Hilbert Modular Forms of Partial Weight One, Part II

Anyone who spends any time thinking about Hilbert modular forms of partial weight one — see part I — should, at some point, wonder whether there actually exist any examples, besides the “trivial” examples arising as inductions of Grossencharacters. Fred Diamond asked me this very question at Fontaine’s birthday conference in March of 2010. There are various reasons why one should not expect to prove this by pure thought, including the possibility that (for certain levels) there may exist no such forms, and that at any level there may exist only finitely many such forms (more on these heuristics another time). Thus the only way I can really imagine showing that such a beast exists is by explicitly finding an example.

As of today, my students Richard Moy and Joel Specter have found such a form! Here is (roughly) the strategy they use. As with computing weight one classical modular forms, one starts by computing a basis of \(q\)-expansions in some regular weight, divides by some Eisenstein series, takes the intersection of that space with its Hecke translates, and hopes that the resulting space has bigger dimension than the space of CM forms (which one can compute in advance). There are a few hiccoughs which occur along the way, of course. How does one compute \(q\)-expansions of Hilbert modular forms? Since computing uniformizations of surfaces is not realistic, they use the fact that (fortunately!) the \(q\)-expansion of a Hilbert modular form can be recovered from its Hecke eigenvalues. On the other hand, by Jacquet-Langlands, a Hilbert modular eigenform over (say) a real quadratic field corresponds to an eigenform on the arithmetic manifold associated to a quaternion algebra which is ramified at all infinite places, which then allows one to pass from the Hilbert modular variety to an adelic quotient which is now a finite set. Lassina Demebele wrote a magma programme which computes the eigenvalues for Hilbert modular eigenforms by this method, although for some reason the programme requires the level to be squarefree, and the character to be trivial. Using Atkin-Lehner theory, one can construct the entire space of forms by this method.

In practice, Richard and Joel worked with \(F =\mathbf{Q}(\sqrt{5})\), computed the forms of level \(\Gamma_0(N)\) (with \(N\) squarefree) and weight \([4,2]\), then divided by an Eisenstein Series of weight \([1,1]\) level \(\Gamma_1(N)\) and character \(\chi^{-1}\), then computed the Hecke operator \(T_2\) on this space and intersected away. Many (many) bugs later, and various annoying steps overcome (to take a random example, magma can compute the L-values of Hecke characters necessary to find constant terms of Eisenstein series [nice] but only as a complex number, not as an algebraic number [not so nice] so “L-value recognition” had to be coded in), the progams finally worked, and after much grinding away (for all squarefree \(N\) of norm less than \(500\)) they didn’t find anything at all (or at least, anything besides CM forms).

So they started working in weight \([6,2]\), computed away, and eventually found a form \(\pi\) of weight \([5,1]\), level \(\Gamma_1(14)\), and character \(\chi\), where \(\chi\) has conductor \(7\) and is of order \(6\). The coefficient field of the eigenform is, I believe, \(\mathbf{Q}(\sqrt{5},\sqrt{-3},\sqrt{-19})\) (note that it must contain the base field as well as the field of the character). Note that this automorphic form \(\pi\) is Steinberg at \(2\)! In particular, it is not CM, and one doesn’t know whether local-global compatibility holds for the corresponding \(p\)-adic Galois representations even restricted to \(2\).

I should say that finding the form actually turned out to be easier than proving the form exists rigorously. Theoretically, the proof should be easy: one has found a form \(F/E\) for some cuspform \(F\) and some Eisenstein series \(E\) which looks like it is holomorphic. All one needs to do is square it (so it becomes regular), find a candidate form \(G\) of weight \([10,2]\) such that \(G E^2 – F^2 = 0\) (which one can prove since the spaces are finite dimension), and then \(E/F\) has no poles and is thus holomorphic. The problem is that the form \([10,2]\) has non-trivial character, and Lassima’s program only works with trivial character. One can take the \(6\)th power and work with a form of weight \([30,6]\), but this is way beyond what magma can cope with. In the end, Richard and Joel had to come up with a few tricks to do this (which took about three months!), but the final computations are in, and the existence has now been proven.