Let $E$ be an elliptic curve over the rationals $\mathbb{Q}$. We consider the Galois representation attached to $E$ by acting on its $p$-adic Tate module $T_p(E)$, $$ \rho_E: G_{K} \rightarrow \mathrm{Aut}(T_p(E)) \cong \mathrm{GL}_2(\mathbb{Z}_p). $$ Then it cuts out a field $M := \overline{\mathbb{Q}}^{\ker(\rho_E)}$, whose Galois group over $\mathbb{Q}$ is isomorphic to the image of $\rho_E$. A theorem of Serre tells us that if $E$ has no CM, then $\mathrm{im}(\rho_E)$ is a finite index subgroup of $\mathrm{GL}_2(\mathbb{Z}_p)$ and strictly equals to $\mathrm{GL}_2(\mathbb{Z}_p)$ for all but finitely many primes $p$. So for simplicity, we take sufficiently large odd prime $p$ to guarantee $\rho_E$ has full image.

Further $\widetilde{\rho_E}: G_{K} \rightarrow \mathrm{PGL}_2(\mathbb{Z}_p)$. Then similarly consider $K := \overline{\mathbb{Q}}^{\ker(\widetilde{\rho_E})}$. We assume $E$ has no CM and $p$ sufficiently large so that $\mathrm{Gal}(K/\mathbb{Q}) \cong \mathrm{PGL}_2(\mathbb{Z}_p)$.

**Ultimate Goal**: The group of torsion points $E(K)_{\mathrm{tors}}$ is a finite group.

A crucial step in proving this (from my naive thought) is to ensure that

**Claim 1: there exist infinitely many primes "good primes" $q$ such that the residue field for any prime of $K$ lying above $q$ is finite.**

This post is asking how to prove (or disprove) this boldface claim 1?

My "attempt" is to imitate the proof of the following claim in the anticyclotomic $\mathbb{Z}_p$-extension case.

**Claim 2**: Let $F$ be an imaginary quadratic field and $F_{\infty}^{\mathrm{ac}}$ is its anticyclotomic $\mathbb{Z}_p$-extension. Then for any prime of $F$ which is inert in $F/\mathbb{Q}$ must split completely in $F_{\infty}^{\mathrm{ac}}/F$.

Then since there are infinitely many inert primes in $F$ (by density arguments), we have infinitely many primes that "has finite residue field in $F_{\infty}^{\mathrm{ac}}$".

The proof of Claim 2 may date back to Iwasawa in the second section of his article "*On the $\mu$-invariants of $\mathbb{Z}_{\ell}$-extensions (1973)*". A sketch of proof goes like this:

Let $q$ be a prime coprime to $p$ that is inert in $F$, write $\mathfrak{q}$ for the unique prime of $F$ lying above $q$, then $\mathfrak{q}$ is unramified in the $n$-th layer $F_{n}^{\mathrm{ac}}$. Let $\mathfrak{q}_n$ be a prime of $F_{n}^{\mathrm{ac}}$ lying above $\mathfrak{q}$ and $Z_n$ be the decomposition group of it for the Galois extension $F_{n}^{\mathrm{ac}}/\mathbb{Q}$. Then since $q$ is unramified in $F_{n}^{\mathrm{ac}}$ and is inert in $F$, $Z_n$ is a cyclic group of $G_n := \mathrm{Gal}(F_{n}^{\mathrm{ac}}/\mathbb{Q})$ such that $G_n = Z_n H_n$, where $H_n := \mathrm{Gal}(F_{n}^{\mathrm{ac}}/F)$. As $G_n$ is a dihedral group of order $2p^n$, it follows that $Z_n$ is a cyclic group of order two satisfying $Z_n \cap H_n = 1$. However, $Z_n \cap H_n$ is nothing but the decomposition group of $\mathfrak{q}_n$ for the extension $F_{n}^{\mathrm{ac}}/F$, hence $\mathfrak{q}$ splits completely in $F_{n}^{\mathrm{ac}}$.

Then to imitate the proof above, we need to know the parallel facts on general $\mathrm{PGL}_2(\mathbb{Z}_p)$-extensions $K/\mathbb{Q}$, for example:

- Specify all intermediate fields of the $\mathrm{PGL}_2(\mathbb{Z}_p)$-extension $K/\mathbb{Q}$. This is equivalent to classifying all closed subgroups of $\mathrm{PGL}_2(\mathbb{Z}_p)$ and figuring out which of them are open.

