What are the strongest conjectured uniform versions of Serre's Open Image Theorem?

Question

This question concerns the uniform conjectured effective versions and generalizations of these two results of Serre on $\ell$-adic Galois representations $\rho_{E,\ell}$ associated to a non-CM elliptic curve $E$ defined over a number field $K$:

1. For each prime $\ell$, the index $I(E,\ell)$ of the image of $\rho_{E,\ell}$ in $\operatorname{GL}_2(\mathbb{Z}_\ell)$ is finite (and bounded independent of $\ell$), and

2. The representation $\rho_{E,\ell}$ is surjective for all but finitely many primes $\ell$.

Remark: As I understand it, these two statements together say that the representation $\rho_E:G_K \to \operatorname{GL}_2(\hat{\mathbb{Z}})$ made up of all the $\ell$-adic ones has open image, and hence "Serre's Open Image Theorem" refers to both of these statements, although many papers written on this topic seem to only discuss one statement or the other.

Let $p_0(E)$ denote the smallest prime such that $\rho_{E,\ell}$ is surjective for all $\ell > p_0$.

My question: What are the strongest conjectured ("conjectured" means either officially conjectured or in a folklore way, or even someone writing down that they hope it's true) generalizations of Serre's Theorem which are uniform in $E$?

Here are examples of the kind of statements I am referring to:

• Conjecture 1: $p_0(E)$ is bounded independently of $E$, where $E$ ranges over all elliptic curves defined over $\mathbb{Q}$ (and the bound is 37).

• Conjecture 2: $I(E,\ell)$ is bounded independently of $E$ and $\ell$, where $E$ ranges over all elliptic curves defined over $\mathbb{Q}$ (and the bound is...?).

• Conjecture 1': Like Conjecture 1, but for elliptic curves defined over an arbitrary number field $K$, and the bound only depends on $K$.

• Conjecture 2': Like Conjecture 2, but for elliptic curves defined over an arbitrary number field $K$, and the bound only depends on $K$.

• Conjecture 1'': Like Conjecture 1', but the bound only depends on $[K:\mathbb{Q}]$.

• Conjecture 2'': Like Conjecture 2', but the bound only depends on $[K:\mathbb{Q}]$.

I am aware that a lot of progress has been made on Conjecture 1, and we might be close to knowing that one.

Answer 1

I've seen Conjecture 1 and Conjecture 1' stated in the literature in many places. I don't believe I have seen Conjecture 1'' so stated.

I'd also like to point out that (EDIT: a weaker version of) Conjecture 2'' is true. In particular, if $E$ is a non-CM elliptic curve defined over a number field $K$ and $\ell$ is a prime number, there is a bound on $I(E,\ell)$ that depends only on $[K : \mathbb{Q}]$.

In particular, suppose that $K/\mathbb{Q}$ is a number field and the $\ell$-adic image for $E/K$ is $G \subseteq {\rm GL}_{2}(\mathbb{Z}_{\ell})$. There is a modular curve $X_{G}$ that parametrizes elliptic curves with $\ell$-adic image contained in $\langle G, -I \rangle$. If there are infinitely many elliptic curves over number fields $K$ with $[K : \mathbb{Q}] \leq d$, there are infinitely many points on $X_{G}$ over number fields of degree $\leq d$ and an argument of Abramovich (relying on a result of Faltings about subvarieties of abelian varieties with infinitely many $K$-rational points, see the paper "A linear lower bound on the gonality of modular curves" in IMRN in 1996) implies that the gonality of the curve $X_{G}$ (i.e. the smallest degree map to $\mathbb{P}^{1}$) is $\leq 2d$. Abramovich also shows that the gonality of $X_{G}$ grows linearly with the index of $G$ in ${\rm GL}_{2}(\mathbb{Z}_{\ell})$.

As a consequence, once the index of $G$ in ${\rm GL}_{2}(\mathbb{Z}_{\ell})$ is large enough, there are only finitely many points on $X_{G}$ over any number field of degree $d$. Since there are only finitely many subgroups of ${\rm GL}_{2}(\mathbb{Z}_{\ell})$ of a given index, Serre's result number 1 in the question then implies that $I(E,\ell)$ is bounded only in terms of $[K : \mathbb{Q}]$.