Given a number field $K$, is there a prime number $p_{K}$ such that, for any elliptic curve $E$ over $K$ without complex multiplication, the residual $\pmod p$ Galois representation $\overline{\rho}_{E,p}$ is surjective onto $GL_{2}(\mathbb{F}_p)$, whenever p is a prime larger than $p_{K}$ ? In the case of $\mathbb{Q}$ we believe that $p_{\mathbb{Q}}=37$, but it is not done ! My question is the following : what is the answer of this conjecture in the case of modular forms the integer weight $>2$ if it exists ?
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3$\begingroup$ This question is somewhat unclear as asked. Are you interested in the problem of fixing a weight $k$, and considering the Galois representations attached to newforms of weight $k$ and asking about their images? Are you interested in restricting the Nebentypus of the modular forms? Are you only interested in representations that land in $GL_{2}(\mathbb{F}_{p})$, which only arise from modular forms $f$ for which there is a degree $1$ prime over $p$ in the ring of integers of the coefficient field? $\endgroup$– Jeremy RouseCommented Feb 10, 2017 at 17:37
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$\begingroup$ Consider all newforms of any weight and any level whose Fourier coefficients lie in the number field $K$. I am asking if there is a prime $p_{K}$ such that for any prime $p>p_{K}$, the $\pmod p$ representations associated to the above newforms ( by the work of P.Deligne ) are all surjective ? $\endgroup$– Zakariae.BCommented Feb 11, 2017 at 9:50
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1$\begingroup$ In contrast to the case of elliptic curves, the Galois representations associated to more general modular forms are very rarely surjective (no matter how large $p$ is): there is an obstruction coming from so-called "inner twists". $\endgroup$– David LoefflerCommented Feb 12, 2017 at 9:07
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