[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
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.

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

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

SOHAN GHOSH AND JISHNU RAY

see §2.1) is a subgroup of the classical Selmer group, which is always co-torsion as a Λ-module (cf. [OT09, Theorem 1.7]).

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

The strategy of the proof is to define the S-fine Mordell-Weil group, 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 → M^S(A/L) → R^S(A/L) → Ж^S(A/L) → 0,

where Ж^S(A/L) denotes the S-fine Tate-Shafarevich group (defined in §2.3). Note that the Ж^S(A/L) is the subgroup of the Ш(A/L)[p^∞], the p-primary part of the Tate-Shafarevich group (see Remark 2.7).

For each n ≥ 1, let Φ_n = ((1+T)^{p^n}-1)/((1+T)^{p^{n-1}}-1) ∈ Λ be the p^n-th cyclotomic polynomial in 1+T. Let K_n be the subextension of K_∞ 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_∞/L be the arithmetic Z_p-extension. Assume that E/L has split multiplicative reduction or good reduction at all primes of L and Ш(E/L_n)[p^∞] is finite for all n ≥ 0.

Also, suppose that E has split multiplicative reduction at all primes of S and that Ж^S(E/L_∞) is finite. Then,

Char_Λ(R^S(E/L_∞)^∨) = ( ∏_{e_n≥1, n≥0} Φ_n^{e_n} )

where e_n = (rank E(L_n) - rank E(L_{n-1})) / φ(p^n).

This theorem has the following applications.

First, under the hypothesis that Ж^S(E/K_∞) is finite one can see easily that the μ-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_∞). 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_∞)^∨ and R^S(E/K_∞)^{∨,ι} as Λ-modules are equal, i.e. there is a pseudo-isomorphism R^S(E/K_∞)^∨ ∼ R^S(E/K_∞)^{∨,ι}.

Here ι is the involution on Λ sending a group-like element of Γ = Gal(K∞/K) to its inverse. For any Λ-module M, we write M^ι for the Λ-module which coincides with M as a Zₚ-module, with the action of Γ given by

γ ·ι x = γ⁻¹x for γ ∈ Γ and x ∈ M.

Remark 1.3. We don't know how to prove the result in Corollary 1.2 without the assumption that Ж^S(E/K∞) is finite. Even in the number field case over the cyclotomic Zₚ-extension of Q, such a result is not known without this additional hypothesis that Ж(E/Q_cyc) is finite (cf. [Lei23, discussion after Theorem C]). Infact, it is conjectured by Wuthrich that Ж(E/Q_cyc) should be finite [Wut07, Question 8.3 and Conjecture 8.4]. For classical p^∞-Selmer groups over Q∞, 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 Kₙ be the finite subextension of K∞ such that [Kₙ : K] = pⁿ. Then we know that eₙ = μ · pⁿ + λ · n + O(1) where |Ж^S(A/Kₙ)| = p^{eₙ}. Because of Conjecture A (which is true without the finiteness assumption of Ж^S(A/K∞)), we know that μ = 0. Additionally, if know that the growth of the Tate-Shafarevich group Ш(A/Kₙ)[p^∞] is of the kind p^{μ₁·pⁿ+c} (where c is a constant independent of n), then we can conclude that λ = 0 and hence the growth of Ж^S(A/Kₙ) stays bounded. In this case Ж^S(E/K∞) 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 (ℓ ≠ p case). Let ℓ be a rational prime, with ℓ ≠ p. Consider K = 𝔽(t), where 𝔽 is a finite field of char p. The ℓ^∞-Selmer group (defined in 2.8) of an elliptic curve E coincides with the ℓ^∞-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 μ-invariant of the fine Selmer) over the non-commutative ℓ-adic trivializing extension K∞ of K. The Galois group of K∞ over K is an open subgroup of GL₂(Z_ℓ) (cf. [Pal14, §4]). In particular we show the following result.

Theorem 1.5. (Theorem 6.3) Let K_cyc be the unique Z_ℓ-extension of K and K∞ be the trivialising extension of K, such that G = Gal(K∞/K) is pro-p. Assume E to be a non-isotrivial elliptic curve over the function field K. Then,
μ_G((S(E/K∞)∨)) = μ_Γ((S(E/K_cyc)∨)) = 0.

1.3. Isogeny and μ-invariants (ℓ = p case). We again return to the setting of §1.1. Unlike the ℓ ≠ p case, here the Selmer and fine Selmer groups are distinct. Our goal in this section is to give a lower bound of μ-invariant of

the Selmer group over the unramified 𝑍ₚ-extension 𝐾_∞ over 𝐾. The strategy that we adopt is to compare the μ-invariants of isogenous elliptic curves. More precisely, we show the following result.

Theorem 1.6. (Theorem 5.1) Let 𝐸₁ and 𝐸₂ be two non-isotrivial elliptic curves over the function field 𝐾 and let φ : 𝐸₁ → 𝐸₂ be an isogeny of degree pʳ for some r > 0. Also assume that H²_fl(𝐾_∞, 𝐸ᵢ[p^∞]) = 0, for i = 1, 2.

Then,
μ(S(𝐸₂/𝐾_∞)^∨) − μ(S(𝐸₁/𝐾_∞)^∨) = ordₚ(χ_fl(Spec(𝐾), 𝐴)),

where 𝐴 = ker(φ) and χ_fl(Spec(𝐾), 𝐴) is the Euler characteristic for flat cohomology defined in §2.7.

It follows that if ordₚ(χ_fl(Spec(𝐾), 𝐴)) is positive, one can obtain examples of elliptic curves with positive μ-invariant over 𝐾_∞.

Remark 1.7. (i) In [LLS⁺21], the authors calculate the change in μ-invariant with isogeny for Selmer groups of elliptic curves having semi-stable reduction everywhere, for the unramified Zₚ-extension 𝐾_∞/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 μ-invariant with pʳ-isogeny of elliptic curves, with the additional assumption that H²_fl(𝐾_∞, 𝐸ᵢ[p^∞]) = 0. Infact our technique is much more general and can also be adapted in the ℓ ≠ p case both for the unramified cyclotomic Z_ℓ-extension and for a non-commutative GL₂(Z_ℓ) extension (see Appendix A). However we don’t get any new result in the ℓ ≠ p case because of the stronger result that we showed earlier concerning the triviality of the μ-invariant (see Theorem 1.5).

(ii) The assumption H²_fl(𝐾_∞, 𝐸ᵢ[p^∞]) = 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 ℓ = p case, the Selmer group and the S-fine Selmer group over the unramified Zₚ-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 Φₙ appearing in Theorem 1.1 is eₙ, unlike the number field case where the power is given by eₙ − 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 \(\mathcal{M}_H(G)\) 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 \(G = \mathrm{Gal}(K_\infty/K)\) and \(H = \mathrm{Gal}(K_\infty/K_{\mathrm{cyc}})\)) 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) Denote by \(B[p^n]\) the \(p^n\)-torsion points of \(B\) and let \(B[p^\infty] = \bigcup_n B[p^n]\).

(2) Let \(T_p B := \varprojlim_k B[p^k]\), the Tate module of \(B\).

(3) The \(p\)-adic completion of \(B\) is defined as \(B^* = \varprojlim_n B/p^n B\).

