fine) Selmer groups under isogeny, and thus
confirming the validity of Theorem 1.5.

Consider the following commutative diagram:

${0}$${S(E_{2}/\mathcal{K})}$${H^{1}(G_{S}(\mathcal{K}),E_{2}[\ell^{\infty}])}$${\underset{v\in
S}{\prod}J^{1}_{v}(E_{2}/\mathcal{K})}$${0}$${0}$${S(E_{1}/\mathcal{K})}$${H^{1}(G_{S}(\mathcal{K}),E_{1}[\ell^{\infty}])}$${\underset{v\in
S}{\prod}J^{1}_{v}(E_{1}/\mathcal{K})}$${0}$$\scriptstyle{g_{1}}$$\scriptstyle{g_{2}}$$\scriptstyle{g_{3}=\underset{v\in
S}{\prod}g_{3,v}}$ (13)

Again, our goal is to consider the change in the $\mu$-invariant of
$\ell^{\infty}$-Selmer groups of $E_{1}$ and $E_{2}$ over $K_{\mathrm{cyc}}$
and $K_{\infty}$.

######  Theorem A.1.

Consider $\mathcal{K}=K_{\mathrm{cyc}}\text{ or }K_{\infty}$. Assume $E_{1}$
and $E_{2}$ are two non-isotrivial elliptic curves over the function field
$K$, and ${\varphi}:E_{1}\longrightarrow E_{2}$ is an isogeny. Also, suppose
that $S(E_{i}/\mathcal{K})^{\vee}$ is a torsion $\Lambda(G)$-module for
$i=1,2$. Then,

$\mu(S(E_{1}/\mathcal{K})^{\vee})=\mu(S(E_{2}/\mathcal{K})^{\vee}).$

###### Proof.

Let $G=\mathrm{Gal}(\mathcal{K}/K)$. We start by calculating
$\mu(\ker(f_{1})^{\vee})-\mu(\operatorname{coker}(f_{1})^{\vee})$. Note that
by our assumption that $S(E_{i}/\mathcal{K})^{\vee}$ is torsion
$\Lambda(G)$-module for $i=1,2$, we get by Proposition 2.18 that

$\mu(\ker(f_{1})^{\vee})-\mu(\operatorname{coker}(f_{1})^{\vee})=\mu(S(E_{1}/\mathcal{K})^{\vee})-\mu(S(E_{2}/\mathcal{K})^{\vee})$

Recall that for a finitely generated $\Lambda$ module $M$,
$\mu(M)=\operatorname{ord}_{p}(\chi(\Gamma,M[p^{\infty}]^{\vee}))$ (Corollary
2.17). Now, suppose $\ker(g_{1})$ and $\operatorname{coker}(g_{1})$ are both
finite and therefore annihilated by a power of $p$. Then, it suffices to
calculate the logarithm to the $p$-base of $\frac{\chi(G,\
\operatorname{coker}{g_{1}})}{\chi(G,\ \ker{g_{1}})}$.

By applying snake lemma to equation (13), we obtain the following exact
sequence:

$0\longrightarrow\ker(g_{1})\longrightarrow\ker(g_{2})\longrightarrow\ker(g_{3})\longrightarrow\operatorname{coker}(g_{1})\longrightarrow\operatorname{coker}(g_{2})\longrightarrow\operatorname{coker}(g_{3})\longrightarrow
0.$

Now, we claim that for $i=2,3$, $\ker(g_{i})$ and
$\operatorname{coker}(g_{i})$ are finite, which shows that $\ker(g_{1})$ and
$\operatorname{coker}(g_{1})$ are finite. We establish the finiteness of
$\ker(g_{1})$ and $\operatorname{coker}(g_{1})$ and calculate $\frac{\chi(G,\
\operatorname{coker}{g_{1}})}{\chi(G,\ \ker{g_{1}})}$ separately for
$K_{\mathrm{cyc}}$ and $K_{\infty}$. Note that the equation (A) also implies
that $\dfrac{\chi(G,\ \operatorname{coker}{g_{1}})}{\chi(G,\
\ker{g_{1}})}=\dfrac{\chi(G,\ \operatorname{coker}{g_{2}})}{\chi(G,\
\ker{g_{2}})}\times\dfrac{\chi(G,\ \ker{g_{3}})}{\chi(G,\
\operatorname{coker}{g_{3}})}.$

