Joël Bellaïche

Very sad to hear that Joël Bellaïche has just died. He got his PhD at the same time as me, and I first got to know him during the Durham conference in 2004 and later at the eigenvarieties semester at Harvard (was that in 2005 or 2006?).

Joël was an original mathematician, and his papers (many written with Gaëtan Chenevier) contain many really good ideas. As a postdoc, I was totally immersed in thinking about Galois deformations of reducible representations when the paper lisseté de la courbe de Hecke de \(\mathrm{GL}_2\) aux points Eisenstein critiques appeared on the arXiV. In that paper, they study the ideal of reducibility for certain Galois deformation rings (or pseudo-deformation rings). By studying the ring-theoretic properties of this ideal, they proved the Eigencurve was smooth at the evil Eisenstein points. It clarified immediately a number of the phenomena I had been thinking about, but it was also simply the “right” way to think about these things. I also learnt from Joël at Durham the problem of proving the non-vanishing of p-adic zeta values like \(\zeta_p(3) \ne 0\), which remains 18 years later one of my favourite problems.

Another really beautiful idea was the approach by Joël and Gaëtan to Bloch-Kato type conjectures (including the Selmer group part of the Birch–Swinnerton-Dyer conjecture) via the geometry of eigenvarieties (including those associated to \(U(3)\)). This is of course related to the ideal of reducibility. Their joint asterisque paper Families of Galois representations and Selmer groups is a very nice read on this topic, as are Joël’s notes for the Clay summer school as well as his recent book on Eigenvarieties.

In more recent times, Joël had been exploring ideas in some interesting directions, including his intriguing work on self-correspondences on curves. What was consistent about his research was that his primary motivation always seemed to be rooted in coming to an original understanding of interesting math rather than simply making incremental improvements on work of others.

Last but not least, one should not forget his sense of humor with a decidedly irreverent streak. This is probably best appreciated with a beer or a glass of wine in a summer evening in Luminy, but to take a quote from on of Joël’s own papers:

Let \(p\) be a prime number that, we shall assume, splits in \(E\). We shall also assume that \(p \ne 13\). I don’t think this is really useful, but who knows?

My thoughts are with his family.