Persiflage

Today I would like to talk about a beautiful new theorem of my student Eric Stubley (see also here). The first version of Eric’s result assumed (unknown) cases of the general Ramanujan conjecture for Hilbert modular forms, and relied on a beautiful idea due to Hida. The final argument, however, is unconditional, and goes beyond Hida’s ideas in a way (I hope) that he would be delighted to see.

Suppose that \(F\) is a real quadratic field in which \(p = vw\) splits. If \(f\) is a Hilbert modular form of (paritious) weight \((1,2k+1)\) and level prime to \(p\), then the corresponding Galois representation (really only defined up to twist):

\(\rho_f: G_F \rightarrow \mathrm{GL}_2(\overline{\mathbf{Q}}_p)\)

has the property that, for exactly one of the places \(v|p\), the restriction \(\rho_f |_{G_v}\) is unramified. Forms of partial weight one are slippery objects — one can construct such forms which are CM, but the existence of any such form which is not CM was open until an example was found by my students Richard Moy and Joel Specter (see here, here, and here). They behave in many ways like tempered cohomological automorphic forms for groups without discrete series, more specifically Bianchi modular forms or cohomological forms for \(\mathrm{GL}(3)/\mathbf{Q}\). In each of these cases, the invariant \(l_0\) as considered in Calegari-Geraghty (see for example section 2.8 of this paper) is equal to \(1\). Following work of Ash-Stevens and Calegari-Mazur, one might consider whether or not \(f\) deforms into a family of classical forms. For example, the form \(f\) will be ordinary at \(v\), and so it lives in a Hida family \(\mathcal{H}\) over \(\Lambda = \mathbf{Z}_p[[\mathcal{O}^{\times}_v(p)]] \simeq \mathbf{Z}_p[[T]]\) where we keep the weight and level at \(w\) fixed and consider (nearly) ordinary forms at \(v\). The specialization of this family to regular paritious weights will give a space of classical Hilbert modular forms. What can one say about the other specializations in partial weight one?

Theorem [Stubley]: only finitely many partial weight one specializations of the one variable \(v\)-adic Hida family \(\mathcal{H}\) associated to \(f\) are both classical and not CM.

This gives a completely general rigidity result for all partial weight one Hilbert modular forms in the split case. Over the past decade or so, the prevailing philosophy is that the only algebraic automorphic forms which are not exceedingly rare are either those coming from automorphic forms which are discrete series at infinity, or come from such forms on lower rank groups by functoriality. In this setting, this predicts that non-CM forms of partial weight one should be rare. It might even be plausible to conjecture that, up to twisting, there are only finitely many such forms of fixed tame level. However, such conjectures are completely open, and Stubley’s result is one of the first general theorems which points in that direction. (Stronger results for very specific \(F\) and \(p\) and tame level were obtained by Richard Moy and are discussed in some of the links above.)

One way to think about this theorem is in terms of the Galois representation associated to \(\mathcal{H}\). Assume for convenience of exposition that the family is free of rank one over \(\Lambda\). The Galois representation \(\rho_f\) extends to a family:

\(\rho: G_F \rightarrow \mathrm{GL}_2(\mathbf{Z}_p[[T]])\)

where \(\Lambda = \mathbf{Z}_p[[T]]\) represents weight space, so \(T = 0\) corresponds to the original specialization, and \(T = \zeta – 1\) for a \(p\)-power root of unity \(\zeta\) corresponds to a specialization to partial weight one with non-trivial level structure at \(v\). These representations are all nearly ordinary at \(v\). Is it possible that they could be split locally at \(v\) for infinitely many specializations to partial weight one? Since a non-zero Iwasawa function has only finitely many zeros, this would actually force the local representation to split for all \(T.\) Moreover, it should imply (and does in many cases) that the specializations \(T = \zeta – 1\) are all classical by modularity lifting theorems. Thus, by Stubley’s theorem, this can only happen when the family \(\rho\) is CM. In particular, Stubley’s result implies a theorem (assuming some Taylor-Wiles hypothesis) that a family of Galois representations which is (say) nearly ordinary at \(w\) of fixed weight and level and nearly ordinary at \(v\) is locally split at \(v\) if and only if it is CM.

Experts should recognize the similarity between the Galois theoretic version of Stubley’s theorem and the work of Ghate-Vatal, who prove that an ordinary family over \(\mathbf{Q}\) cannot be locally split unless it is CM. The main ingredient in their proof is the fact that there are only finitely many weight one forms of fixed tame level (up to twist) which are not CM, since these correspond either to \(A_4, S_4, A_5\) extensions of \(\mathbf{Q}\) unramified outside a fixed set of primes, which are clearly finite, or real multiplication forms, whose finiteness comes down to the finiteness of the ray class group of conductor \(N \mathfrak{p}^{\infty}\) for a split prime \(\mathfrak{p}\) in a real quadratic field. However, the analogous statement for partial weight one forms is completely open as mentioned above, so Stubley’s theorem requires a quite different argument.