Case 1: Take $\mathcal{K}=K_{\mathrm{cyc}}$. Then,
$G=\Gamma=\mathrm{Gal}(K_{\mathrm{cyc}}/K)$. Let $A=\ker({\varphi}(\ell))$,
the $\ell$-primary part of $\ker({\varphi})$. From the long exact sequence
corresponding to the short exact sequence $0\longrightarrow A\longrightarrow
E_{1}[\ell^{\infty}]\longrightarrow E_{2}[\ell^{\infty}]\longrightarrow 0$ and
using the fact $H^{2}(K_{\mathrm{cyc}},E_{2}[\ell^{\infty}])=0$([GJS22,
Proposition 2.10]), we get that

$\operatorname{coker}(g_{2})=H^{1}(G_{S}(K),A)\text{ and
}\ker(g_{2})=\dfrac{H^{1}(G_{S}(K),A)}{\Big{(}\frac{{E_{2}[\ell^{\infty}](K_{\mathrm{cyc}})}}{{(E_{1}[\ell^{\infty}](K_{\mathrm{cyc}})/H^{0}(G_{S}(K),A)}}\Big{)}}.$

Since, $E_{1}$ and $E_{2}$ are non-iostrivial, hence we get that
$E_{1}[\ell^{\infty}](K_{\mathrm{cyc}})$ and
$E_{2}[\ell^{\infty}](K_{\mathrm{cyc}})$ are finite ([BL09b, Lemma 3.2]).
Hence, $\ker(g_{2})$ and $\operatorname{coker}(g_{2})$ are finite.

As $K_{\mathrm{cyc}}$ is unramified at every prime $v$, we obtain that for any
prime $v$ and $w\mid v$,
$\mathrm{Gal}(\overline{K_{\mathrm{cyc},w}}/K_{\mathrm{cyc},w})\cong\underset{r\neq\ell}{\prod}\mathbb{Z}_{r}$.
Therefore, $\ker g_{3}$ and $\operatorname{coker}g_{3}$ are trivial.

As $\ker g_{i}$ and $\operatorname{coker}g_{i}$ are finite for $i=1,2$, we get
that $\chi(\Gamma,\ker g_{j})$ and $\chi(\Gamma,\operatorname{coker}g_{j})$
exists for $j=1,2,3$. Therefore,

$\dfrac{\chi(\Gamma,\operatorname{coker}(g_{2})}{\chi(\Gamma,\ker(g_{2}))}=\dfrac{\chi(\Gamma,H^{2}(G_{S}(K_{\infty}),A))\times\chi(\Gamma,E_{1}[\ell^{\infty}](K_{\infty}))\times\chi(\Gamma,H^{0}(G_{S}(K_{\infty}),A))}{\chi(\Gamma,H^{1}(G_{S}(K_{\infty}),A)\times\chi(\Gamma,E_{2}[\ell^{\infty}])}$
(14)

As $E_{1}[\ell^{\infty}](K_{\infty})$ and $E_{2}[\ell^{\infty}](K_{\infty})$
and $A$ are all finite. Therefore,
$\chi(\Gamma,E_{1}[\ell^{\infty}](K_{\mathrm{cyc}}))=\chi(\Gamma,E_{2}[\ell^{\infty}](K_{\mathrm{cyc}}))=\chi(\Gamma,H^{0}(G_{S}(K_{\mathrm{cyc}}),A))=1$.

Now, using the Hochschild-Serre spectral sequence,

$H^{i}(\Gamma,H^{j}(G_{S}(K_{\mathrm{cyc}}),A))\implies H^{i+j}(G_{S}(K),A),$

in equation (14). Therefore,

$\dfrac{\chi(\Gamma,\operatorname{coker}(f_{2}))}{\chi(\Gamma,\ker(f_{2}))}=\chi(\mathrm{Gal}(K_{S}/K),A).$

By [Mil86, Remark I.5.2], we get that $\chi(\mathrm{Gal}(K_{S}/K),A)=1$. This
concludes Case 1.

