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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 surjectibe. 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 $N(E)$.

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I heard a talk by Ribet at MSRI on something similar a few years ago but can not seem to recollect nor find the notes from that talk. Sorry to be unhelpful. –  Arijit May 3 '13 at 13:59
    
That's already helpful: now I can ask Ribet. Thanks. –  Joël May 3 '13 at 15:22
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2 Answers

up vote 6 down vote accepted

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 $11$.

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I should add that I don't know about GRH-conditional bounds. But it feels a bit awkward to assume such a strong conjecture when instead you could assmue the conjecture $M(E)\leq 37$; which may well be proven one day. –  Chris Wuthrich May 13 '13 at 10:50
    
Thanks Chris. Actually, since I got no answer I has asked Pierre Parent. He pointed me to the same paper by A.C. Cojocaru for the best unconditional bound he would know of. As for the best bound under GRH, according to Cojocaru, the best one at the time she was writing at least are the old ones derived by Serre at the end of his paper "Quelques applications du théorème de densité de Chebotarev". –  Joël May 13 '13 at 14:02
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So I think I can now answer my own question about the best known bound under GRH for M(E) in terms of N(E). The accepted answer of Chris refers to a recent paper by Cojoacru which in its introduction reviews quickly the history and cites Serre's result (in his paper on Chebotarev) as the best. This result is (theorem 22, under GRH) $$ M(E) \ll \log N(E) (\log \log 2N(E))^3.$$ This result is deduced from an other one:

Theorem 21': If $E$ and $E'$ are two non-isogenous elliptic curves over $\mathbb Q$, which can be deduce from each other by torsion by a quadratic character, then there exists, under GRH, $$p \ll (\log N(E,E'))^2 (\log \log 2 N(E,E'))^6$$ such that $a_p \neq a'_p$. Here $N(E,E')$ is the products of prime of ramifications of $E$ or $E'$, and $a_p=1+p-|E(\mathbb F_p)|$, $a'_p=1+p-|E(\mathbb F_p)|$ as usual.

Without the hypothesis that $E$ is obtained from $E'$ by torsion, Serre proves the same result with the exponent 6 replaced by 12 (Theorem 21).

Now I say that those results of Serre are not optimal anymore. More precisely, the $\log \log$ factors can be removed (under GRH) in Theorem 21, 21', and 22. The proof uses the modularity of $E$, $E'$, which replaces those curves by modular forms. One then can use the Rankin-Selberg result, which gives Theorem 21 and 21' without $\log \log$ factors, cf. Iwaniec-Kowalski, Proposition 5.22 page 118. Theorem 22 without the \log\log factors follows by the same argument that Theorem 22 followed from theorem 21' with the \log \log factor. So in short, we have under GRH $$M(E) \ll \log N(E).$$

I learnt about this when I submitted an article improving (among many other things fortunately) the exponent of $\log \log$ in Serre's theorem 22. The referee told me that the $\log \log$ factor was not needed. I guess that when no modularity is available, say for elliptic curve over number fields, where you can prove similar results with similar methods, then the best results are still Serre-like, that is with a $\log \log$-factor (which my method allows to reduce the exponent of, but not to completely remove).

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