(4) Also, define \(B^\bullet := B^* \otimes \mathbb{Q}_p\) 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

\[
J^1_v(A/L) := \prod_{w|v} \frac{H^1_{fl}(L_w, A[p^\infty])}{\operatorname{im}(\kappa_w)} \text{ and } K^1_v(A/L) := \prod_{w|v} H^1_{fl}(L_w, A[p^\infty]).
\]

Here \( H^i_{fl}(-,-) \) 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 \( \Sigma_K, S \) and \( K \subset L \subset K_S \) be as above. Then the Selmer group \( S(A/L) \) is defined as:

\[
S(A/L) := \ker \left(H^1_{fl}(L, A[p^\infty]) \longrightarrow \prod_{v\in\Sigma_K} J^1_v(A/L)\right).  \tag{1}
\]

We define the S-fine Selmer group as:

\[
R^S(A/L) := \ker \left(H^1_{fl}(L, A[p^\infty]) \longrightarrow \bigoplus_{v\in S} K^1_v(A/L) \prod_{v\in\Sigma_K\setminus S} J^1_v(A/L)\right)
\]
\[
\cong \ker \left(S(A/L) \longrightarrow \bigoplus_{w|v,\ v\in S} A(L_w) \otimes \mathbb{Q}_p/\mathbb{Z}_p\right).  \tag{2}
\]

**Remark 2.3 (Dependence on \( S \)).** Let \( A, S, K, U \) be as above and let \( \mathcal{A} \) be the Néron model of \( A \) over \( C_K \). Recall the following equivalent definition [KT03] of the Selmer group:

\[
S(A/K) := \ker \left(H^1_{fl}(U, \mathcal{A}[p^\infty]) \longrightarrow \bigoplus_{v\in S} J^1_v(A/K)\right).  \tag{3}
\]

Using definitions 2 and 3, we have

\[
R^S(A/K) := \ker \left(H^1_{fl}(U, \mathcal{A}[p^\infty]) \longrightarrow \bigoplus_{v\in S} K^1_v(A/K)\right).  \tag{4}
\]

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]:

\[
0 \to H^1_{fl}(U, \mathcal{A}[p^\infty]) \longrightarrow H^1_{fl}(K, A[p^\infty]) \longrightarrow \bigoplus_{v\in U} H^1_{fl}(K_v, A[p^\infty])/H^1_{fl}(\mathcal{O}_v, A[p^\infty]), \tag{5}
\]

where \( \mathcal{O}_v \) is the valuation ring of \( K_v \) and \( H^1_{fl}(\mathcal{O}_v, A[p^\infty]) \cong A(K_v) \otimes \mathbb{Q}_p/\mathbb{Z}_p \).

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

**Remark 2.4.** Consider a \( p \)-adic Lie extension \( L_\infty/K \) where \( G = \operatorname{Gal}(L_\infty/K) \) 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 $P(t) = a_nt^n + \cdots + a_0 \in \mathbb{F}[t]$. We define the Carlitz polynomial $[P(t)](X)$ with coefficients in $\mathbb{F}[t]$ recursively as follows:

$1 = X,$
$t = X^p + tX,$
$t^n = t)$ and
$a_nt^n + \cdots + a_1t + a_0 = a_nt^n + \cdots + a_1t + a_0(X).$

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

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 $\widetilde{K} := \bigcup_{n \geq 1} K(\Lambda_{\mathfrak{P}^n})$, 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 $d \geq 1$ 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 $k \geq 1$ and an extension $L/K$, the $p^k$-$S$-fine Mordell-Weil group of $A/L$, $M_{p^k}^S(A/L)$ is defined as:
$$M_{p^k}^S(A/L) = \ker \left( A(L)/p^k \to \bigoplus_{v|S} A(L_v)/p^kA(L_v) \right)$$

The $p$-primary $S$-fine Mordell-Weil group of $A/L$ is given by:
$$\mathcal{M}^S(A/L) = \varinjlim_k M_{p^k}^S(A/L) = \ker \left( A(L) \otimes \mathbb{Q}_p/\mathbb{Z}_p \to \bigoplus_{v|S} A(L_v) \otimes \mathbb{Q}_p/\mathbb{Z}_p \right).$$

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:

$$\mathfrak{X}^S(A/L) := \frac{R^S(A/L)}{M^S(A/L)}.$$

Hence, we have the following short exact sequence

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

Remark 2.7. Let k \geq 1 and L be a finite extension of K. Note that $M^S(A/L)$ 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 $\mathfrak{X}^S(A/L) \subset \Sha(A/L)[p^\infty]$.

Remark 2.8. By Remark 2.7, we observe that for any separable extension L of K, the finiteness of $\mathfrak{X}^S(A/L)$ follows from finiteness of $\Sha(A/L)[p^\infty]$. The finiteness of $\Sha(A/L)$ (i.e., the BSD conjecture) has been established 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 $n \geq 0$, define $\omega_n = (1+T)^{p^n}-1$ and $\omega_{0,-1} := T$.

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

$$\mathfrak{G}:= \varprojlim_n \left(\frac{X}{\omega_n X}[p^\infty]\right).$$

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

Let $\Phi_n := \frac{\omega_n}{\omega_{n-1}}$ for $n \geq 1$ and $\Phi_0 = X$. The functor $\mathfrak{G}$ has the following properties [Lee20, Lemma A.2.9].

(1) $\mathfrak{G}(\Lambda) = 0$. Hence, $\mathfrak{G}(X)$ is a torsion $\Lambda$-module.
(2) $\mathfrak{G}(\Lambda/g^e) = \Lambda/g^e$ if g is coprime to $\omega_n$ for all n.
(3) For $m \geq 0$,
$$\mathfrak{G}(\Lambda/\Phi_m^e) = \begin{cases}
\Lambda/\Phi_m^{e-1}, & e \geq 2 \\
0, & e = 1
\end{cases}$$

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

Consider a finitely generated \(\mathbb{Z}_p\)-module \(M\). Denote \(M_{\mathrm{div}}\) 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]:

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 \(E_{v,\mathrm{ns}}(\mathbb{F}_v)\) be the set of non-singular points of \(E_v(\mathbb{F}_v)\) and define,
\[
E_0(K_v) := \{P \in E(K_v) \mid P_v \in E_{v,\mathrm{ns}}(\mathbb{F}_v)\}
\]

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

Remark 2.11. (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.

\(0 \rightarrow A = A[p^\infty] \rightarrow B[p^\infty] \rightarrow C[p^\infty] \rightarrow 0\)

\(X_{\mathrm{div}} \rightarrow Y_{\mathrm{div}} \rightarrow Z_{\mathrm{div}} \rightarrow W \rightarrow 0\)

is exact.

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 \( d \geq 1 \) and \( B \) be a finite p-primary \( G \)-module. Then,

\[
|H^1(G, B)| \leq |B|^d \text{ and } |H^1(G, B)| \leq |B|^{\frac{d(d-1)}{2}}.
\]

Remark 2.13. Let \( K_d \) be a \( \mathbb{Z}_p^d \)-extension of \( K \) for some \( d \geq 1 \). Then the group \( E[p^\infty](K_d) \) is finite [BL09a, Lemma 4.3]. Therefore, by Lemma 2.12, we get that \( H^1(G, E[p^\infty](K_d)) \) is bounded by \( |E[p^\infty](K_d)|^d \).