Case 2: Take $\mathcal{K}=K_{\infty}$ and
$G=G_{\infty}=\mathrm{Gal}(K_{\infty}/K)$. Again, let
$A=\ker({\varphi}(\ell))$, the $\ell$-primary part of $\ker({\varphi})$.
Similar to Case 1, we obtain from the long exact sequence corresponding to the
short exact sequence $0\longrightarrow A\longrightarrow
E_{1}[\ell^{\infty}]\longrightarrow E_{2}[\ell^{\infty}]\longrightarrow 0$ and
$H^{2}(K_{\infty},E_{2}[\ell^{\infty}])=0$ ([BV14, Proposition 4.5]) that

$\operatorname{coker}(g_{2})=H^{1}(G_{S}(K),A)\text{ and
}\ker(g_{2})=H^{0}(G_{S}(K),A).$

Let $S_{\text{un}}\subset S$ be the set of primes of $K$ is $S$ that are
unramified in $K_{\infty}$. For $v\in S_{\text{un}},\ \ker(g_{3,v})=0$, by
arguments similar to Case 1. And for $v\in S\setminus S_{\text{un}},\
\ker(g_{3,v})=\underset{w\mid v}{\prod}H^{1}(K_{\infty,w},A)$ and
$\operatorname{coker}(g_{3,v})\subset\underset{w\mid
v}{\prod}H^{2}(K_{\infty,w},A)$. As $cd_{\ell}(K_{\infty,w})=1$ ([NSW13,
Theorem 7.1.8]), we get that $\operatorname{coker}(g_{3,v})=0$. Therefore,
$\ker(g_{3})=\underset{v\in S\setminus S_{\text{un}}}{\prod}\ker(g_{3,v})$ and
$\operatorname{coker}(g_{3})=0$.

For each prime $v\in S$ we fix a choice of prime $K_{\infty}$ lying above,
which we will also denote by $v$. It follows from Shapiro’s lemma that

$H^{j}(G_{\infty},\ker(g_{3,v}))\cong
H^{j}(G_{\infty,v},H^{i}(K_{\infty,v},A)).$

Thus, $\chi(G_{\infty},\ker({\varphi}_{3}))=\underset{v\in S\setminus
S_{\text{un}}}{\prod}\chi(G_{\infty,v},H^{1}(K_{\infty,v},A))$.

Again, using the Hochschild-Serre spectral sequence, we obtain

$H^{i}(G_{\infty,v},H^{j}(K_{\infty,v},A))\implies H^{i+j}(K_{v},A)$

Thus,

$\dfrac{\chi(G_{\infty},\ker({\varphi}_{3}))}{\chi(G_{\infty},\operatorname{coker}({\varphi}_{3}))}=\dfrac{1}{\underset{v\in
S\setminus
S_{\text{un}}}{\prod}\chi(\mathrm{Gal}(\overline{K}_{v}/K_{v}),A)}.$

By [Mil86, Chapter 1,Lemma 2.9],we know that
$\chi(\mathrm{Gal}(\overline{K}_{v}/K_{v}),A)=1$. Moreover, using calculations
similar to Case 1, we obtain that
$\dfrac{\chi(G_{\infty},\operatorname{coker}(g_{2}))}{\chi(G_{\infty},\ker(g_{2}))}=1.$

Hence, in both cases, we obtain that $\dfrac{\chi(G,\
\operatorname{coker}{g_{1}})}{\chi(G,\ \ker{g_{1}})}=1,$ for $G=G_{\infty}$ or
$\Gamma$. This concludes the proof of the theorem. ∎

######  Remark A.2.

  1. (i)

For an arbitary abelian variety $A$, $S(A/K_{\mathrm{cyc}})^{\vee}$ is a
finitely generated torsion $\Lambda$-module [Pal14, Theorem 3.12].

  2. (ii)

By [Wit20, Corollary 4.38], $S(E/K_{\mathrm{cyc}})^{\vee}$ is a finitely
generated $\mathbb{Z}_{\ell}$-modules, then by [Pal14, Theorem 4.4]
$S(E/K_{\infty})^{\vee}$ is a torsion $\Lambda(G)$-module.

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