The special semester on Group Actions in Number Theory, organized by Ph. Michel and myself, started last week with our Winter School.
We were lucky to have J.-P. Serre give a short course on equidistribution, with emphasis on questions related to the Sato-Tate conjecture and its variants. Here are a few things I’ve learnt (from the lectures and discussions afterwards):
(1) Bourbaki writes
for the unit circle, the projective plane and the affine 3-space respectively, because only for the third it is true that
(2) Contrary to popular (e.g., mine…) belief, there is one canonical finite field besides the fields of prime order. It is , which is canonical because there is a unique irreducible quadratic polynomial over , so that
(3) For the same reason, Bourbaki regrets the notation
for all finite fields, because it was their tradition to use bold fonts exclusively for objects which are completely canonical. Serre gave an example of a statement in his paper on Propriétés galoisiennes des points de torsion des courbes elliptiques which, if read too quickly, could give the impression of leading to a contradiction or a mistake — but only if one believes that is an unambiguous notation…
And finally — any links to mathematical brilliance will be left for the reader to contemplate — I learnt that around the 1940’s in Southern France, wine was the usual drink in middle and high-school lunches and dinners.