Before discussing the proof, we first need to discuss a result of Hida (see this paper) about fields of definition of ordinary forms in families. Consider an ordinary family over \(\mathbf{Q}\), and consider specializations in some fixed weight, amounting (with some normalization) to specializing \(T\) to \(\zeta – 1\) for a \(p\)th power root of unity. The coefficient field will automatically contain \(\mathbf{Q}(\zeta)\). Suppose that for any prime \(q\), the degrees \([\mathbf{Q}(a_q,\zeta):\mathbf{Q}(\zeta)]\) are bounded for infinitely many specializations. Then Hida proves the family has to be a CM family. Let \(\alpha_q\) be one of the corresponding Frobenius eigenvalues. Hida’s key insight is to note that \(\alpha_q\) is a Weil number, and that Weil numbers over extensions of \(\mathbf{Q}(\zeta)\) of uniformly bounded degree are extremely restricted, and in particular given an infinite collection of such numbers then infinitely many of them have to be of the form \(\alpha \zeta\) for a fixed \(\alpha\). Using a rigidity lemma fashioned for this very purpose, he then deduces that \(\alpha_q\) in the Iwasawa algebra more or less has to equal \(\alpha (1+T)^s\) for some \(s \in \mathbf{Z}_p\), and this puts enough restrictions on \(a_q\) for him to be able to deduce the family is CM.

Stubley’s first idea is to use Hida’s result in the context of partial weight one forms. The key fact that is different in partial weight one is that when \(a_{v} \ne 0\), the form \(f\) is automatically ordinary at \(v\), and hence the \(\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}(\zeta))\) conjugates of \(f\) will still be ordinary at \(v\)! This is completely false in regular weights. However, in partial weight one, the only possible (finite) slope of any form at a split prime is \(0\). As a consequence, the boundedness assumption of Hida’s theorem is always going to be satisfied, because all of the conjugates have to lie on one of the finitely many Hida families which all have bounded rank over \(\Lambda\).

There is, however, a problem. Hida’s argument crucially uses the fact that \(\alpha_q\) is a Weil number, which uses the Ramanujan conjecture for forms of regular weight. The Ramanujan conjecture is completely open for partial weight one forms, since we have no idea how to prove they occur motivically (nor prove modularity of their symmetric powers). This is where Stubley’s second idea comes in. Instead of the Ramanujan conjecture, one does have standard bounds on the coefficients \(a_q\). This is not enough to deduce that \(\alpha_q\) has the form \(\alpha \zeta\) for some fixed \(\alpha\). Instead, Stubley shows that it does allow one to show that the trace of \(a_q\) (together with the trace of any if its powers) to \(\mathbf{Q}(\zeta)\) (which has uniformly bounded degree) can be written as a finite sum of roots of unity where the number of terms does not depend on \(\zeta\). Again for convenience of exposition and to avoid circumlocutions with traces, let us suppose that the rank of the Hida algebra is one and so \(\mathbf{Q}(\zeta,f) = \mathbf{Q}(\zeta)\). Then Eric shows that infinitely many of the \(a_q\) satisfy:

\(a_q = \alpha_1 \zeta_1 + \alpha_2 \zeta_2 + \ldots + \alpha_N \zeta_N\)

for varying \(p\)-power roots of unity \(\zeta_i\), but where \(\alpha_i\) and \(N\) are fixed. Then Stubley proves a new rigidity theorem in this context (not unrelated to results of Serban) showing that one must have an equality

\(a_q = \alpha_1 (1+T)^{s_1} + \alpha_2 (1+T)^{s_2} + \ldots + \alpha_N (1 + T)^{s_n}\)

over the Iwasawa algebra. This is probably enough to show the family has to be CM using ideas similar to Hida, but even that is not necessary — by using this formula for specializations in regular weight one deduces that the \(\alpha_i\) are in \(\overline{\mathbf{Q}}\), and then applying Hida’s theorem in this fixed regular weight one deduces that the family is CM.

Stubley’s theorem is the first result that gives general theoretical evidence towards the conjecture (if one is so bold to make such a conjecture) that there are only finitely many non-CM partial weight one forms of fixed tame level up to twist. It also shows that certain \(v\)-ordinary deformations of a non-CM partial weight one form \(f\) will not be classical. But there is also a second possible way to deform \(f\), namely, to vary the weight at \(w|p\) instead (or as well). For example, if the form \(f\) was also ordinary at \(w|p\), one could look at the ordinary at \(w\) Hida family. One might also conjecture that this family only contains finitely many non-CM points, but this is still open. (Boxer has raised this question.) I think this is an interesting but very hard question!