Let $p$ be a prime, and $F$ a $p$-adic Hida family of ordinary modular forms (of some tame level $N \ge 1$). I'd like to know whether, for $q$ a prime factor of $N$, the actions of the Atkin--Lehner involutions $W_q$ on the classical specializations of $F$ interpolate $p$-adically. (Equivalently, I'd like to know if there's an operator $W_q$ on the space of ordinary $\Lambda$-adic cusp forms of tame level $N$, which corresponds with its classical namesake under the specialization map at each classical weight.)

I've convinced myself that this is true -- I can bash out a conditional proof modulo some plausible-looking integrality statements -- but I'd be happier if I had a complete proof. Does anyone know a reference for this? (I've consulted various papers of Hida without finding anything.)

I'd also be interested in the corresponding question for Coleman families of non-zero slope.


Like yourself, I have found a reference for these results hard to find, even in the case of Hida families. These are a few pointers.

In the case of Hida families, once the Atkin-Lehner involutions are given an appropriate group-theoretic or geometric definition, the same proof that modular forms interpolate implies that the involutions interpolate. The geometric argument is given for instance in Mazur-Wiles (84) page 237 though of course the application to Hida theory is not worked at there for obvious historical reasons. The group-theoretic argument in the case $q=p$ (which you exclude, but which is strictly harder) is given in Nekovar-Platter (2000) section 1.6. It seems to me there might be everything you need in Ohta (Composition 99) section 3.4

That said, my experience has been that there are so many conflicting normalizations about everything in the literature that it is probably safer not to look at any reference and just reprove the results needed with one's own choices of normalizations (this is what I ended up doing in a slightly more general framework in my article at Compositio).

For Coleman families, the results of Bellaïche-Chenevier (2009) section 7.4 plus an appropriate definition of the Atkin-Lehner operators purely in terms of local automorphic representations seem a good place to start (presumably, Buzzard's formalization does the job equally well but I am less familiar with it). Just plug in in your test objects the supplementary requirement that there are operators satisfying the appropriate properties and the uniqueness of the eigenvariety will then imply that on sufficiently small affinoids, there exists similar operators on the sheaf of admissible $\mathbf{G}(\mathbb A_{\mathbb Q})$-representations. It seems also probable that that Mazur-Wiles calculations for the Igusa tower plus the construction of the eigencurve through the Igusa tower yields the results, so I guess that Pilloni (Ann. Institut Fourier 2013?) plus Mazur-Wiles might also be good starting points.

Please do provide your own answer if you ever find the perfect reference...

  • $\begingroup$ Thanks! I will look at the references you suggest. But I am puzzled by what you say about including the case $q = p$ as well; the $W_p$ operator cannot possibly interpolate in Hida families, because it does not preserve the ordinary subspace (it interchanges forms which are ordinary for $U_p$ with forms that are ordinary for $U_p^*$). $\endgroup$ Aug 8 '14 at 7:14
  • $\begingroup$ What I meant is that in order to prove these kind of results, you need a definition of the operators which is compatible with inverse limit on the level. This is possible at $p$ and the fact that it interchanges different ordinary spaces is a virtue, not a vice: it is absolutely crucial in the correct definition of twisted self-dual Hida families (but someday I'll tell you an amusing story about this my advisor told me once). $\endgroup$
    – Olivier
    Aug 8 '14 at 8:17
  • $\begingroup$ There is, as far as I'm aware, no sensible definition of the $W_p$ operator which is compatible with inverse limit on the level: there are two natural degeneracy maps from level $Np^{r + 1}$ to level $Np^r$, and $W_p$ interchanges these. $\endgroup$ Aug 8 '14 at 13:47
  • $\begingroup$ Yes, exactly. I have a feeling we are talking past each other, so let me try to be clearer: what you will find in Nekovar-Platter is a group-theoretic computation and a proof of the compatibility of this computation with inverse limits (of course in the sense that one projection is turned in the other) that is strictly parallel (though slightly harder because of the interchanging) to the one you need for $q\neq p$. Both cases (of course with the appropriate modification when $q=p$) are treated (maybe) in Ohta and my article. $\endgroup$
    – Olivier
    Aug 8 '14 at 14:31

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