# Is it expected that the mod $p$ representation determines a normalized Hecke newform of fixed weight for p large enough?

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 $$\ell-1$$. Then we know (by Mazur) that there exists a newform of weight $$2$$ and level $$\Gamma_0(\ell)$$ whose residual semi-simple 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).

• 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. – user130124 Oct 23 '18 at 1:56
• 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. – Emmanuel Lecouturier Oct 23 '18 at 2:15