Intersection of field extensions of torsion points of non-isogenous elliptic curves - MathOverflow

Intersection of field extensions of torsion points of non-isogenous elliptic curves

2011-09-08T16:54:18Z

Let $E$ and $E'$ be non-isogenous elliptic curves over a field $k$ (characteristic 0) such that $Gal(k(E[p^{\infty}])/k)=Gal(k(E'[p^{\infty}])/k) = SL_2(\mathbb{Z}_p)$ with $p \geq 5$ (where $E[p^{\infty}]$ is the set of $p^n$ torsion points of $E$ for all $n$). Then is it true that $k(E[p^{\infty}])\cap k(E'[p^{\infty}]) = k$, or can someone provide a counterexample?

Answer by ACL for Intersection of field extensions of torsion points of non-isogenous elliptic curves

2011-09-08T20:10:13Z

Since both fields $K(E_{l^\infty})$ and $K(E'_{l^\infty})$ contain the $l$-adic cyclotomic extension of $K$, your expectation cannot hold. However, this is almost the only obstruction.

In Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15, 259--331 (1972), J-P. Serre proved the following Theorem (Theorem 6$''$, p. 325).

Let $K$ be a number field, let $K^{\rm cycl}$ be the (cyclotomic) extension of $K$ generated by all roots of unity. Let $E$ and $E'$ be two elliptic curves such that, over $\bar K$,

(i) $E$ and $E'$ have no complex multiplication;

(ii) $E$ and $E'$ are not isogeneous.

Then, the extensions $K(E_\infty)$ and $K(E^\prime_\infty)$ of $K^{\rm cycl}$ are almost disjoint: $K(E_\infty)\cap K(E'_\infty)$ is finite over $K^{\rm cycl}$.

(By Faltings, hypothesis (ii) is equivalent to the one given by Serre.)

Answer by JSE for Intersection of field extensions of torsion points of non-isogenous elliptic curves

2011-09-09T01:54:43Z

By the way, I think that under your hypotheses, your question is really about group theory, not about algebraic geometry. Namely: the action of Galois on E[p^infty] x E'[p^infty] gives you a homomorphism

G_K -> SL_2(Z_p) x SL_2(Z_p).

Call the image H. By your hypothesis, H projects surjectively onto both copies of SL_2(Z_p). You also know that H is not contained in any conjugate of the diagonal (if it were, E[p^infty] and E'[p^infty] would be isomorphic Galois representations and I'm presuming you're in a situation where Faltings rules that out -- you'd better be, if you want an affirmative answer to your question.)

Now what you have to prove is that a subgroup of SL_2(Z_p) x SL_2(Z_p) which projects surjectively onto each direct summand and which is not conjugate to a subgroup of the diagonal must be finite-index in SL_2(Z_p) x SL_2(Z_p). This is true for SL_2(F_p) by Hall's lemma and I think you can induct from there (but didn't think about it carefully.)