Weibo Fu wrote an interesting paper on upper bounds for spaces of Bianchi modular forms, pushing previous results of Simon Marshall and Yongquan Hu to get more or less optimal results in the weight aspect. More generally, for any number field \(F\) which is not totally real, and for the space of regular algebraic cuspidal automorhpic representations of fixed level and parallel weight \(k\), he obtains the bound (see Theorem 1.2):
\[ \mathrm{dim} S_k = O(k^{d-1})\]
where \(d = [F:\mathbf{Q}]\) (The “easy” bound is \(O(k^d)\)). This is a great result! I do however have one tiny quibble. Fu makes the remark that
If \(F\) only admits one complex place … it seems like [the bound above] gives a sharp upper bound by heuristics from the Calegari-Emerton conjecture.
I never, unfortunately, had the time to examine this paper in any detail, but I do disagree with this comment. The basic point is that the codimension of completed cohomology over the non-commutative Iwasawa algebra is closely related to the growth of mod-\(p\) cohomology, but the growth of mod-\(p\) cohomology only gives an upper bound on cohomology in characteristic zero, and they don’t have much to do with each other unless the cohomology is torsion free. If you take a number field \(F\) whose only totally real subfield is \(\mathbf{Q}\), then I think the most natural guess for the dimension of the space of forms is \(O(k)\), which only coincides with this upper bound for imaginary quadratic fields.
Are you going to write a blog post about your new paper? Would love to have a high-level, distilled summary of what’s going on there.
Well, I have several new papers, but if you are talking about my 200+ page paper with Dimitrov and Tang then I would say we made a great effort to make the paper accessible to a wider audience than usual, and almost the first 40+ pages of the paper is exactly an exposition on what is going on for those who are interested enough to read it. So honestly your comment just sounds lazy.