Serre's open image theorem for products of elliptic curves over function fields via specialization - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T22:37:22Z http://mathoverflow.net/feeds/question/76526 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76526/serres-open-image-theorem-for-products-of-elliptic-curves-over-function-fields-v Serre's open image theorem for products of elliptic curves over function fields via specialization Adam Harris 2011-09-27T15:35:06Z 2011-09-30T16:44:50Z <p>In Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15, 259--331 (1972), Serre proved the following (Theorem 6 ′′, p. 325):</p> <p>Let $K$ be a number field and let $K^{cycl}$ be the cyclotomic extension of $K$ generated by all roots of unity. Let $E$ and $E'$ be two elliptic curves such that, over $\bar{K}$,</p> <p>(i) $E$ and $E'$ have no complex multiplication;</p> <p>(ii) The $l$-adic representations $(\rho_l)$, $(\rho'_l)$ attached to $E$ and $E'$ don't become isomorphic over any finite extension of $K$.</p> <p>Then $K(E_{tors}) \cap K(E'_{tors})$ is finite over $K^{cycl}$.</p> <p>My question is whether this holds for $E$ and $E'$ defined over a function field? If this hasn't already been considered somewhere with an argument specific to the function field case, then maybe a specialization argument might work? Could anyone please provide a reference where there are similar specialization arguments used, or a standard reference for the basic theory of these specialization theorems?</p> http://mathoverflow.net/questions/76526/serres-open-image-theorem-for-products-of-elliptic-curves-over-function-fields-v/76639#76639 Answer by Felipe Voloch for Serre's open image theorem for products of elliptic curves over function fields via specialization Felipe Voloch 2011-09-28T13:35:26Z 2011-09-28T13:35:26Z <p>If $K$ is a function field over an algebraically closed field and one of your elliptic curves is constant (which does not necessarily violate your hypotheses unless the constant field is the algebraic closure of a finite field) then the answer is no. What kind of constant field are you interested in? You might want to add some non-isotriviality condition. The person to ask is probably Chris Hall, but I don't think he reads MO.</p>