If $E$ is a non-CM elliptic curve over $Q$, then it is a famous theorem of Serre
that there is some integer $M(E)$ such that for any prime $\ell > M(E)$, the image of the
Galois representations $\rho_{E,l}: G_{\mathbb Q} \rightarrow Gl_2(\mathbb F_\ell)$
on the $\ell$-torsion points of $E$ is surjective. If we define $M(E)$ as the smallest integer
having this property, then Serre's bounded uniformity conjecture is that $M(E)$ is bounded above by an absolute constant (41 perhaps) when $E$ varies over all non-CM-elliptic curve over $\mathbb Q$.
Let $N(E)$ be the product of all primes where $E$ has bad reduction.
My question is:
What are the current best bounds (if any) of $M(E)$ in terms of $N(E)$, both unconditionally and under GRH (the second being the case that interests me most)?
I kind of get lost in the immense literature on the subject. As is well-known, there are four types of proper maximal subgroups of $Gl_2(\mathbb F_\ell)$, which are (1) Borel subgroups,
(2) (resp. (3)) Normalizer of split (resp. non-split) Cartan subgroups, (4) exceptional ones (icsoahedral, dodecahedral, etc), so if $\rho_{E,\ell}$ fails to be surjective, it lands in a
subgroup of one of those type, and we can define four integers $M^i(E)$ for $i=1,..,4$
,as the smallest integer such that for $\ell$ larger that $M^i(E)$, $\rho_{E,\ell}$ does not
fall into a subgroup of type (i). Please tell me if I am not correct (I am troubled by Lemma 17 page 197 of this paper of Serre)
, but it is known
that $M^4(E)$,$M^1(E)$,$M^2(E)$ are bounded by an absolute constant (independent of $E$)
due to results of Serre, Mazur, and Bilu-Parent (respectively and in chronological order,
the last one being very recent). So the only problem that remains for Serre's
uniformity conjecture would be to bound $M^3(E)$ uniformly, and for my question to bound it at least in terms of $N(E)$.
Best Answer
Since no specialist has replied to this question, I will add a long comment about the little I know.
The unconditional bound depending on the conductor which was used in the implementation in sage comes from Theorem 2 in
A.C. Cojocaru, On the surjectivity of the Galois representations associated to non-CM elliptic curves, http://homepages.math.uic.edu/~cojocaru/cojocaru-CanMathBull-2005.pdf
But I could imagine that there are better bounds out there. The author of that paper may be a good guess of who to ask. Or Drew Sutherland; his talk http://math.mit.edu/~drew/JMM2013.pdf on computing the image of $\rho_{E,\ell}$ gives a list of what possible images were found yet.
The Lemme 17 of Serre you are referring to just excludes that the image is in the non-split Cartan for $\ell>2$. It does not exclude that it is in the normaliser of a non-split Cartan. There are indeed examples where this happens for $\ell=5, 7 ,11$ for instance.
I believe, too, that the only absolute bound we do not know is $M^3(E)$. For the others we know that $M^1(E)\leq 37$, $M^2(E)\leq 13 $ and $M^4(E)\leq 13$. These are optimal with the exception of $M^2(E)$, which we believe to be at most $7$, see a recent paper by Bilu-Parent-Rebolledo. Finally for $M^3(E)$, we could conjecture that it is at most $11$.