Read my NSF proposal

Since this is NSF season, I took the opportunity to go back and look at some of my old proposals. I am definitely too shy to put my *most recent* proposal online, but I thought it might be interesting to share the very first proposal I ever submitted back in 2006. You can find it here. Honestly, it’s not as bad as I might have imagined. Here are some first impressions:

• The first thing that strikes me is that there is no “results from prior support section.” In particular, there is a pretty limited discussion of my previous work. It looks like I don’t even try to name drop my paper with Matt in Inventiones which I had been recently accepted before writing this grant; how virtuous.
• I attribute a theorem to “Taylor” which is really a theorem of Taylor and Harris-Soudry-Tayor. Sorry Michael! (I do reference [HST] later on in the proposal.)
• What is claimed in Theorem 3 is not entirely accurate — this was later fixed by my student Vlad Serban in this paper.
• It’s less than the full 15 pages — Possibly this is an incomplete draft?
• Already in 2006, I had started thinking about the modularity of elliptic curves over imaginary quadratic fields. Many ideas are missing. There is at least one reasonable idea here, however, namely, that if one can prove that the “half” Hida families (taking limits for one prime above \(p = \pi \pi’\) but not the other) are flat over \(\mathbf{Z}_p)\), then one is effectively in an \(\ell_0 = 0\) situation. Of course, even today, nobody has any idea how to prove this flatness. The problem is that one can sometimes show that it is pure of co-dimension one over the Iwasawa ring, but then one has to deal with a \(\mu\)-invariant type question proving that the support over \(\Lambda\) does not contain \((p)\). GB and I occasionally discussed whether it was reasonable even to conjecture this. I think I am more bullish that it should always be flat, but the question remains open.
• Using poles of (as yet unconstructed) \(p\)-adic L-functions to prove lifting criteria from smaller groups is a great idea! I’m sure I discussed this with Matt. If you don’t want to find it in the PDF, here is the basic idea. Given an autormophic form \(\pi\), Langlands explains how (morally) to determine whether it arises via functoriality from a smaller group by considering \(L(\pi,\rho,s)\) for every representation \(\rho\) and determining the order of vanishing (or the order of poles) of this L-function at \(s=1\). This is the automorphic analog of the group theoretic fact that one can determine a representation \(V\) of a group \(G\) by knowing not only the dimension of the invariant subspace of \(V\) but also of \(S(V)\) for every Schur functor applied to \(V\). Actually, it’s more than just an analogy, since both are just consequences of the Tannakian formalism (which only conjecturally applies to automorphic forms). For example, a completely concrete example of this is that a cuspidal \(\pi\) for \(\mathrm{GSp}(4)\) should arise as an induction from \(\mathrm{GL}(2)/F\) for a quadratic extension \(F\) if and only if \(L(\pi \otimes \chi,\rho,s)\) has a pole at \(s=1\) where \(\rho\) is the standard 5-dimensional representation and \(\chi\) is the quadratic character of \(F\). I believe this is even a theorem in this case. The point made in the proposal is that this formalism should apply equally to ordinary Siegel modular forms of non-classical weight, where the consequence of course is the weaker claim that \(\pi\) comes via induction from a non-classical ordinary form \(\varpi\) for \(\mathrm{GL}(2)\). Here is a nice example which suggests that this picture is consistent. Start with a classical ordinary \(\varpi\) for \(\mathrm{GL}(2)\) over an imaginary quadratic field (with some Galois invariance condition on the central character). After inducing, we obtain an ordinary Siegel modular form \(\pi\) such that \(L(\pi \otimes \chi,\rho,s)\) has a pole. This should also be true more or less for the \(p\)-adic L-function, defined correctly. But now as we vary \(\pi\) over the ordinary family, the locus where the \(p\)-adic L-function has a pole should have codimension one. Thus the philosophy predicts a one-dimensional family of ordinary deformations of \(\varpi\). And this is indeed something that Hida proved. But everything we know strongly suggests that this will be a non-classical family in general, so this lifting criterion is something that is really completely different from the classical analog. It also suggests and even partially implies corresponding results for lifting torsion classes as well. I think that this project is definitely something worth pursuing, but I’ve never learnt enough about \(p\)-adic L-functions to do so. Whenever I have talked to someone who has constructed such functions, they are always working in some context where normalizations have been made to ensure that the L-functions are Iwasawa functions and certainly don’t have poles. Anyway, I think this remains the most attractive open problem in this proposal.
• Question 2 has been answered (and much more) by Ian Agol. Agol (et. al.) pretty much put an end to the cottage industry of using number theory to answer various special cases of these Thurston conjectures. Interesting problems still remain, of course.
• I haven’t had anything really interesting to say about the geometry of the Eigencurve since writing this proposal. But Hansheng Diao and Ruochuan Liu did end up proving that the Eigencurve is indeed proper in this paper.
• I redacted some stuff! There’s an idea in this proposal that I might want to give to a graduate student — so I blacked it out (no peeking using secret technologies)
• The broader impact section suffers from the fact that this was my first year as a tenure track assistant professor. But the panel understands that there is only so much you can do at this point. The more senior you are, the more you should be doing.

In the end, I think this was not a bad proposal from a young researcher. There are some good ideas and some good problems. Probably the part on the geometry of the eigencurve is the weakest bit, and that is not unrelated to the fact that I stopped thinking about these types of questions. I think I accomplished less of what I set out to do than for some of my more recent proposals. This is not entirely surprising from looking at the proposal — a (forgivable) weakness is that it’s somewhat speculative and optimistic. What did I end up doing instead? Probably my most interesting result in the next cycle was my result with Matt on bounds for spaces of tempered automorphic forms using completed cohomology. This proposal was (in the end) funded — I think I certainly must have benefited from the fact that panels look generously on proposals from people within 5 years (or is it six?) from their PhD (“early career researchers”).