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I am interested in the set of number fields $K$ having the property that for \emph{any semistable} elliptic curve $E$ defined over $K$, there exists a constant $c(E,K)>0$ such that $$p>c(E,K)\Longrightarrow \rho_{E,p}:\mathrm{Gal}(\overline{K}/K)\longrightarrow \mathrm{Aut}(E[p])\textrm{ is irreducible}.$$ Since such a constant $c(E,K)$ exists if and only if $\mathrm{End}_K(E)=\mathbf{Z}$, an equivalent formulation of the above property is~: for any elliptic curve $E/K$, we have $$\mathrm{End}_K(E)\not=\mathbf{Z}\Longrightarrow E\textrm{ has bad reduction at a finite place of }K.$$ There are lots of examples of such number fields (e.g. number fields which do not contain the Hilbert class field of some imaginary quadratic field), but I wonder whether there exists a nice characterization of the whole set.

Many thanks in advance for your answers!

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I added tag "nt.number-theory" -- there's a convention that MO questions should have at least one Arxiv subject area tag. – David Loeffler May 9 '11 at 10:04
So you wish to understand the field over which cm-curves achieve good reduction everywhere. Maybe Serre-Tate's "Good reduction of abelian varieties" helps. – Chris Wuthrich May 9 '11 at 12:25

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