This question is a variant of this one.

Let $f$ be as in the other question, but suppose that we look at the $\ell$-adic representiation attached to $f$: $$ \rho_f : G_{\mathbb Q} \to \operatorname{GL}_2(E), $$ where $E$ is a finite extension of $\mathbb Q_\ell$, with $\ell$ a prime that does not divide $N$. Let $p \neq \ell$ be another prime that does not divide $N$.

What can be said about $\rho_{f,p}$, the restriction of $\rho_f$ to a decomposition subgroup at $p$?


The following is (very) well known. Let's assume $f$ is a normalised newform of weight $k \geq 2$, nebentype $\chi$ with $p$th Fourier coefficient $a_p$. Then your representation $\rho_{f,p}$ (obtained as a piece of the cohomology of the modular curve) is unramified, and the characteristic polynomial of Frobenius is $X^2 - a_p X + \chi(p)p^{k-1}$. This is the Eichler-Shimura relation, proved in this generality following the argument in Deligne's original paper.

In weight $k = 1$, the Galois representation has finite image and so Frobenius is always semisimple. In weight $k \ge 2$, it is conjectured that the polynomial $X^2 - a_p X + \chi(p)p^{k-1}$ always has distinct roots, and hence Frobenius is always expected to be semisimple. This is known in weight $k = 2$, but unknown in general, although it does follow from the Tate conjecture; see http://www.wstein.org/people/coleman/papers/coleman-edixhoven.pdf

  • $\begingroup$ By the Petersson conjectures, you know the eigenvalues are different, so if you increase $E$ you can diagonalize it. $\endgroup$ – A. Pacetti Jul 8 '14 at 2:46
  • $\begingroup$ Can I just ask @TomLovering -- was this latest edit by you? $\endgroup$ – Yemon Choi Jul 10 '14 at 1:55
  • $\begingroup$ No. I originally posted saying that I wasn't sure what is known about semisimplicity in general (I had a vague idea it was known in some cases and conjectured in the rest but not sure which). Thanks to "community" for clearing it up. $\endgroup$ – Tom Lovering Jul 14 '14 at 21:35
  • $\begingroup$ In weight one, $\rho_f:G_\mathbb{Q} \rightarrow \mathbb{GL}_2(\mathbb{C})$ is unramified outside the level $N$ of $f$ and there is an exemple where the Frobenius at $p \nmid N$ is not semi-simple (p-irregular weight one form) , see the article of Dimitrov-Ghate. $\endgroup$ – Adel BETINA May 10 '16 at 9:35

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