# Le Haut Commissariat qui surveille rigoureusement l'alignement de ses Grandes Pyramides

Yesterday I came across the following one-paragraph summary of the history of the Law of Quadratic Reciprocity in Roger Godement's Analyse mathématique, IV, p.313 (perhaps the only treatise on Analysis which contains a statement of the Law in question).

Legendre a deviné la formule et Gauss est devenu instatanément célèbre en la prouvant. En trouver des généralisations, par exemple aux anneaux d'entiers algébriques, ou d'autres démonstrations a constitué un sport national pour la dynastie allemande suscité par Gauss jusqu'à ce que le reste du monde, à commencer par le Japonais Takagi en 1920 et à continuer par Chevalley une dizaine d'années plus tard, découvre le sujet et, après 1945, le fasse exploser. Gouverné par un Haut Commissariat qui surveille rigoureusement l'alignement de ses Grandes Pyramides, c'est aujourd'hui l'un des domaines les plus respectés des Mathématiques.

Which Haut Commissariat is he referring to ? Or is it just a joke ?

-
I would guess it's a joke, although Godement certainly must have something in mind when he used these words; perhaps he was simply referring to Langlands' program. –  Franz Lemmermeyer Sep 21 '13 at 7:05
BTW there are quite a few textbooks on complex analysis that state and prove the quadratic reciprocity law, –  Franz Lemmermeyer Sep 21 '13 at 7:06
Yes, he's surely referring to the Langlands Programme, but it is somewhat funny to call it the Haut Commissariat of something or the other. Godement is a good friend of Langlands, by the way, and one of the three people who are gratefully mentioned in the acceptance speech when Langlands received the Grande Médaille d'Or of the Académie des sciences : publications.ias.edu/sites/default/files/discours-ps.pdf –  Chandan Singh Dalawat Sep 21 '13 at 7:57
Langlands writes "... je veux nommer trois mathématiciens qui se donnèrent la peine de persuader le jeune homme [the young Langlands], bien plus modeste que moi, qu’il valait quelque chose: Salomon Bochner, né je crois à Cracovie en Pologne, Harish-Chandra, né à Kanpur en Inde, tous les deux devenus mathématiciens américains, et Roger Godement, mathématicien français. Il me serait impossible d’exprimer en quelques phrases courtes combien lui, il leur devait, et combien moi, je leur dois toujours." –  Chandan Singh Dalawat Sep 21 '13 at 8:05
For those who might have missed it, Roger Godement has graced MathOverflow once (thanks to Anton for helping find the link) : mathoverflow.net/questions/91385/… –  Chandan Singh Dalawat Sep 22 '13 at 4:48

I disagree with Michael Grünewald's interpretation, which by the way doesn't answer the initial question: who Godement is he referring too? I think this is a joke made without acrimony. "Thought police", "innovation preventing", are much too strong phrases to translate Godement's light ironical quotation.

To a french-spaking ear, "Haut Commissariat" in this context evokes the "Commissariat Général au Plan", created by the administration led by de Gaulle in 1946 (and including a large political spectrum, from right wing to communists). It was an institution without real power but which was supposed to prepare non-compulsory "plans" to develop the economy for the next five years, the idea being to take advantage of whatever was thought efficient in soviet-like planning while staying essentially a free-market economy. (Of course there are other institutions with that name, like UN's "haut-commissariat aux réfugiés", but really that the plan one that comes to mind).

So back to quadratic reciprocity, I may be completely wrong but I imagine that the Haut-Commissaire in question might be R.P. Langlands and his huge program that has provided a non-compulsory, but hugely influential, planning for the research in "higher class field theory" since more than 40 years.

-
I read after writing my answer the comments under the questions, to which I agree. Godement was very close to this "haut-commissariat", perhaps he even considered himself a member of it :-). After all he was Jacquet's advisor. There was a touching fear in the groups of mathematicians to which Godement belonged: the fear to become what they called a "mandarin", an installed mathematician detaining (and this detained by) a large power on the developments. Self-irony (or irony aimed at friends and students) was seen as a way to protect oneself against such an evolution. –  Joël Sep 21 '13 at 14:40
I think this is a joke made without acrimony. I also do! "Thought police", "innovation preventing", are much too strong phrases to translate Godement's light ironical quotation. You are definitely right, but irony is one of the hardest thing to deal with for non native speakers. This is why I choosed to rephrase the exceirpt without any subtlety. –  Michael Grünewald Sep 21 '13 at 16:10

This “Haut Commissariat” is not a formal organisation, but a fictional organisation he invented to make an ironical statement. Here is how I would rephrase his statment in a non-ironical way:

This subject [the legacy of Legendre, Gauss, Takagi and Chevalley] felt under control of a Thought Police preventing any innovation in the field.

(Thought Police is referring to Orwell's novel 1984, but is probably clear enough by itself.)

One could almost understand that participants of this “Thought Police” take care of pushing new comers apart, to make sure that old respectable problems are not resolved by anybody bu themselves—if some—but this would really be one step further.

I base my lecture on the usual opposition between innovation, imagination and freedom on the one hand, conservatism, respect and police on the other hand. I also read several books by Godement, so I may hope I do not abuse his statement too much!

-
I'm sure he didn't mean any such thing, and it is not true at all that there is a "Thought Police preventing any innovation in the field". The new ideas in this field have no parallel in the history of mankind. –  Chandan Singh Dalawat Sep 21 '13 at 7:26
I think "thought police" is putting it a little bit too strong, but I would rather agree with the idea that he meant the remark as somewhat caustic towards the "establishment" of number theory and its inclination to defend a certain orthodox view of (pure) mathematics.I don't see how it could be viewed otherwise since "Haut Commissariat" refers to a high-level official agency of the French government, which he certainly didn't put close to his heart. –  Jean Raimbault Sep 21 '13 at 11:58
Comments under the question, posted 6 and 7 hours ago now, make these comments posted under this answer in the past 6 hours seem unlikely, as they suggest Godement is personally friendly to Langlands (who would surely be in any "establishment" of number theory) and involved in promoting his work. And it is hard to guess what "establishment" Godement would say is merely preserving old ideas. –  Colin McLarty Sep 21 '13 at 14:26
@ColinMcLarty I always will love and help my friends, which does not forbid me to sometimes disagree with or make fun of their positions or deeds. –  Michael Grünewald Sep 21 '13 at 16:23
@MichaelGrünewald You have a broad and warm sense of friendship, but that by itself does not make a persuasive argument for your reading of Godement. –  Colin McLarty Sep 22 '13 at 1:20