# Surjectivity in Deligne-Serre

Let $$f$$ be a newform of weight $$k$$ and level $$N$$ with integer coefficients. Deligne-Serre theorem theorem says there exist a nice associated representation $$\rho_{f}^{(\ell)}:\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \text{GL}_2(\mathbb{Q}_{\ell}),$$ for any prime $$\ell$$ not dividing $$N$$. In particular, this induces a representation $$\rho_{f,\ell}:\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \text{GL}_2(\mathbb{F}_{\ell}).$$ Due to Serre's open image theorem, both adelic and mod $$\ell$$ representations are surjective for all but finitely many $$\ell$$, when $$f$$ has weight $$2$$ and comes from an elliptic curve. For the general case (especially when the weight is more than $$2$$ ?) is the mod $$\ell$$ representation coincides with the matrices having determinant a $$(k-1)^{th}$$ power in $$\mathbb{F}_{\ell}^*$$, for all but finitely many $$\ell$$ ? If not, what's stopping it to be ? and what we can say about the images ?

## 1 Answer

In the setting that $$f$$ has integer Hecke eigenvalues, then the result is true, so long as $$f$$ has weight $$k\ge 2$$ and does not have complex multiplication. See Theorem 3.1 of this paper by Ribet: in this setting, $$R=\mathbb Z$$.

In general, the Hecke eigenvalues of $$f$$ will all lie in a finite extension $$E$$ of $$\mathbb Q$$, and $$\rho_f$$ will be valued in $$\mathrm{GL}_2(E_{\lambda})$$ for a prime $$\lambda$$ above $$\ell$$. Let $$\mathbb F_\lambda$$ be the residue field of $$E_\lambda$$. If $$k\ge 2$$ and $$f$$ does not have CM, then, for all but finitely many primes, the image of the mod $$\lambda$$ representation $$\overline{\rho}_f$$ in $$\mathrm{GL}_2(\mathbb F_\lambda)$$ will contain a subgroup conjugate to $$\mathrm{SL}_2(\mathbb F_\ell)$$ (but not necessarily $$\mathrm{SL}_2(\mathbb F_\lambda)$$).

To say anything further, we have to be a lot more careful: I'd suggest looking at the answers to this question.

• Yes, I am happy when eigenvalues are integer.s In the Ribet's paper, are you talking about (2) at page 186 ? but that's only said for level 1, no ? – dragoboy May 20 '20 at 12:42
• @dragoboy I'm referring to Theorem 3.1 on p191. You're right that the result on p186 only applies to level $1$ forms. – Ariel Weiss May 20 '20 at 12:44
• Great! thanks... – dragoboy May 20 '20 at 13:04