Mazur's conjecture on the image of Galois representations of Elliptic curves states that for $N$ large enough there is a unique elliptic curve $E$ over $\mathbb{Q}$ giving rise to a fixed mod $N$ Galois representation $\bar{\rho}: G_{\mathbb{Q}}\rightarrow GL_2(\mathbb{Z}/N\mathbb{Z})$. Is there a similar expectation for normalized Hecke newforms of a fixed weight? In greater detail, let $p$ be a prime, $\bar{\rho}: G_{\mathbb{Q}}\rightarrow GL_2(\mathbb{Z}/p\mathbb{Z})$ be a fixed irreducible Galois representation and $k\geq 2$ a fixed integer, is the set of normalized Hecke newforms $f$ with weight $k$ and rational coefficients whose associated residual Galois representation coincides with $\bar{\rho}$ expected to be a finite set when $p$ is large enough? If not, is there a heuristic why it isn't the case?
1 Answer
This is not true. Take $k=2$, and $p \geq 5$. Let $\ell$ be a prime such that $p$ divides $\ell1$. Then we know (by Mazur) that there exists a newform of weight $2$ and level $\Gamma_0(\ell)$ whose residual semisimple representation is $\overline{\rho} = 1 \oplus \overline{\chi}_p$ where $\overline{\chi}_p$ is the modulo $p$ cyclotomic character. The set of such $\ell$ is infinite. Note however that the newform we get almost never have rational coefficients (I think except if $p=5$ and $\ell=11$ in which case $X_0(11)$ is an elliptic curve).

1$\begingroup$ Thank you, Emmanuel, this is indeed the correct answer, I had anticipated that the question needed rephrasing, is it as easy if I insist that the new form have rational coefficients? I'm actually happy that there is an abundance of newforms which realize the representation since it has to to with a certain question I've been thinking about. I hope you don't mind that I'm editing the question. $\endgroup$– user130124Oct 23, 2018 at 1:56

$\begingroup$ Indeed, if you want the newforms to have rational coefficients then it becomes much more interesting. I don't know what to expect in this case. $\endgroup$ Oct 23, 2018 at 2:15