Definition 2.14. [How02] Let \( M \) be a finitely generated \( \Lambda(G) \)-module then we define

\[
\mu(M) := \sum_{i \geq 0} \mathrm{rank}_{\Omega(G)}(p^i(M[p^\infty])/p^{i+1}),
\]

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 \( i \geq 0 \), then the \( G \)-Euler characteristic of \( M \) is defined and it is given by:

\[
\chi(G, M) := \prod_{i \geq 0} (\# H^i(G, M))^{(-1)^i}.
\]

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 \( \operatorname{ord}_p(\alpha) \) 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 Λ(G)-modules, one has
\[ 0 \to A \to B \to C \to 0, \]
we have μ(B) ⩽ μ(A) + μ(C) with equality holding if B, and hence also A and C, is Λ(G)-torsion.

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
\[
\chi_{fl}(S, N) := \prod_i (\# H^i_{fl}(S, N))^{(-1)^i}.
\]
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_∞ be an unramified \(\mathbb{Z}_p\)-extension with Γ = Gal(K_∞/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]
\[
E^{a, b}_2 := H^a(\Gamma, H^b_{fl}(K_\infty, A)) \implies H^{a+b}_{fl}(K, A),
\]
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
\[
\chi_{fl}(\operatorname{Spec}(K), A) = \frac{\chi(\Gamma, H^2_{fl}(K_\infty, A)) \times \chi(\Gamma, H^0_{fl}(K_\infty, A))}{\chi(\Gamma, H^1_{fl}(K_\infty, A))}
\]

2.8. The ℓ ≠ p case. Choose and fix a rational prime ℓ ≠ 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 ℓ^∞-Selmer groups of an elliptic curve E over certain compact ℓ-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 ℓ^∞-fine Selmer group coincides with the ℓ^∞-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_∞ and the places where E has a bad reduction. Let K_S be the maximal separable extension of K, which is unramified outside S. For each v ∈ S, put
\[
J^1_v(E/L) := \prod_{w \mid v} H^1(L_w, E[\ell^\infty]).
\]

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:

$$S(E/L) := \ker(H^1(G_S(L), E[\ell^\infty]) \longrightarrow \bigoplus_{v \in S} J_v^1(E/L)), \tag{7}$$

where $G_S(L) := \operatorname{Gal}(K_S/L)$.

The definition of $S(E/L)$ extends to an infinite extension $L/K$ 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 $G_\infty := \operatorname{Gal}(K_\infty/K)$ 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-isotrivial 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:

$$S_n^A : S(A/K_n) \longrightarrow S(A/K_\infty)^{\Gamma_n}.$$

Remark 3.1. The following theorem of [Tan10] shows that $\ker(S_n^A)$ and $\operatorname{coker}(S_n^A)$ 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 $\Gamma_d := \operatorname{Gal}(K_d/K)$. 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 res $K_d/F : S(A/F) \to S(A/K_d)^{\Gamma_F}$, where $\Gamma_F = \operatorname{Gal}(K_d/F)$ are finite. Furthermore, if $d=1$, then the orders of the kernel and the cokernel of res $K_d/F$ are bounded as $F$ varies.

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

$$
Ш(A/K) := \ker(H^1(K, A) \to \prod_{v \in \Sigma_K} H^1(K_v, A))
$$

**Proposition 3.3.** Let $A/K$ be an abelian variety and $S$ be as above. Assume that $\ker(S_n^A)$ and $\operatorname{coker}(S_n^A)$ are finite and bounded independently of $n$. Further, if the groups $A(K_n)$ and $Ш(A/K_n)[p^\infty]$ are finite for each $n > 0$, then there exists $\mu, \lambda, \nu \geq 0$, such that
$$
\#S(A/K_n) = \#Ш(A/K_n)[p^\infty] = p^{e_n}
$$
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,

(i) $X$ is a finitely generated torsion $\Lambda$-module with $\mu=0$ and $\operatorname{char}_\Lambda(X)$ is coprime to $\omega_n$ for all $n$.

(ii) The modules $A(L_\infty)[p^\infty]/p^n A(L_\infty)[p^\infty]$ and $A(L_\infty)[p^\infty]/\omega_n A(L_\infty)[p^\infty]$ are finite and bounded independent of $n$.

(iii) For the natural maps
$$
MW_n^A : A(L_n)\otimes \mathbb{Q}_p/\mathbb{Z}_p \to (A(L_\infty)\otimes \mathbb{Q}_p/\mathbb{Z}_p)^{\Gamma_n}
$$
and
$$
S_n^A : S(A/L_n) \to S(A/L_\infty)^{\Gamma_n}
$$
the groups $\ker(MW_n^A)$ and $\ker(S_n^A)$ 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
$$
\frac{X}{pX} \cong (A(L_\infty)[p])^\vee, \ \frac{X}{\omega_n X} \cong (A(L_n)[p^\infty])^\vee
$$
$A(L_\infty)[p]$ 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:

\[
\ker(MW_n^A) \hookrightarrow \ker(S_n^A) \hookrightarrow \frac{A(L_\infty)[p^\infty]}{\omega_n A(L_\infty)[p^\infty]}
\]

The second injectivity follows from the fact that \(\ker(S_n^A) \subset H^1(\Gamma_n, A(F_\infty)[p^\infty])\).
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) \(\mathcal{G}((A(L_\infty) \otimes \mathbb{Q}_p/\mathbb{Z}_p)^\vee) = 0\).
(2) There is a \(\Lambda\)-linear injection
\[
(A(L_\infty) \otimes \mathbb{Q}_p/\mathbb{Z}_p)^\vee \hookrightarrow \Lambda^r \oplus \left( \bigoplus_{i=1}^u \frac{\Lambda}{\Phi_{c_i}} \right) \qquad (8)
\]
with finite cokernel for some \(r, u, c_i \geq 0\).

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
\[
MW_n^A : A(L_n) \otimes \mathbb{Q}_p/\mathbb{Z}_p \longrightarrow (A(L_\infty) \otimes \mathbb{Q}_p/\mathbb{Z}_p)^{\Gamma_n}
\]

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 \(\varinjlim_n (C_n)_{\mathrm{div}} = 0\).

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
\[
0 = (A(L_n) \otimes \mathbb{Q}_p/\mathbb{Z}_p)_{\mathrm{div}} \longrightarrow ((A(L_\infty) \otimes \mathbb{Q}_p/\mathbb{Z}_p)^{\Gamma_n})_{\mathrm{div}} \longrightarrow (C_n)_{\mathrm{div}} \longrightarrow 0.
\]
by applying Lemma 2.9(iii). Taking direct limits to the above sequence, we get that
\[
\varinjlim_n ((A(L_\infty) \otimes \mathbb{Q}_p/\mathbb{Z}_p)^{\Gamma_n})_{\mathrm{div}} = 0.
\]
Taking Pontryagin dual, we obtain that
\[
\mathcal{G}((A(L_\infty) \otimes \mathbb{Q}_p/\mathbb{Z}_p)^\vee) = \varprojlim_n \frac{(A(L_\infty) \otimes \mathbb{Q}_p/\mathbb{Z}_p)^\vee}{\omega_n(A(L_\infty) \otimes \mathbb{Q}_p/\mathbb{Z}_p)^\vee}[p^\infty] = 0.
\]

Recall from the description of functor 𝒢, given before the statement of Lemma 2.9, that 𝒢(Λ) = 0 and \((\frac{\Lambda}{\omega_{e+1,e}}) = 0\), if e = 1 and \(= \frac{\Lambda}{\omega_{e-1,e}}\), if e ≥ 2.

