Although I don’t think about it so much anymore, the eigencurve of Coleman-Mazur was certainly one of my first loves. I can’t quite say I learnt about \(p\)-adic modular forms at my mother’s knee, but I did spend a formative summer before starting university thinking about (with Matthew Emerton) what in effect was the \(2\)-adic eigendecomposition of the (inverse) hauptmodul \(f = q \prod (1 +q^n)^{24}\) of \(X_0(2)\). I remember that we had a massive file called “tee-hee” which contained an absolutely huge number of Fourier coefficients which tested the memory limits of the University of Melbourne computer system (it was 10MB).
Jumping forward in time, I learnt about Kevin Buzzard’s Arizona Winter School project on a special case of his slope conjectures. This turned out to be closely related to the explicit computations I had done when I was younger. I got in touch and we managed to solve the first special case of his conjectures. Kevin and I continued collaborating over the next few years on a number of papers related to the geometry of the eigencurve.
In the abstract theory of the eigencurve, it is not important how overconvergent a modular form is but merely that it is overconvergent. However, it has always seemed to me that the analytic theory of overconvergent modular forms deep into the supersingular annuli has many unrevealed mysteries. One problem Kevin and I thought about was whether the eigencurve was “proper” in the sense of whether any punctured disc of finite slope eigenforms could be filled in at the central point. (Coleman and Mazur raise this question in their original paper.) At one point we thought we had proved it — the idea was that (by Buzzard’s analytic continuation theorem) any finite slope eigenform would converge uniformly far into the supersingular annuli, and since this property would hold uniformly for all points on the punctured disc of finite slope eigenforms it would follow that the limiting form at the centre was also highly overconvergent. However, if that form had infinite slope, it would lie in the kernel of the \(U\) operator, and now there was an elementary argument to show that any such form had a natural radius which was not (as) highly overconvergent. Done! Except there was a problem: the results on overconvergence were only proved for forms of integral weight since they relied on geometric constructions, particularly on the fact that one could make sense of \(\omega^k\) (for an integer \(k\)) on the entire modular curve. Coleman’s definion of overconvergent forms of general weight used a trick involving Eisenstein series. The notion of radius of convergence arising from this construction ended up being related to ratios of certain Eisenstein series in weight zero, and these ratios are not very overconvergent for the most general weights. This meant that (for general weights) the radius only made sense in a small overconvergent region — in particular smaller than the radius necessary to rule out elements in the kernel of \(U\) — and the idea didn’t work. In some cases (for example the \((N,p)=(1,2)\) eigencurve) there were workarounds one could make to give ad hoc definitions of the radius in order to push things through (a proof of concept as it were), but the situation was otherwise not so great.
Some time later, Hansheng Diao and Ruochuan Liu proved that the eigencurve was indeed proper. There argument was completely different, and used local arguments and period rings. It was a very nice result, and possibly their argument should even be considered the “correct” one. However, to my delight, Lynnelle Ye has just posted to the arXiv a new proof of the properness of the eigencurve which does indeed proceed by exploiting the radius of overconvergence for finite slope forms and proving that it inconsistent with the radius of convergence of elements in the kernel of \(U\). As mentioned above, the immediate stumbling block for Kevin and I was that the definition of overconvergent forms of a general weight \(\kappa\) was not geometric. However, thanks to Pilloni and Andreatta-Iovita-Stevens there now are such definitions available. Ye takes these constructions and then pushes them further into the supersingular annuli. These efforts are then indeed enough to turn what was merely a heuristic into a completely rigorous proof!
This is great!