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Let A be an abelian variety defined over a number field F. For a prime number$\ell$, we consider the field extension of F generated by the $\ell$-poweredtorsion points of A. According to a conjecture made by Rasmussen and Tamagawa,if we require these fields to be both a pro-$\ell$ extension of$F(\mu_{\ell^{\infty}})$ and unramified away from $\ell$, examples are quiterare. Indeed, it is expected that for a fixed dimension and field ofdefinition, there exists such an abelian variety for only a finite number ofprimes.  We prove a uniform version of the conjecture in the case where the abelianvarieties are elliptic curves with complex multiplication. In addition, weprovide explicit bounds in cases where the number field has degree less than orequal to 100.

A Uniform Version of a Finiteness Conjecture for CM Elliptic Curves