Using the fact that \((A(L_\infty)\otimes \mathbb{Q}_p/\mathbb{Z}_p)^\vee\) is a finitely generated Λ module and 𝒢\(( (A(L_\infty)\otimes \mathbb{Q}_p/\mathbb{Z}_p)^\vee ) = 0\), we get the following Λ-linear pseudo-isomorphism:

\[
(A(L_\infty)\otimes \mathbb{Q}_p/\mathbb{Z}_p )^\vee \longrightarrow \Lambda^r \oplus \left(\bigoplus_{i=1}^u \frac{\Lambda}{\Phi_{c_i}}\right)
\]

for some r, u, \(c_i \geq 0\).

The injectivity follows from the fact that \((A(L_\infty)\otimes \mathbb{Q}_p/\mathbb{Z}_p)^\vee\) has no non-trivial finite Λ-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 Ш\((E/L_n)[p^\infty]\) 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

\[
r = \lim_{n\to \infty} \frac{\mathrm{rank}_\mathbb{Z} A(L_n)}{p^n} \text{ and } e_n = \frac{\mathrm{rank}_\mathbb{Z} E(L_n) - \mathrm{rank}_\mathbb{Z} E(L_{n-1})}{\varphi(p^n)} - r;
\]
for \(n \geq 1\) and \(e_n\) is defined as the number between \(1 \leq i \leq t\) 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 Λ-modules
\[
\mathcal{M}^S(A/L_\infty)^\vee \sim \bigoplus_{i=1}^u \frac{\Lambda}{\Phi_{c_i}},
\]
for some \(u, c_i \geq 0\).

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 \(E(L_n) \to E(L_\infty)\).

(a) The kernel of \(m_n\) is finite, with order bounded independently of n.

Proof. Consider the following commutative diagram:

\[
\begin{array}{ccccccccc}
0 &\to& R^S(E/L_\infty)^{\Gamma_n} &\to& H^1(L_\infty, E[p^\infty])^{\Gamma_n} &\to& \left( \prod_{v \in S} K^1_v(E/L_\infty) \prod_{v \in \Sigma_K \setminus S} J^1_v(E/L_\infty) \right)^{\Gamma_n} \\
 & & \phantom{\downarrow} \downarrow f_n & & \phantom{\downarrow} \downarrow g_n & & \phantom{\downarrow} \downarrow h_n \\
0 &\to& R^S(E/L_n) &\to& H^1(L_n, E[p^\infty]) &\to& \prod_{v \in S} K^1_v(E/L_n) \prod_{v \in \Sigma_K \setminus S} J^1_v(E/L_n)
\end{array}
\]

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:

\[
0 \longrightarrow \ker(f_n) \longrightarrow \ker(g_n) \longrightarrow \ker(h_n) \cap \mathrm{image}(\lambda_{L_n}) \longrightarrow \mathrm{coker}(f_n) \longrightarrow 0.
\]

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 \( J^1_v(E/L_n) \rightarrow J^1_v(E/L_\infty)^{\Gamma_{n, v}} \), 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 \( \ker(h_n) \cong \bigoplus_{v \in S} H^1(\Gamma_{n, v}, E(L_{\infty, v})[p^\infty]) \).

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:

\[
\begin{array}{ccccccccc}
0 &\to& M^S(E/L_\infty)^{\Gamma_n} &\to& R^S(E/L_\infty)^{\Gamma_n} &\to& \mathfrak{X}^S(E/L_\infty)^{\Gamma_n} \\
 & & \phantom{\downarrow} \downarrow m_n & & \phantom{\downarrow} \downarrow f_n & & \phantom{\downarrow} \downarrow \gamma_n \\
0 &\to& M^S(E/L_n) &\to& R^S(E/L_n) &\to& \mathfrak{X}^S(E/L_n) &\to& 0
\end{array}
\]

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 \( \mathfrak{X}^S(E/L_n) \) is finite. Then, (b) follows from the fact that \( \mathrm{coker}(f_n) \) is finite.

\qed

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 $\mathfrak{X}^S(E/L_n)$ is finite for each $n$. Then there is a $\Lambda$-isomorphism
\[
T_p \mathcal{M}^S(E/L_n) \longrightarrow \bigoplus_{c_i \leq n} \frac{\Lambda}{\Phi_{c_i}},
\]
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, $\bigoplus_{c_i \leq n} \Lambda/\Phi_{c_i}$. The proof now follows from [Lei23, Remark 2.4]. $\Box$

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:
\[
0 \longrightarrow E(L_\infty) \otimes \mathbb{Q}_p/\mathbb{Z}_p \longrightarrow S(E/L_\infty) \longrightarrow \mathrm{III}(E/L_\infty)[p^\infty] \longrightarrow 0.
\]
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 $\mathrm{III}(E_q/L_n)[p^\infty]$ 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 $\mathrm{III}(E/L_n)[p^\infty]$ is finite for all $n \geq 0$. Also, suppose that $E$ has split multiplicative reduction at all primes of $S$. Then,
\[
\mathrm{Char}_\Lambda(\mathcal{M}^S(E/L_\infty))^\vee = \left(\prod_{e_n \geq 1, n \geq 0} \Phi_n^{e_n}\right),
\]
where
\[
e_n = \frac{\mathrm{rank}\, E(L_n) - \mathrm{rank}\, E(L_{n-1})}{\varphi(p^n)}.
\]

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 \( f_n : E(L_n)^{\bullet} \to E(L_{n,v})^{\bullet} \) 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 \( v \in S \), \( E_0(L_{n,v}) \cong \mathcal{O}^{\times}_{n,v} \), 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 \( (\operatorname{Image} f_n)[\Phi_n] = 0 \).

Now, consider the following short exact sequences,

\[
0 \to M^S_{p^k}(E/L_n) \to E(L_n)/p^k \to \bigoplus_{v \in S} E(L_{n,v})/p^k
\]

Taking inverse limit as \( k \) varies and tensoring with \( \mathbb{Q}_p \), we get

\[
0 \to V_p M^S_{p^k}(E/L_n) \to E(L_n)^{\bullet} \xrightarrow{f_n} \bigoplus_{v \in S} E(L_{n,v})^{\bullet}
\]

Using Corollary 4.3, we get that

\[
V_p M^S_{p^k}(E/L_n) \cong \bigoplus_{m=0}^{n} (\mathbb{Q}_p[X]/\Phi_m)^{s_m},
\]

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 \( (\operatorname{Image} f_n)[\Phi_n]=0 \), we get that \( e_m = s_m \). This completes the proof.

\(\Box\)

Remark 4.7. Note that the characteristic ideal of \( M^S(E/L_\infty) \) 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 \( (\operatorname{Image} f_n)[\Phi_n]=0 \). 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 \( n \geq 0 \).

