I was shocked and saddened to learn that Joël Bellaïche passed away last week. Joël was a wonderful mathematician, a fantastic expositor, and an ever-cheerful smiling figure in the number theory community. In this post I want to highlight some of the beautiful mathematics of Joël.

When I was writing my thesis, learning about “rigid analytic modular symbols” was rather difficult, due to a scarcity of written references. The main references were a gnomic preprint of Stevens, and a rather forbidding manuscript of Ash-Stevens. However, there was also a wonderful new paper of Joël, Critical p-adic L-functions, then available as a preprint. This paper was a total revelation: Joël simply and elegantly reconstructs the eigencurve using rigid analytic modular symbols, and gives a beautifully clear construction of p-adic L-functions varying over it, relating their behavior directly to the geometry of the eigencurve near classical points. This paper, together with Joël’s more leisurely exposition of this circle of ideas in his course notes, was a critical inspiration and reference for me for several projects. It’s only a slight exaggeration to say that my paper with JB is a student exercise, whose point is to check that Joël’s ideas generalize to the setting of Hilbert modular forms.

One of Joël’s main interests was the interplay between the local geometry of eigenvarieties, especially their (non)smoothness at classical points, p-adic L-functions, and Galois cohomology. One major effort in this direction was his pioneering and highly influential book with Chenevier, which was an instant classic and is still an invaluable reference for many different things. Also very beautiful is his paper with Chenevier, showing that the eigencurve is smooth at critical Eisenstein points. In a complementary direction, Joël showed that U(3) eigenvarieties can exhibit surprisingly unpleasant behavior near classical points, giving an example of such a point where the local ring is not even a UFD. It’s a fascinating question to understand the singularities of eigenvarieties, and I think this example of Joël’s was perhaps the first indication of the true richness of this question.

Finally, I also want to mention Joël’s wonderful notes on the Bloch-Kato conjecture. Despite the title, these notes are actually a stroll through a large portion of modern arithmetic geometry, and should be read by every graduate student.

Rest in peace.

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