Reciprocity law for number fields defined by torsion points of modular elliptic curves - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T03:08:26Z http://mathoverflow.net/feeds/question/32607 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/32607/reciprocity-law-for-number-fields-defined-by-torsion-points-of-modular-elliptic-c Reciprocity law for number fields defined by torsion points of modular elliptic curves Chandan Singh Dalawat 2010-07-20T09:08:42Z 2010-07-24T03:05:42Z <p>Shimura (Crelle 221, 1966) considers the elliptic curve $E:y^2+y=x^3-x^2$ (although he doesn't use this equation) of conductor $11$ whose associated modular form is $$q\prod_{k=1}^{+\infty}(1-q^k)^2(1-q^{11k})^2=\sum_{n=1}^{+\infty}c_nq^n$$ where $q=e^{2i\pi\tau}$ and $\tau$ is in the upper half of $\bf C$. For a prime $l$, he denotes by $K_l$ the extension of $\bf Q$ obtained by adjoining the $l$-torsion points of $E$ and shows that if $l\in[7,97]$, then ${\rm Gal}(K_l|{\bf Q})$ is isomorphic to ${\rm GL}_2({\bf F}_l)$. </p> <p><strong>Question.</strong> Is ${\rm Gal}(K_l|{\bf Q})$ now known to be isomorphic to ${\rm GL}_2({\bf F}_l)$ even for $l>97$ ?</p> <p>Even if the faithful representation ${\rm Gal}(K_l|{\bf Q})\rightarrow{\rm GL}_2({\bf F}_l)$ fails to be surjective for a few $l>97$, does the recent proof of Serre's modularity conjecture not imply the</p> <p><strong>Statement</strong>. For every prime $l>5$ and every prime $p\neq11,l$, the characteristic polynomial of ${\rm Frob}_p$ (thought of as an element of ${\rm GL}_2({\bf F}_l)$) is $\equiv X^2-c_pX+p \pmod l$ ?</p> <p>Shimura shows this only for $l\in[7,97]$. </p> <p><strong>Addendum.</strong> (2010/07/24) Looking at Shimura's paper beyond the first page shows that he actually proves (Section 3) that the characteristic polynomial for the action of ${\rm Frob}_p$ on the $l$-adic Tate module $T_l(E)$ is </p> <p>$X^2-a_pX+p\in{\bf Z}_l[X]$ </p> <p>for all primes $l$ and $p\neq11, l$ and (Section 6) that $a_p=c_p$ for all $p\neq11$. And yes, he does use the Eichler-Shimura relation. </p> http://mathoverflow.net/questions/32607/reciprocity-law-for-number-fields-defined-by-torsion-points-of-modular-elliptic-c/32616#32616 Answer by Chris Wuthrich for Reciprocity law for number fields defined by torsion points of modular elliptic curves Chris Wuthrich 2010-07-20T11:26:04Z 2010-07-20T14:10:03Z <p>The first question is answered in Serre's <a href="http://dx.doi.org10.1007/BF01405086" rel="nofollow">[Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, <em>Invent. Math.</em> <strong>15</strong>:4 (1972) 259--331]</a>, on page 304, section 5.2, exactly for this curve. In general this paper give a good way to determine for which $\ell$ the mod-$\ell$ representation is not surjective. <em>Sage</em> can do that efficiently for a given curve.</p> <p>For every prime $p$ different from $\ell$ and $11$, the characteristic polynomial of $\rm{Frob}_p$ is indeed $T^2 - c_p T +p$ in $\mathbb F_{\ell}[T]$. The isomorphism $\rm{Gal}(K _{\ell}/\mathbb Q) \to \rm{Aut}(E[\ell])=\rm{GL} _2(\mathbb F _{\ell})$ sends $\rm{Frob}_p$ to the Frobenius endomorphism $\phi:E[\ell] \to E[\ell]$ on $E/\mathbb{F}_p$. Your $c_p$ is the trace of $\phi$ and $p$ is the determinant of it since the Eichler--Shimura relation shows that $c_p$ is the Fourier coefficent of the associated modular form. See <a href="http://mathoverflow.net/questions/19390/intuition-behind-the-eichler-shimura-relation/19399#19399" rel="nofollow">this answer</a> for why it is so.</p>