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

Char_Λ(R^S(E/L_∞)^∨) =  ( ∏_{e_n ≥ 1, n ≥ 0} Φ_n^{e_n} )

Proof. Consider the following commutative diagrams:

Here, λ_i denotes the maps:

λ_i : H^1(K_∞, E_i[p^∞]) → ∏_{v∈Σ_K} J_v^1(E_i/K_∞).

0 → S(E_2/K_∞) → H^1(K_∞, E_2[p^∞]) →^{λ_2} Im(λ_2) → 0
    ↑f_1                    ↑f_2            ↑f_3
0 → S(E_1/K_∞) → H^1(K_∞, E_1[p^∞]) →^{λ_1} Im(λ_1) → 0           (9)

0 → Im(λ_2) → ∏_{v∈Σ_K} J_v^1(E_2/K_∞) → coker(λ_2) → 0
    ↑f_3               ↑f_4              ↑f_5
0 → Im(λ_1) → ∏_{v∈Σ_K} J_v^1(E_1/K_∞) → coker(λ_1) → 0           (10)

Let Γ = Gal(K_∞/K). As S(E_i/K_∞)^∨ is torsion Λ-module for i = 1, 2, we get by Proposition 2.18 that

μ(ker(f_1)^∨ ) - μ(coker(f_1)^∨ ) = μ (S(E_1/K_∞)^∨ ) - μ (S(E_2/K_∞)^∨ )

Theorem 5.1. Let E_1 and E_2 be two non-isotrivial elliptic curves over the function field K and let φ: E_1 → E_2 be an isogeny of degree p^r for some r > 0. Also assume that H^1_fl(K_∞, E_i[p^∞]) = 0, for i = 1, 2.

Then,

μ(S(E_2/K_∞)^∨ ) - μ(S(E_1/K_∞)^∨ ) = ord_p (χ_fl(Spec(K), A)),

where A = ker(φ).

We start by showing that ker(f₁) and coker(f₁) are both finite and therefore annihilated by a power of p. By Corollary 2.17, we know that for a finitely generated Λ module M, μ(M) = ord_p(χ(Γ, M[p^∞]^∨)). Therefore it suffices to calculate the logarithm to the p-base of χ(Γ, coker f₁) / χ(Γ, ker f₁), once we establish the finiteness of ker(f₁) and coker(f₁).

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

0 → ker(f₁) → ker(f₂) → ker(f₃) → coker(f₁) → coker(f₂) → coker(f₃) → 0.

Now, we claim that for i = 2, 3, ker(f_i) and coker(f_i) are finite. This in turn will show that ker(f₁) and coker(f₁) are finite. Following this, we will conclude our calculation of χ(Γ, coker f₁)/χ(Γ, ker f₁), by using the relation

χ(Γ, coker f₁) / χ(Γ, ker f₁) = (χ(Γ, coker f₂) / χ(Γ, ker f₂)) × (χ(Γ, ker f₃) / χ(Γ, coker f₃)),

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(φ(p)), where φ(p) is the map between E₁[p^∞] and E₂[p^∞], which is induced by φ.

We claim that χ(Γ, ker(f₃)) = χ(Γ, coker(f₃)) = 1. Consider the long-exact sequence:

0 → S(E₁/K_∞)^Γ → H¹(K_∞, E₁[p^∞])^Γ → (im(λ₁))^Γ → H¹(Γ, S(E₁/K_∞))
→ H¹(Γ, H¹(K_∞, E₁[p^∞])) → H¹(Γ, im(λ₁)) → 0.

The assumption H²_fl(K_∞, E₁[p^∞]) = 0 implies that H¹(Γ, H¹(K_∞, E₁[p^∞])) = 0 ([BV18, Proposition 2.2]). Hence, H¹(Γ, Im(λ₁)) = 0.

Again, consider the exact sequence

0 → (Im(λ₁))^Γ → ( ∏_{v∈Σ_K} J^1_v(E₁/K_∞))^Γ → (coker(λ₁))^Γ → H¹(Γ, Im(λ₁))
→ H¹(Γ, ∏_{v∈Σ_K} J^1_v(E₁/K_∞)) → H¹(Γ, coker(λ₁)) → 0

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

coker((Im(λ₁))^Γ → ( ∏_{v∈Σ_K} J^1_v(E₁/K_∞))^Γ) = 0.

This along with the fact that H¹(Γ, Im(λ₁)) = 0 implies that (coker(λ₁))^Γ = 0. Hence, (ker(f₅))^Γ = 0.

As \( K_\infty \) is unramified at every prime \( v \), we obtain that for any prime \( v \) and \( w \mid v \), \( \operatorname{Gal}(K_{\infty,w}/K_{\infty,w}) \cong \prod_{\ell \neq p} \mathbb{Z}_\ell \). 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 \( H^2_{\mathrm{fl}}(K_\infty, E_2[p^\infty]) \) implies that \( H^1(\Gamma, \operatorname{Im}(\lambda_2)) = 0 \) as well. Now, from the long exact sequence corresponding to the short exact sequence

\( 0 \to \ker(f_3) \to \operatorname{Im}(\lambda_1) \to \operatorname{Im}(\lambda_2) \to \operatorname{coker}(f_3) \to 0, \)

we obtain that \( H^1(\Gamma, \operatorname{coker}(f_3)) = 0 \). This proves our 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 \( 0 \to A \to E_1[p^\infty] \to E_2[p^\infty] \to 0 \) and using the fact \( H^2_{\mathrm{fl}}(K_\infty, E_2[p^\infty]) = 0 \), we get

\[
\frac{\chi(\Gamma, \operatorname{coker}(f_1))}{\chi(\Gamma, \ker(f_1))} = \frac{\chi(\Gamma, \operatorname{coker}(f_2))}{\chi(\Gamma, \ker(f_2))} = \frac{\chi(\Gamma, H^2_{\mathrm{fl}}(K_\infty, A)) \times \chi(\Gamma, E_1[p^\infty](K_\infty)) \times \chi(\Gamma, H^0_{\mathrm{fl}}(K_\infty, A))}{\chi(\Gamma, H^1_{\mathrm{fl}}(K_\infty, A)) \times \chi(\Gamma, E_2[p^\infty])}
\tag{11}
\]

Since, \( E_1 \) and \( E_2 \) are non-isotrivial, 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, \( \chi(\Gamma, E_1[p^\infty](K_\infty)) = \chi(\Gamma, E_2[p^\infty](K_\infty)) = \chi(\Gamma, H^0_{\mathrm{fl}}(K_\infty, A)) = 1 \).

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, \( \operatorname{Fr}_q \). This is a purely inseparable isogeny. As we know that the order of the kernel of isogeny equals its separable degree, we get that \( \mu(\mathrm{S}(E_2/K_\infty)^\vee) = \mu(\mathrm{S}(E_1/K_\infty)^\vee) \).

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

Let \( p \) be an odd prime and \( \ell \geq 5 \) 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 \( G = \mathrm{Gal}(K_\infty/K) \) and \( H = \mathrm{Gal}(K_\infty/K_{\mathrm{cyc}}) \).

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 \( i \geq 1 \).