Maybe we can try some congruence subgroups? Let $\Gamma_n$ be the group of matrices in $\mathrm{GL}_2(\mathbb{Z}_p)$ that are congruence to identity matrix modulo $p^n$, and let $\overline{\Gamma_n}$ be its image of $\mathrm{PGL}_2(\mathbb{Z}_p)$. Do these groups work and are these all such subgroups? (I got hints from here.)

- The properties of the Galois groups of intermediate fields are good for us to run a similar argument as the anticyclotomic case. In this process, we hope to get a sufficient condition $(\star)$ for such "good primes" (that satisfying Claim 1).

Follow Greenberg's hint, maybe a common property for finite dihedral groups and $\mathrm{PGL}_2(\mathbb{Z}_p)$ is that $g$ and $g^{-1}$ are conjugate for any group element $g$. Unfortunately, I cannot see how this property is used in the above proof of anticyclotomic case either.

- There are infinitely many primes satisfying the sufficient condition $(\star)$ in 2.

Following Greenberg's hint, I guess that the primes $q$ such that $E$ has good supersingular reduction satisfies the abstract sufficient condition $(\star)$. Then a result of N. Elkies shows that there are infinitely many good supersingular primes. Yet I cannot see how such good supersingular properties are used in the abstract discussion of $\mathrm{PGL}_2(\mathbb{Z}_p)$-extensions. Since I am not able to do 2, figuring out 3 is impossible.

So though there is a rough three-step roadmap, I got stuck on each step. So I am here to ask if there is any way out.

*Some further remarks*:

- This entire problem is motivated by solving Exercise 1.12 of Ralph Greenberg's IAS/Park City note "
*Introduction to Iwasawa Theory for Elliptic Curves*". The hints above by Greenberg is taken from here. - The
**Claim 2**seems to work for anticyclotomic $\mathbb{Z}_p$-extension $\mathcal{F}_{\infty}^{\mathrm{ac}}$ for any CM field $\mathcal{F}/\mathcal{F}^{+}$. - Even if all the problems in this post are solved, we still cannot get the full proof of Greenberg's Exercise 1.12 that $E(K)_{\mathrm{tors}}$ is finite since we haven't touched the CM elliptic curves. An exercise in Silverman's book tells us that in this case, $\rho_E$ has abelian image in $\mathrm{GL}_2(\mathbb{Z}_p)$, and hence $\widetilde{\rho_E}$ has abelian image in $\mathrm{PGL}_2(\mathbb{Z}_p)$. Then in this CM elliptic curve case, it seems that we need a classification of abelian subgroups of $\mathrm{PGL}_2(\mathbb{Z}_p)$ as "Step 0" and try to run the argument above.

Or am I so stupid that missed some easy solution to this exercise? Actually, I feel like I am quite good at making things unnecessarily complicated and as a result, obtaining nothing valuable during my Ph.D. study up to now. Quite frustrated. :(

So sorry for such a long post, and thank you all for commenting and answering! :)

*EDIT: Even Further Remarks*: I found Greenberg's Exercise 1.16 was focusing on the CM case.

- In Greenberg's Exercise 1.16, he considered a particular elliptic curve $y^2 = x^3 - x$ with CM $\mathbb{Z}[\sqrt{-1}]$. Let $F=\mathbb{Q}(\sqrt{-1})$. Then he asked the reader to show $K$ contains $F_{\infty}^{\mathrm{ac}}$ and $[K:F_{\infty}^{\mathrm{ac}}] < \infty$. This example (though I am still trying to prove) may inspire us to show that maybe (I am not sure at all) $K$ contains an anticyclotomic extension of finite index and then use the
**Claim 2**directly to conclude? - When searching for references, I found arXiv: 2008.04960, where the authors said in the second paragraph of the introduction that
**the $\mathrm{PGL}(2)$-extension does not contain the cyclotomic extension**. This (though still I am trying to prove it) may further provide some evidence on that certain "anticyclotomic line" appears in $K/\mathbb{Q}$?

whythe PGL2 extensions are studied: because they are the first obvious example of p-adic Lie extensions which are "essentially" nonabelian, having no nontrivial abelian extensions inside them. $\endgroup$