# GANT beginnings

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.

### Kowalski

I am a professor of mathematics at ETH Zürich since 2008.

## 4 thoughts on “GANT beginnings”

1. Chandan Singh Dalawat says:

I never write F_p^r unless r=1; I just say “Let k be a finite field with p^r elements” or even “Let k be *the* finite field of cardinality p^r”. Maybe I should allow myself to write F_2^2.

By the way, I have heard from someone who had heard from someone that there is a completely canonical construction of an algebraic closure of F_2. I have never tried to work it out. Does anyone have a such construction ?

2. Wait. Did you get the information on wine at lunch from an eminent Fields medallist and an element of Bourbaki who happened to be present at the winter school in Lausanne? I suspect an issue of bias here.

3. Bias in what sense? I have no reason to doubt the information, and do not speculate whether the wine had any actual effect on mathematical brilliance in the students of these high-schools…

4. I meant that this was a non-scientific sample. If wine at lunch before maths talks does contribute to brilliance and longevity, perhaps this is the anthropic principle at work.

Can you enlighten me on why F_{p^2} is not canonical? Surely all fields with p^2 elements are isomorphic? The fact that there is more than one quadratic irreducible polynomial modulo p simply implies that the choice of *generator* alpha (F_{p^2} = F_p(\alpha)) is in general non-canonical.