**Proof.** Recall that the maps
\[
\lambda(\mathcal{K}) : H^1(G_S(\mathcal{K}), E[\ell^\infty]) \to \prod_{v \in S} J^1_v(E/\mathcal{K})
\]
are surjective for \( \mathcal{K} = K_\text{cyc} \) 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:
\[
0 \rightarrow S(E/K_\infty)^H \rightarrow H^1(G_S(K_\infty), E[\ell^\infty])^H \rightarrow \prod_{v \in S} J^1_v(E/K_\infty)^H \rightarrow 0
\tag{12}
\]

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 \( i \geq 1 \). 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
\[
0 \rightarrow S(E/K_\infty) \rightarrow H^1(G_S(K_\infty), E[\ell^\infty]) \rightarrow \prod_{v \in S} J^1_v(E/K_\infty) \rightarrow 0,
\]
and considering the corresponding long exact sequence. \( \Box \)

## Proposition 6.2.
Let \( E/K \) be a non-isotrivial elliptic curve and \( G = \operatorname{Gal}(K_\infty/K) \) a pro-\( p \) group. Then, \( \mu_G(S(E/K_\infty)^\vee) = \mu_\Gamma(S(E/K_\text{cyc})^\vee) \).

**Proof.** By [Pal14, Theorem 4.4], we obtain that the kernel of the map \( S(E/K_\text{cyc}) \xrightarrow{f} S(E/K_\infty)^H \)
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 \( S(E/K_\text{cyc})^\vee \) 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]. \( \Box \)

As discussed in [GJS22, §2.1], we know that \( S(E/K_\text{cyc})^\vee \) 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 \( K_\text{cyc} \) 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,

\[
\mu_G(S(E/K_\infty)^\vee) = \mu_\Gamma(S(E/K_\text{cyc})^\vee) = 0.
\]

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 \( K_\text{cyc} \) or the trivialising extension \( K_\infty \).

APPENDIX A. BEHAVIOUR OF \( \mu \)-INVARIANTS UNDER ISOGENY (\( \ell \neq p \) CASE)

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.

Again, our goal is to consider the change in the \( \mu \)-invariant of \( \ell^\infty \)-Selmer groups of \( E_1 \) and \( E_2 \) over \( K_\text{cyc} \) and \( K_\infty \).

Consider the following commutative diagram:

\[
\begin{tikzcd}
0 \arrow{r} & S(E_2/K) \arrow{r} \arrow{u}{g_1} & H^1(G_S(\mathcal{K}), E_2[\ell^\infty]) \arrow{r} \arrow{u}{g_2} & \prod_{v \in S} J^1_v(E_2/K) \arrow{r} \arrow{u}{g_3 = \prod_{v \in S} g_{3,v}} & 0 \\
0 \arrow{r} & S(E_1/K) \arrow{r} & H^1(G_S(\mathcal{K}), E_1[\ell^\infty]) \arrow{r} & \prod_{v \in S} J^1_v(E_1/K) \arrow{r} & 0
\end{tikzcd}
\]
(13)

Theorem A.1. Consider \( \mathcal{K} = K_\text{cyc} \) 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 \( G = \text{Gal}(\mathcal{K}/K) \). 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 Λ module M, μ(M) = ord_p(χ(Γ, M[p^∞]^∨)) (Corollary 2.17). Now, suppose ker(g₁) and coker(g₁) are both finite and therefore annihilated by a power of p. Then, it suffices to calculate the logarithm to the p-base of χ(G, coker g₁) / χ(G, ker g₁).

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

0 → ker(g₁) → ker(g₂) → ker(g₃) → coker(g₁) → coker(g₂) → coker(g₃) → 0.

Now, we claim that for i = 2, 3, ker(g_i) and coker(g_i) are finite, which shows that ker(g₁) and coker(g₁) are finite. We establish the finiteness of ker(g₁) and coker(g₁) and calculate χ(G, coker g₁) / χ(G, ker g₁) separately for K_cyc and K_∞. Note that the equation (A) also implies that χ(G, coker g₁) / χ(G, ker g₁) = (χ(G, coker g₂) / χ(G, ker g₂)) × (χ(G, ker g₃) / χ(G, coker g₃)).

Case 1: Take K = K_cyc. Then, G = Γ = Gal(K_cyc/K). Let A = ker(φ(ℓ)), the ℓ-primary part of ker(φ). From the long exact sequence corresponding to the short exact sequence 0 → A → E₁[ℓ^∞] → E₂[ℓ^∞] → 0 and using the fact H²(K_cyc, E₂[ℓ^∞]) = 0 ([GJS22, Proposition 2.10]), we get that

coker(g₂) = H¹(G_S(K), A) and ker(g₂) = H¹(G_S(K), A) / (E₂[ℓ^∞](K_cyc) / (E₁[ℓ^∞](K_cyc) / H⁰(G_S(K), A))).

Since, E₁ and E₂ are non-isotrivial, hence we get that E₁[ℓ^∞](K_cyc) and E₂[ℓ^∞](K_cyc) are finite ([BL09b, Lemma 3.2]). Hence, ker(g₂) and coker(g₂) are finite.
As K_cyc is unramified at every prime v, we obtain that for any prime v and w | v, Gal(K_cyc,w/K_cyc,w) ≅ ∏_{r≠ℓ} Z_r. Therefore, ker g₃ and coker g₃ are trivial.

As ker g_i and coker g_i are finite for i = 1, 2, we get that χ(Γ, ker g_j) and χ(Γ, coker g_j) exists for j = 1, 2, 3. Therefore,

χ(Γ, coker(g₂)) / χ(Γ, ker(g₂)) = (χ(Γ, H²(G_S(K_∞), A)) × χ(Γ, E₁[ℓ^∞](K_∞)) × χ(Γ, H⁰(G_S(K_∞), A)))
/ (χ(Γ, H¹(G_S(K_∞), A)) × χ(Γ, E₂[ℓ^∞]))

As E₁[ℓ^∞](K_∞) and E₂[ℓ^∞](K_∞) and A are all finite. Therefore, χ(Γ, E₁[ℓ^∞](K_cyc)) = χ(Γ, E₂[ℓ^∞](K_cyc)) = χ(Γ, H⁰(G_S(K_cyc), A)) = 1.

Now, using the Hochschild-Serre spectral sequence,

H^i(Γ, H^j(G_S(K_cyc), A)) ⇒ H^{i+j}(G_S(K), A),

in equation (14). Therefore,

χ(Γ, coker(f₂)) / χ(Γ, ker(f₂)) = χ(Gal(K_S/K), A).

By [Mil86, Remark I.5.2], we get that χ(Gal(K_S/K), A) = 1. This concludes Case 1.

Case 2: Take \(\mathcal{K} = K_\infty\) and \(G = G_\infty = \text{Gal}(K_\infty/K)\). 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 \(0 \rightarrow A \rightarrow E_1[\ell^\infty] \rightarrow E_2[\ell^\infty] \rightarrow 0\) and \(H^2(K_\infty, E_2[\ell^\infty]) = 0\) ([BV14, Proposition 4.5]) that

\[
\text{coker}(g_2) = H^1(G_S(K), A)\ \text{and}\ \ker(g_2) = H^0(G_S(K), A).
\]

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 \(v \in S \setminus S_\text{un}\), \(\ker(g_{3,v}) = \prod_{w|v} H^1(K_{\infty,w}, A)\) and \(\text{coker}(g_{3,v}) \subset \prod_{w|v} H^2(K_{\infty,w}, A)\). As \(\text{cd}_\ell(K_{\infty,w}) = 1\) ([NSW13, Theorem 7.1.8]), we get that \(\text{coker}(g_{3,v}) = 0\). Therefore, \(\ker(g_3) = \prod_{v \in S \setminus S_\text{un}} \ker(g_{3,v})\) and \(\text{coker}(g_3) = 0\).

