GANT beginnings – E. Kowalski's blog

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
$\mathbf{S}_1,\quad \mathbf{P}_2,\quad \mathbf{A}^3,$
for the unit circle, the projective plane and the affine 3-space respectively, because only for the third it is true that
$\mathbf{A}^m=(\mathbf{A}^1)^m\ldots$

(2) Contrary to popular (e.g., mine…) belief, there is one canonical finite field besides the fields of prime order. It is $\mathbf{F}_4$ , which is canonical because there is a unique irreducible quadratic polynomial over $\mathbf{F}_2$ , so that
$\mathbf{F}_4\simeq \mathbf{F}_2[X]/(X^2+X+1).$

(3) For the same reason, Bourbaki regrets the notation
$\mathbf{F}_q$
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 $\mathbf{F}_{p^2}$ 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.

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Kowalski