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I'm wondering about the relationship between two formulations of Ihara's lemma for $\text{GL}_2$ I've seen:

(1) the "concrete" version given in, for example, Darmon, Diamond, and Taylor, which says that the sum of the two $p$-degeneracy maps from level $N$ to level $Np$ is injective away from the Eisenstein locus with $l$-adic coefficients (or the dual statement, which I think is the one in there as lemma 4.24 or nearby in DDT). I like the hands-on feel of this one, because of things like Ribet's article on how to raise the level in lifting modular forms.

(2) the automorphic formulation as given in, say, Shephard-Barron, Harris, and Taylor, which asserts that the mod $l$ local component at $p$ of the automorphic rep $\pi_p$ associated to a non-Eisenstein global automorphic rep has no finite-dimensional subobjects, i.e. admits a Whittaker model. I thought it might be clear how it's related, since the Whittaker model has the interpretation in terms of oldforms, but I admit it defeated me, probably because I'm not very comfortable yet with the automorphic formalism. So my primary question is:

1) How does one see that these are equivalent? Or maybe they're not quite equivalent?

I'd also like to clarify for myself:

2) What exactly does "generic" mean in automorphic representation theory? Is it admitting a Whittaker model? Not having a f.d. subobject? (Are these equivalent?) In what sense are these known to be "generic" for more general reductive groups?

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  • $\begingroup$ @Kimball The question is about mod $\ell$ representations, while words like "tempered" and "packet" are more usually defined for representations with characteristic 0 coefficients. Is it clear what these words should mean in finite characteristic? $\endgroup$
    – David Loeffler
    Feb 8, 2018 at 9:15
  • $\begingroup$ @DavidLoeffler You're right, I was thinking about characteristic 0 reps. $\endgroup$
    – Kimball
    Feb 8, 2018 at 12:46

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