Hilbert Modular Forms of Partial Weight One, Part I

Let \(\pi\) be an algebraic Hilbert modular cuspform for some totally real field \(F^{+}\). Then, associated to \(\pi\), one has a compatible family of Galois representations:

\(r_{\lambda}(\pi): G_{F^{+}} \rightarrow \mathrm{GL}_2(\mathcal{O}_{\lambda})\)

which are unramified outside finitely many primes (this is the work of many people). The expectation is that this representation should satisfy local global compatibility at all primes. This is known if \(\pi\) has regular weight, and also if \(\pi\) has parallel weight one. However, this is not known, even  for the case \(p \ne \ell\) (Here \(\ell\) is the characteristic of \(\mathcal{O}/\lambda\)). The problem is that these representations are constructed via congruences, not from geometry. Deforming in families does give some control, and indeed one can prove that, for \(v|p\) and \(p \ne \ell\),

\(\mathrm{WD}(r_{\lambda}(\pi)|_{G_v})^{\mathrm{F}\text{-}\mathrm{ss}} \prec \mathrm{rec}(\pi_v)\)

which is a way of saying you get the correct answer up to the monodromy operator \(N\), and moreover the monodromy operator on the Galois side can only be more degenerate than the automorphic side. In English, if (for example) \(\pi_v\) is Steinberg, then one may deduce (as expected) that the image of inertia on the Galois side is unipotent, but not necessarily that it is non-trivial. In fact, by solvable base change, this is really the only problem one has to worry about (so we shall assume we are in this case below).

The usual methods for computing the monodromy \(N\) are all geometric (nearby cycles), and, as it seems hopeless to try to construct any (conjectural) motive associated to \(\pi\), there doesn’t seem to be much one can do.

One does, however, have the following strategy, which I learnt from Martin Luu, which should suffice for all but finitely many primes \(\lambda\) for which \(r_{\lambda}(\pi)\) is ordinary. Namely, take the \(\lambda\)-adic Galois representation associated to \(\pi\), and prove that it is potentially automorphic using extensions of the Buzzard-Taylor idea (which has been employed by Sasaki, Kassaei, Pilloni and others in the case of Hilbert modular forms of parallel weight one, but should also apply in this context). The result is that one shows that \(r_{\lambda}(\pi) |G_{E^{+}}\) for some totally real extension \(E^{+}/F^{+}\) is now associated to a cuspidal automorphic form \(\Pi\) of the right level. How does this help? Well, now using what we know from local global compatibility (which is ok in the unramified case), we deduce that \(\Pi_w\) for some \(w|v\) is associated to the corresponding local Galois representation \(r_{\lambda}(\pi)|_{G_w}\). Now this representation has the property that it looks unipotent on inertia mod \(\lambda^n\) for all \(n\), but, assuming local-global compatibility fails, is actually unramified at \(p\). In particular, the semi-simplification is given by two characters whose ratio is the cyclotomic character, whereas \(\Pi_w\) is an unramified principal series. This implies that the Satake parameters \(\{\alpha_w,\beta_w\}\) satisfy \(\alpha_w/\beta_w = N(w)\), which contradicts Ramanujan. We are not done yet, because one  doesn’t have purity in partial weight one. However,  one can appeal to bounds coming from Rankin-Selberg, and this is enough to obtain a contradiction.

The only obvious examples of partial weight one HMF (which are not of parallel weight one) are CM, and since those are potentially unramified, the monodromy operator will always be trivial on the autormophic side (and hence also on the Galois side). So this suggests (but does not beg) the question: do there actually exist any partial weight one (but not parallel weight one) Hilbert modular forms which are not CM? Stay tuned for part II!