Let $D$ be a totally definite quaternion algebra over a totally real number field $F$. Let $U$ be an open compact subgroup of $D(\mathbb{A}_F)^\times$, maximal compact almost everywhere. Consider automorphic forms on $D$ as functions on the usual adelic double quotient (which here is a finite set). There are Hecke operators etc. as in the $GL_2$ setting.

Suppose $f$ is a Hecke eigenform, giving rise to an automorphic representation $\pi_f$ with local components $\pi_{f,\nu}$.

What is knows about these $\pi_{f,\nu}$?

I've read claims that they are always Steinberg (or unramified?). What would be a good reference for this?

More specific (but kind of sloppy): Let $p$ be a fixed rational prime, $\rho_f$ be the mod $p$ Galois representation associated to $f$ and call $\nu$ a Taylor-Wiles prime of $F$, if

  • $\rho_f$ is unramified at $\nu$ with eigenvalues $\alpha\neq\beta$ of $\rho_f(Frob_\nu)$
  • $D$ splits at $\nu$
  • $U_\nu$ is contained in the Iwahori subgroup and $det(U_\nu)\equiv 1 $ in the $p$-power quotient of $k_\nu^\times$
  • the Hecke operator at $\nu$ has eigenvalue $A_\nu$ on $f$; where $A_\nu$ is a lift of one of $\alpha$.

Then, supposedly, at Talor-Wiles primes $\nu$ it holds that $\pi_{f,\nu}$ is not Steinberg but tamely ramified principal series.

How is this? And how does this fit together with the unramifiedness asserted above?

  • 3
    $\begingroup$ Where did you read about the claims that the local components at $\nu$ are always Steinberg or unramified? That doesn't seem right. And by the way, in this statement, what is $\nu$? any finite place of $F$? any finite place where $D$ is split? $\endgroup$
    – Joël
    Jun 19, 2013 at 17:33
  • 3
    $\begingroup$ Dear Konrad, Remember that at all but finitely many primes, $D^{\times}$ is just $\GL_2$, so the local components can be whatever they want to be. In particular, the TW primes will always be primes where $D$ is split, and then we add level structure that, from the automorphic viewpoint, amounts to considering automorphic reps. whose local comps. at the TW primes are tame principal series. At the ramified primes, the local components have to be reps. of the local quat. alg., which are ... whatever they are. (Perhaps you want to know what they correspond to on the GL_2 side of... $\endgroup$
    – Emerton
    Jun 19, 2013 at 19:44
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    $\begingroup$ ... Jacquet--Langlands? One of the basic properties of JL is that its image consists of those GL_2 reps. that are discrete series, i.e. twist of Steinberg or supercuspidal.) The short answer is that your comment about unramified or Steinberg is completely wrong. Regards, $\endgroup$
    – Emerton
    Jun 19, 2013 at 19:45
  • $\begingroup$ Thank you both. It seems I should first thoroughly read up on JL. Still I don't see, how the added level at TW primes rules out the possibility of Eisenstein-ness. $\endgroup$
    – Konrad
    Jun 20, 2013 at 9:02
  • 2
    $\begingroup$ Dear Konrad, If $\overline{\rho}$ is a mod $p$ Galois rep'n coming from level prime to $q$, then there are some conditions (level-raising conditions, due to Ribet) which must be satisfied if $\overline{\rho}$ can also arise from a modular form which is Steinberg at $q$. When you choose the TW primes, you choose them in such a way that these level-raising conditions are not satisfied. In this way, the possibility of Steinberg at the TW primes is ruled out. (Look in any discussion of the TW method, e.g. the one in the Fermat's Last Theorem book, and you will see this explained.) Regards, $\endgroup$
    – Emerton
    Jun 20, 2013 at 13:54


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