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

2010-07-20T09:08:42Z

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 ${\rm GL}_2({\bf F}_l)$.

Question. Is ${\rm Gal}(K_l|{\bf Q})$ now known to be isomorphic to ${\rm GL}_2({\bf F}_l)$ even for $l>97$ ?

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

Statement. 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 $\equiv X^2-c_pX+p \pmod l$ ?

Shimura shows this only for $l\in[7,97]$.

Addendum. (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

$X^2-a_pX+p\in{\bf Z}_l[X]$

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.

Answer by Chris Wuthrich for Reciprocity law for number fields defined by torsion points of modular elliptic curves

2010-07-20T11:26:04Z

The first question is answered in Serre's [Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15:4 (1972) 259--331], 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. Sage can do that efficiently for a given curve.

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 this answer for why it is so.

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

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

Answer by Chris Wuthrich for Reciprocity law for number fields defined by torsion points of modular elliptic curves