I spent part of last weekend reading Alice Silverberg’s blog, which is simultaneously depressing and hilarious. Everyone should read it, but you probably won’t enjoy reading it unless you enjoy Coen brothers movies or the short stories of Kafka. Anyway, the following thoughts have been in my head for a few months, but I decided to record them here after reading this.
The cohomology of non-basic local Shimura varieties is described by the “Harris-Viehmann conjecture”, which is formally stated as Conjecture 8.4 in Rapoport-Viehmann’s paper. This story started with a daring and beautiful conjecture of Harris (conjecture 5.2 here), whose formulation however turned out to be slightly incorrect in general, cf. Example 8.3 in RV. The conjecture was then modified by Viehmann, and Rapoport named this modified conjecture the Harris-Viehmann conjecture (footnote 5 in RV).
Unfortunately, Conjecture 8.4 in RV is still not correct as stated: the Weil group action on the summands appearing on the right-hand side needs to be modified by certain half-integral Tate twists. As far as I know, Alexander Bertoloni-Meli is the only person who has publicly pointed out the need for this modification, and Conjecture 3.2.1 in his very cool paper is the only correct formulation of the Harris-Viehmann conjecture in print.
Since the need for these Tate twists was overlooked by a lot of very smart people, it only seems fair to me that Alexander should get credit for his contribution here. The obvious way to do this would be to refer to the Harris–Viehmann–Bertoloni-Meli conjecture, or the Bertoloni-Meli–Harris–Viehmann conjecture. You could pick the second option if you’re a stickler for alphabetical name orders in mathematics, or the first option if you feel (as I do) that Harris’s contribution here deserves priority.
But it gets even worse, because Harris also formulated another conjecture along similar lines (conjecture 5.4 in his article linked above), which has gotten somewhat less attention but which is nevertheless extremely interesting.* It turns out that one can formulate a unified conjecture which encompasses both Harris’s conjecture 5.4 and the Harris–Viehmann–Bertoloni-Meli conjecture. What should it be called? The Harris–Viehmann–Bertoloni-Meli—Harris conjecture? I guess not.
*Here’s a comment from MH: “I was (and am) much more attached to this conjecture than to the one that is called the Harris-Viehmann conjecture, because it required some work to find the right formalism (the parabolics that transfer between inner forms), whereas the other conjecture (independently of the incorrect formulation in my paper) was just the obvious extension of Boyer’s result.”