7-2-51. H. Poincaré to Felix Klein, English translation

Caen, 27 June 1881

Sir,

When I received your postcard I was just about to write you, to thank you for the parcel and announce its arrival. If it was delayed it was due to a postal error, which routed it first to the Sorbonne and then to the Collège de France, even though it bore the correct address.

As far as Mr. Fuchs and the naming of Fuchsian functions is concerned, clearly I would have chosen a different name if I had known of Mr. Schwarz’s work; however, I knew it only from your letter and thus after the publication of my results, so that I can now no longer change the name I gave these functions without showing a lack of respect for Mr. Fuchs.

I have begun reading your offprints which have great interest for me, mainly the one entitled “Ueber elliptische Modulfunktionen”. I will ask your permission to pose a few questions about it.

1° Have you determined the Fundamentalpolygone of all the Untergruppen that you call Congruenzgruppen, and in particular of the following:

$$\begin{array}[]{cc}\alpha\equiv\delta\equiv 1&\beta\equiv\gamma\mod n.\end{array}$$

2° In my memoir on Fuchsian functions, I classified Fuchsian groups according to various principles, among others by means of a number I call their genus. You similarly classify Untergruppen by means of a number you call their Geschlecht. Are the genus (as I understand it) and the Geschlecht one and the same number? I haven’t been able to find out since I do not know what the Geschlecht im Sinne der Analysis situs is. All I see is that these numbers cancel each other out. Would you please, therefore, do me the favor of telling me what Geschlecht im Sinne der Analysis situs means, or, if the definition is too long to be included in a letter, in which work I might find it?

In your last letter you ask me if I have confined myself to the special case where “Du gruppe der linearen Transformationen ist dadurch particularisirt, dass sie in einer doppelt so grossen Gruppe von Operationen enthalten ist, welche neben linearen Transformationen auch Spiegelungen umfasst.” I have not limited myself to this case, but I have assumed that all the linear transformations preserve a certain fundamental circle. Furthermore, I believe I can address the most general case by means of a similar method.

Concerning this point, it seems to me that all the Untergruppen relevant to modular functions do not fall under this special case.

On the subject of the discontinuous group you spoke of, obtained via Spiegelungen and by the Vervielfältigung of a polygon bounded by arcs of circles tangent in pairs.

It seems to me that there is a supplementary condition you have not mentioned, although it undoubtedly did not escape you: any two extended circular arcs must not intersect.

Would it abuse your indulgence if I asked you yet another question?

You write “in diesem Falle ist die Existenz der Funktion durch Arbeiten von Schwarz sicher gestellt”, and you add: “sofern man nicht auf die allgemeinen Riemann’schen Principien rekurriren will”. What do you mean by this.

I recently wrote to Mr. Hermite; I communicated succinctly the content of your letters, and conveyed to him your compliments as requested.

Please be assured, sir, of my consideration and respect.

Poincaré

PTrL. Translated by S.A. Walter from the original French (§ 4-47-6).

Time-stamp: " 8.08.2021 14:03"

Notes

• 1 Klein (1880), reed. Fricke et al. (1923, 169–178).
• 2 That is, “genus”, or what Poincaré terms “genre”.
• 3 “Genus in the sense of Analysis Situs.”
• 4 Klein was among the first to use the term Geschlecht (genus) to characterize surfaces, including Riemann surfaces (Lê, 2020).
• 5 “The group of linear transformations is characterized by the fact that it is contained in a group of operations twice as large, which, besides linear transformations, also contains reflections”.
• 6 Reproduction via symmetries.
• 7 You write “in this case, the existence of the function has been established by the works of Schwarz”, and you add “so far as no appeal is made to general Riemannian principles”.