Timeline for Reciprocity law for number fields defined by torsion points of modular elliptic curves

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Apr 13, 2017 at 12:58	history	edited	CommunityBot		replaced http://mathoverflow.net/ with https://mathoverflow.net/
Jul 20, 2010 at 14:49	vote	accept	Chandan Singh Dalawat
Jul 20, 2010 at 14:49	comment	added	Chandan Singh Dalawat		I think the best reference is Cor.1 on p.308 of Serre's paper you mention. It says that if $E$ is a semistable elliptic curve over ${\bf Q}$ and if $p$ is the smallest prime where $E$ has good reduction, then ${\rm Gal}(K_l\|{\bf Q})\to{\rm GL}_2({\bf F}_l)$ is surjective for every prime $l>(\sqrt p+1)^2$.
Jul 20, 2010 at 14:27	comment	added	Chandan Singh Dalawat		His method seems to need the restriction $l\in[7,97]$ for the surjectivity ${\rm Gal}(K_l\|{\bf Q})\to{\rm GL}_2({\bf F}_l)$, though.
Jul 20, 2010 at 14:27	comment	added	Chandan Singh Dalawat		Many thanks for for the reference to Serre: I should have looked it up before asking the question. What I found strange was that Shimura was claiming the "Statement" only for $l\in[7,97]$, whereas it seems to follows for all $l$ from the fact that the representation $\rho_l:{\rm Gal}(K_l\|{\bf Q})\to{\rm GL}_2({\bf Z}_l)$ is unramfied at every $p\neq11,l$, that for these $p$ the characteristic polynomial of ${\rm Frob}_p\in{\rm GL}_2({\bf Z}_l)$ is $T^2-a_pT+p$ where $a_p$ is defined by ${\rm Card}(E({\bf F}_l))=1-a_p+p$, and finally the fact that $a_p=c_p$.
Jul 20, 2010 at 14:10	history	edited	Wadim Zudilin	CC BY-SA 2.5	typos fixed
Jul 20, 2010 at 11:26	history	answered	Chris Wuthrich	CC BY-SA 2.5