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,

\[
\chi(G_\infty, \ker(\varphi_3)) = \prod_{v \in S \setminus S_\text{un}} \chi(G_{\infty,v}, H^1(K_{\infty,v}, A)).
\]

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,

\[
\frac{\chi(G_\infty, \ker(\varphi_3))}{\chi(G_\infty, \text{coker}(\varphi_3))} = \frac{1}{\prod_{v \in S \setminus S_\text{un}} \chi(\text{Gal}(\overline{K_v}/K_v), A)}.
\]

By [Mil86, Chapter 1, Lemma 2.9], we know that \(\chi(\text{Gal}(\overline{K_v}/K_v), A) = 1\). Moreover, using calculations similar to Case 1, we obtain that \(\frac{\chi(G_\infty, \text{coker}(g_2))}{\chi(G_\infty, \ker(g_2))} = 1\).

Hence, in both cases, we obtain that \(\frac{\chi(G, \text{coker } g_1)}{\chi(G, \ker g_1)} = 1\), for \(G = G_\infty\) or \(\Gamma\). This concludes the proof of the theorem.

## Remark A.2.

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

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

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(Ghosh) Harish Chandra Research Institute, A CI of Homi Bhabha National Institute, Chhatnag Road, Jhunsi, Prayagraj (Allahabad) 211 019 India
Email address: ghoshsohan4@gmail.com

(Ray) Harish Chandra Research Institute, A CI of Homi Bhabha National Institute, Chhatnag Road, Jhunsi, Prayagraj (Allahabad) 211 019 India
Email address: jishnuray@hri.res.in; jishnuray1992@gmail.com


[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
Arbitrarily accurate fault-tolerant (FT) universal quantum computation can be carried out using the Clifford gates Z, S, CNOT plus the non-Clifford T gate. Moreover, a recent improvement of the Solovay-Kitaev theorem by Kuperberg implies that to approximate any single-qubit gate to an accuracy of ϵ > 0 requires O(log^c[1/ϵ]) quantum gates with c > 1.44042. Can one do better? That was the question asked by Nielsen and Chuang in their quantum computation textbook. Specifically, they posted a challenge to efficiently approximate single-qubit gate, fault-tolerantly or otherwise, using Ω(log[1/ϵ]) gates chosen from a finite set. Here I give a partial answer to this question by showing that this is possible using O(log[1/ϵ] log log[1/ϵ] log log log[1/ϵ] · · ·) FT gates chosen from a finite set depending on the value of ϵ. The key idea is to construct an approximation of any phase gate in a FT way by recursion to any given accuracy ϵ > 0. This method is straightforward to implement, easy to understand, and interestingly does not involve the Solovay-Kitaev theorem.

Introduction—While Clifford gates alone cannot perform universal quantum computation, adding one more non-Clifford gate, namely, the T gate, to the list can. (Definitions of the gates mentioned here will be given later.) For instance, using single-qubit Z, S and T gates together with two-qubit CNOT gate are sufficient to approximate any unitary operator in U(2) to arbitrary precision as well as performing universal computation [1–4]. As the operator e^{iα} can be executed fault-tolerantly, I will restrict most of the discussions below to SU(2). This approach of approximating any (special) unitary operator to arbitrary precision using a fixed finite set of quantum gates is analogous to hardware gates and machine language software in classical computers—all classical digital computer hardware and programs are composed of a finite set of classical gates and low-level machine codes that is powerful enough to perform universal classical computation.

There is an important difference between the classical and quantum situations. As SU(2) is a continuous group, any unitary operator generated by a finite set of quantum gates is at most dense in rather than equals SU(2). Thanks to the Solovay-Kitaev theorem [1, 2], one may approximate any SU(2) operator to arbitrary precision by composition of a finite set of quantum gates obeying certain technical conditions. Specifically, let S be a subset of SU(2) such that the set generated by S is dense in SU(2) and that G ∈ S implies G^{-1} ∈ S. Then, for any ϵ > 0 and U ∈ SU(2), there exists a S ∈ SU(2) composing of a finite product of gates taken from S such that the operator norm ||U − S|| < ϵ [1–5]. Since CNOT plus the set of all SU(2) gates can simulate any special unitary gate [6], it means that all special unitary operators can be approximated to arbitrary accuracy. (Throughout this paper, the terms accuracy or precision between two unitary operators U and S always refer to the operator norm ||U − S||. Besides, the symbol ϵ is implicitly assumed to be in the interval (0, 1].)

The Solovay-Kitaev theorem further tells us that the approximation can be done very efficiently. The version of Solovay-Kitaev theorem reported in Ref. [2] showed that for any fixed ϵ, δ > 0, there is an algorithm returning S that makes up of an instruction sequence of length O(log^{3+δ}[1/ϵ]) such that ||U − S|| < ϵ. The run time of this algorithm is O(log^{3+δ}[1/ϵ]) and the approximation quantum circuit does not involve ancilla. Furthermore, if ancillas are allowed, both the instruction sequence length and the run time scale like O(log^2[1/ϵ] log log[1/ϵ]). Recently, Kuperberg generalized the Solovay-Kitaev theorem in a several directions. In particular, he proved that the instruction sequence length can be reduced to O(log^c[1/ϵ]) with c = log_{(1+√5)/2} 2 + δ ≈ 1.44042 + δ without using ancilla [7].

How to execute gates in the set S in a fault-tolerant (FT) manner? Although this is an open question for a general set S, FT implementations of Z, H, X, S, T and CNOT are known using the so-called Steane-like code [3, 4]. Consider a classical linear self-orthogonal binary code C, namely, the one whose codeword space is a subset of its dual. Then, the CSS code constructed from C, namely,

|ℓ⟩_L = (1 / |C|^{1/2}) ∑_{a∈C} |ℓ + a⟩

for all ℓ in the coset C^⊥/C [3, 4, 8–10] is called Steane-like because the Steane’s seven qubit code [10] is perhaps the most famous example of this type of code. (In what follows, a state ket with the subscript L denotes a logical state of a quantum code.) Using Steane-like code, the logical Z, H, X, S and CNOT are automatically FT. Besides, T gate has a FT implementation using gate teleportation [3, 4]. Here I remark that Harrow et al. gave a non-constructive proof showing that the instruction sequence length can be lowered to O(log[1/ϵ]) using a certain instruction set making up of three elements [11]. Nonetheless, no FT execution of the gates used in their construction is known to date.

