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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.

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    $\begingroup$ As you know, "Conjecture 1" here was stated by Serre not as a conjecture, but as a question (what is commonly known as Serre's uniformity question). As it turns out, I asked Serre last week if he thought this was still a "question" (knowing what we know now about the split case, i.e., Bilu-Parent) or whether he was willing to upgrade it to a "conjecture"... He answered that it is definitely still just a question. $\endgroup$ – Álvaro Lozano-Robledo Apr 29 '15 at 20:42
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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 $[K : \mathbb{Q}]$.

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  • $\begingroup$ Thank you very much for bringing this argument fact (re Conjecture 2'') to my attention! I guess this is the reason for the focus in recent literature on conjectures 1, 1', etc. In your opinion, is there a good reason not to believe in Conjecture 1''? It would be nice if one could say all bounds in the OIT only depend on $[K:\mathbb{Q}]$. $\endgroup$ – Bobby Grizzard Apr 27 '15 at 15:04
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    $\begingroup$ In my opinion there isn't a good reason not to believe Conjecture 1", but there isn't a good reason to believe it either. It's just too far from what we know. $\endgroup$ – Jeremy Rouse Apr 27 '15 at 16:11
  • $\begingroup$ Wait -- but the bound we get this way also depends on $\ell$ (unlike in the statement of conjecture 2, 2', 2'' above), right? Specifically, the "gonality grows linearly" statement and/or the "only finitely many subgroups of a given index" statement are dependent on $\ell$, yes? $\endgroup$ – Bobby Grizzard May 2 '15 at 19:37
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    $\begingroup$ Sorry - the bound obtained in this situation does depend on $\ell$, but not on $E$. $\endgroup$ – Jeremy Rouse May 2 '15 at 22:01

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