How should I think about Ihara's Lemma?

Solution 1:

My preferred way is to think in an automorphic/rep'n theoretic way, since this is how to phrase the conjectural generalization to $GL_n$ (as discussed say in various papers of Taylor and his collaborators).

The first thing to know is that if $f$ is a cuspidal Hecke eigenform, then it gives rise to (indeed, generates, if one phrases things appropriately) an irrep. of $GL_2(\mathbb A_f)$ (where $\mathbb A_f$ is the ring of finite adeles). (In fact one can even get a rep'n of $GL_2(\mathbb A)$ for the full adelic group, but we don't need to consider the action of $GL_2(\mathbb R)$ in what follows.) This is a tensor product of irreps. of $GL_2(\mathbb Q_p)$ for each prime $p$. These are traditionally denoted $\pi_p$.

Concretely, the rep'n $\pi_p$ can be obtained as follows: think about all the oldforms generated by $f$ by adding all powers of $p$ to the level (not just $\Gamma_0$ $p$-power level, but full $p$-power level, i.e. passing to the (disconnected) modular curve of full level $p^n$, plus whatever level structure $f$ had to begin with).

This space of $p$-power old forms has a natural action of $GL_2(\mathbb Q_p)$, and is is precisely $\pi_p$.

Now a basic theorem is that $\pi_p$ contains no finite-dim'l subreps. This can be proved via looking at $q$-expansions, for example. (Basically, a finite dim'l subrep. could only have constant $q$-expansion, but this is not possible for a cuspform.)

Ihara's lemma is the analogous statement when we work with mod $\ell$ cuspforms for some prime $\ell$ (and we take $p \neq \ell$ in the above discussion).

One can either work directly with mod $\ell$ modular forms, as in the papers of Serre, Swinnerton--Dyer, etc., or instead one can use Eichler--Shimura (at least philosophically) to replace modular forms by cohomology, and then use cohomology with mod $\ell$ coefficients.

Rather than just working with cuspforms in the mod $\ell$ setting, it is important to work with the $\pi_p$ generated by a non-Eisenstein eigenform.

The different statements you can find in the literature are not all literally equivalent, but they all imply my formulation here, and this formulation is the key point.

Now "contains no finite-dim'l subreps.'' has a more theoretical interpretation as "admits a Whittaker model'', which is sometimes also phrased as being "generic''. (This is a rep'n theoretic condition which is something like admiting at least one non-zero non-constant Fourier coefficient.) It makes sense if we replace $GL_2$ by $GL_n$, and is true for cuspidal automorphic reps. of $GL_n$.

The analogous mod $\ell$ statement is not known in general (though it is conjectured); the problem (or at least one problem) is that when $n > 2$, it it not enough to eliminate finite-dimensional subreps.; there can be infinite-dim'l irred. reps. that are not generic (i.e. don't admit Whittaker models). (There are heirarchies of infinite dimensionality for reps. of $GL_n(\mathbb Q_p)$, and those that admit Whittaker models are at the top --- they are the "most" infinite dimensional; for $GL_2$ this is the only member of the heirarchy, but when $n > 2$ there other "less" infinite-dimensional reps.)

The first paper of Clozel--Harris--Taylor on Sato--Tate was contingent on Ihara's Lemma for $GL_n$ (and if you look at Taylor's ICM talk you can see a formulation of Ihara's Lemma close to the one I have here). In his follow-up solo paper, Taylor found a way around it (via a method people in the field often call "Ihara avoidance").

The role that Ihara's Lemma plays in modularity theorems is that implies then when there are congruences between $\ell$-adic Galois reps. of different conductor, and the one of lower conductors is modular, then the one of higher conductor is also modular (one passes to an oldform mod $\ell$, which we can find because of the genericity, i.e. infinite-dimensionality, that Ihara's Lemma guarantees, and then lifts it to a form that is actually new at the higher conductor). So Ihara's Lemma is what lets you prove modularity theorems for $\ell$-adic reps. by induction on the prime-to-$\ell$ conductor. (In [DDT], you will see that it is used to go from the minimal case to the non-minimal case, which is one formulation of the induction I am discussing.)

Taylor's Ihara avoidance allowed him to make a form of this inductive argument anyway (the argument uses a subtle comparison of the geometry of the generic and special fibres of various Galois def. spaces).

Still, there is considerable interest in proving the $GL_n$ form of Ihara's Lemma. E.g. I think the recent work of Clozel and Thorne uses some forms of Ihara. Also, David Helm and I described how to interpolate local Langlands for $GL_2(\mathbb Q_p)$ in $\ell$-adic families, but to know that this is what actually occurs in $\ell$-adic families of automorphic forms for $GL_n$, we would need to have Ihara (since our interpolation uses generic representations).