Semistable Elliptic Curves and irreducible Galois representations - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T12:10:45Z http://mathoverflow.net/feeds/question/64363 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/64363/semistable-elliptic-curves-and-irreducible-galois-representations Semistable Elliptic Curves and irreducible Galois representations Nicolas B. 2011-05-09T08:36:32Z 2011-05-09T10:03:27Z <p>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.</p> <p>Many thanks in advance for your answers!</p>