Note: I feel as a service to the number theory entertainment complex that I should blog more often in these times, even if it means being less coherent than usual. I might even try to get a few guest posts since I won’t be going to any conferences any time soon…
I recently linked to my first NSF proposal here, but just today I stumbled upon my graduate NSF fellowship application from 1998. There is really only one page which involves any proposal (rather than a list of courses I took or references), and I include the mathematical portion here in full (the only changes from the original are one or two spelling errors and changing fake LaTeX (\rho) to real LaTeX (\(\rho\)).
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My research interests center mainly around the study of two dimensional Galois representations, the connection of such representations to Modular forms, and application of these connections to the arithmetic of Elliptic curves. Here are several possible questions which are of interest to me.
1. Serre’s conjectures predict that for any odd, absolutely irreducible Galois representation \(\rho\) into \(\mathrm{GL}_2(\mathbf{F}_q)\), there exists a modular form \(f\) which gives rise to \(\rho\), in the sense of Deligne/Shimura/Deligne-Serre. The characteristic zero representation \(\rho_f\), however, need not be defined into \(\mathrm{GL}_2(W(\mathbf{F}_q))\), (\(W(\mathbf{F}_q)\) = Witt–vectors of \(\mathbf{F}_q\)), but perhaps in \(\mathrm{GL}_2(\mathcal{O})\), for some ramified extension \(\mathcal{O}\) of \(W(\mathbf{F}_q)\). Is there any sense in which one can quantify the ramification of \(\mathcal{O}\) in advance? Is there perhaps a clear cohomological obstruction to a modular lift over \(W(\mathbf{F}_q)\)? Can one quantify this in terms of some \(H^2(G_{\mathbf{Q}},*)\), or perhaps in terms of \(R^{\mathrm{univ}}\), where \(R^{\mathrm{univ}}\) is the universal deformation ring of \(\rho\)? Perhaps if this is too difficult, some qualitative result in this direction?
2. Applications of the above ideas to rational cuspidal eigenforms of Level \(1\). Such forms
are only known to exist if \(\mathrm{dim} S_{2k}(1) = 1\). Can one use ideas from representations to show that no other cuspidal eigenform can be defined over \(\mathbf{Q}\)?
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My first thought is “I guess I haven’t changed that much as a mathematician over 22 years” followed by “not a bad problem but too optimistic.” The funny thing is that I do think of myself as a number theorist with a certain amount of breadth (despite protestations to the opposite from my most dyspeptic collaborator), so I guess I have to claim that I work on a large circle of ideas and sometimes return to very similar points on the circle. There are also echos in the first proposal of future work with Matt where we studied the ramification of \(\mathcal{O}\) for the reducible representation occurring in weight \(2\) and prime level \(N\), as studied by Mazur. The most definitive result in that paper was computing the exact ramification degree when \(p=2\), where in the case that the ring was a DVR one had \(e = 2^{h-1} – 1\), where \(2^h\) was the order of the \(2\)-part of the class group of \(\mathbf{Q}(\sqrt{-N})\). Other progress on this problem more in the spirit of the formulation above has been done by Lundell and Ramakrishna (MR2770582), although I still think there are many open questions around this problem which are of interest.
On the other hand, the second problem is too optimistic. One reason is related to Buzzard’s observation that, in high weight with \(p\) fixed, the representations all seemed to be defined over rings with very little ramification. (He goes on to make a conjecture along these lines for which nobody has made any progress.) So it seems unlikely to rule out forms of large weight with coefficients in \(\mathbf{Q}\) by showing that there are no such forms over \(\mathbf{Q}_p\) because the latter seems to be false in a strong way. The problem of showing there are no more eigenforms over \(\mathbf{Q}\) when the weight is at least 24, which is very close to Maeda’s Conjecture, is something on which virtually no progress has been made since my proposal, so I guess I don’t have to feel bad for not making any progress myself. On the other hand, I don’t actually think it is an impossible problem. Maybe I should work on it!