The length of instruction sequence required to approximate a general SU(2) operator to a precision of ε using a fixed finite set of generators is at least Ω(log[1/ε]) [3]. Can this lower bound be attained using a computationally efficient procedure? This was the challenge posted by Nielsen and Chuang in their book [3]. Here I give a partial answer to this question by reporting an efficient way to approximate any SU(2) operator to an accuracy of ε using O(log[1/ε] log log[1/ε] log log log[1/ε] ··· ) gates with the product terminated when log log ··· log(1/ε) becomes O(1). (In what follows, all products in the form log(1/ε) log log(1/ε) ··· are implicitly assumed to be terminated when log log ··· log(1/ε) becomes O(1).) More importantly, this approximation can be performed in a FT manner.

in a FT way using a procedure introduced in Ref. [3]. (See also Ref. [13] for a more effective alternative procedure discovered recently.) Specifically, one prepares copies of high fidelity |cat⟩ = (|0⟩^⊗n + |1⟩^⊗n)/√2 with n being the length of the Steane-like code using noisy state preparation and stabilizer measurement apparatus. Then these high fidelity |cat⟩ states are used to check if the states prepared by the circuit in the red region of Fig. 1 are error-free by measuring their stabilizer P(α)_L X_L P(−α)_L using noisy apparatus. This checking is carried out a few times and only those states that pass the test more often than fail the test are kept [3]. By using these high fidelity |Θ(α)⟩_L states as inputs, it is straightforward to verify the validity of the gate teleportation circuit in Fig. 2. More importantly, this gate teleportation circuit is FT if P(2α)_L is FT as well. (Actually, for α = π/4, this circuit is the one used to execute the T gate in a FT manner as there is a FT implementation of S_L = P(π/2)_L [3, 4].)

Method— I begin by denoting the computational basis of a two-dimensional Hilbert space by {|0⟩, |1⟩}. Recall that the set of all single qubit gates plus CNOT gate is universal, where CNOT|a, b⟩ = |a, a + b⟩ for a, b = 0, 1. Besides, up to a global phase, any single qubit gate can be decomposed as two rotations along z-axis and one rotation along x-axis, or equivalently generated using two Hadamard gates H and three phase gates P(α) where H|a⟩ = ∑_{b=0}^1 (−1)^{ab} |b⟩/√2 for a = 0, 1 and [6, 12]

P(α) = |0⟩⟨0| + exp(iα)|1⟩⟨1|. (2)

So it remains to show how to approximate any P(α) in a FT way through a computationally efficient algorithm. Observe that ||P(α) − P(α + Δα)|| = [ (1 − cos Δα)^2 + sin^2 Δα ]^{1/2} ≤ |Δα|. Thus, to approximate the P(α) gate in a FT way to an accuracy of ε, it suffices to find a FT implementation of the gate P(α′) with |α − α′| < ε. (Here I mention on passing that the Z, S and T gates are nothing but the special cases of the phase gate with α = π, π/2 and π/4, respectively.)

Pick a Steane-like code and by circuit tracing, it is clear that Fig. 1 prepares the logical state

|Θ(α)⟩_L ≡ (|0⟩_L + e^{iα} |1⟩_L) / √2    (3)

Now I report how to execute P(2α)_L. Let G and H be the generator and parity check matrices a classical self-orthogonal [n, k, d] code C, respectively. Without lost of generality, I write H as [ L | G ] where L corresponds to

the logical operation of the resultant Steane-like quantum code. Surely, H is of full row rank over F_2, the finite field of two elements; and hence it must also be of full row rank over ℝ. Therefore, for any given w ∈ ℝ^{n−k}, the equation Hv = w has solution in ℝ^n though it is not unique for k > 0. In particular, by setting the jth component of w, namely, w_j to δ_{jℓ} for a fixed ℓ = 1, 2,···, n−2k, then any one of the solutions v corresponds to the logical operation P(α)_L acting on the ℓth logical qubit via

P(α)_L = ⊗_{j=1}^n P(α v_j).    (4)

Clearly, P(α)_L can be computed in O(n^3) time by Gaussian elimination. To simplify matter, one could amend k more rows to H that are orthogonal to the ℓth row in L to make the resultant n × n matrix invertible (both in F_2 and ℝ). Then the solution v is unique. Applying this method to the Steane’s seven qubit code [10]

|ℓ⟩_L = 1/(2√2) ∑_{a,b,c=0}^1 |ℓ + a + b + c, ℓ + a + b, ℓ + a + c⟩ ⊗ |ℓ + a, ℓ + b + c, ℓ + b, ℓ + c⟩    (5)

for ℓ = 0, 1 as an example, I obtain the following logical implementation of the P(α) gate

P(α)_L = P(−α) ⊗ P(α) ⊗ 1^⊗4 ⊗ P(α).            (6)

As the S = P(π/2) has a FT implementation [3, 4, 13] and the P(α)_L gate execution in Eq. (4) is transversal, by recursively applying the quantum circuits in Figs. 1 and 2, I can implement any P(πℓ/2^m) gate in a FT manner with m, ℓ ∈ Z^+ and ℓ < 2^m in O(n) times through the recursion P(πℓ/2^m) ⇐ P(πℓ/2^{m−1}) ⇐ P(πℓ/2^{m−2}) ⇐ ··· ⇐ P(πℓ/4) that doubles the rotation angle modulo 2π in each iteration. Actually, the idea of recursively generating gates starting from the Clifford gates had been pointed out by Gottesman in Ref. [4]. Nonetheless, he did not develop this idea further.

Conclusions — In summary, to approximate a phase gate P(α) (with 0 < α < 2π) to a precision of at least ϵ, one simply finds non-negative integers m, ℓ such that |α − πℓ/2^m| < δ with δ ≤ ϵ/⌈log_2(1/ϵ)⌉ and apply the above procedure to execute P(πℓ/2^m) in a FT way. In other words, to approximate any SU(2) operator to ϵ > 0 in operator norm, one decomposes the operator as three phase gates plus two H gates and apply FT implementation to each phase gate with angle of rotation accurate to ϵ log_2(1/ϵ) log_2 log_2(1/ϵ) ··· until log_2 log_2 ··· log_2(1/ϵ) becomes O(1). In this way, one obtains a computational efficient method to approximate any single qubit gate to an accuracy of ϵ using O(ϵ log[1/ϵ] log log[1/ϵ] ···) gates. In essence, this method is built upon approximating an arbitrary phase gate (or alternatively arbitrary rotation angle) to any given precision in a FT manner rather than the Solovay-Kitaev theorem for specific kind of set of operators in SU(2). For any given precision ϵ, the gates needed are drawn from the set {H, P(πℓ/2^m) : m, ℓ ∈ Z^+, ℓ < 2^m} where m = ⌈ϵ/ [log_2(1/ϵ) log_2 log_2(1/ϵ)···]⌉. Although it appears that a huge gate set, whose size increases as ϵ decreases, is needed, this collection is very natural in all major noisy intermediate-scale quantum computers to date. Note that superconducting transmon qubits [14], trapped ions [15] and neutral atoms [16] implementation of quantum computers to date, single qubit gates are executed by control pulses in which the angle of rotation on the Bloch sphere is proportional to the integral of effective Hamiltonian over time. For these quantum computers, one can apply P(α) on each qubit and hence P(α)_L on the logical qubit relatively easily. Whereas for photonic implementation [3, 17–19], P(α) and hence P(α)_L gates can be easily implemented via phase, polarization or amplitude modulators depending on the qubit representation. Finally, I remark that the construction reported here can be extended to FT implementation of qudits although its effectiveness decreases exponentially with the Hilbert space dimension of the qudit.

Acknowledgments — This work is supported by the RGC grant No. 17303323 of the Hong Kong SAR Government. Useful discussions with Rong Wang and Wenyuan Wang are gratefully acknowledged.