2
$\begingroup$

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 $p$?

$\endgroup$
0
$\begingroup$

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

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.