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

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

4 thoughts on “GANT beginnings”

  1. 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.

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