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Does anybody know why the genus (arithmetic or geometric) of a curve was historically denoted by $p$ ($p_a$ and $p_g$)? What does the letter "$p$" stand for?

Any references would be greatly appreciated.

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According to jeff560.tripod.com/g.html, the earliest known use of "genus" was in the paper "Über die Anwendung der Abelschen Funktionen in der Geometrie" by A. Clebsh, published in 1863. Perhaps you can look there. –  Martin Brandenburg Apr 1 '12 at 15:10
I don't think that the russians made that big a contribution to anything having to do with genus in the 19th century, so probably german is the place to look... –  Igor Rivin Apr 1 '12 at 16:32
род is a literal translation of "genus" –  Igor Rivin Apr 1 '12 at 17:18
The letter p is used by Riemann in his 1857 paper on abelian functions: a surface has connectivity 2p+1 if it requires 2p transverse cuts to render it simply connected. So he spoke of closed surfaces of connectivity 2p+1, rather than surfaces of genus p. (He showed an even number of cuts is required.) Then he shows a surface of connectivity 2p+_1 has p independent global holomorphic differentials, and discusses their moduli of periodicity. So if Clebsch introduced the terminology "genus", it seems the letter p preceded the term. The letter p may go back before Riemann. ?? –  roy smith Apr 1 '12 at 20:50
@Igor: I was just kidding in my previous comment (hence the smiley face at the end). Certainly the Russians had nothing to do with creating the notation, but I remember that when I first saw род I was struck that here was a word for genus starting with p, even though it was obviously a coincidence. Anyway, roy's comment suggests that maybe the letter p is related to the word periodische (periodic). –  KConrad Apr 3 '12 at 0:08
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2 Answers

In Bers, genus g is used...


About the word genus, see the comment of Martin Brandenburg, above.

As a complementary information, A.R. Forsyth, Theory of Functions of a Complex Variable, Cambridge, 1918, writes (last paragraph, p.371):

"If the connectivity of a closed surface with a single boundary be 2p+1, the surface is often said to be of genus p"

In the footnote: (genus) Sometimes class. The German word is Geschlecht; French writers use the word genre, and Italians genere.

By the way, in Portuguese, classe or genero.

On p.109, "Laguerre appears to have been the first to discuss the class of transcendental integral functions"

I think p stands for point. In I.M. James, History of Topology, Elsevier, 1999, we can see on pp. 39, last paragraph, "For the sketch of a proof Poincaré collected all types of differential equations on an algebraic curve of given genus p an with given….". On note (38), same page, "p is regular singular point of the differential equation …"

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First, the author is Bers (Lipman is his first name; we don't call Newton "Isaac"). Second, this doesn't really answer the question, which is about why p is used for the genus historically. That paper by Bers is from the 1950s, long past the time of interest in the question. –  KConrad May 11 '12 at 2:56
First, thank you very much for your kindly reply. The author's name was corrected. Second, please define "historically". Last, please read the second phrase in my answer. –  Papiro May 11 '12 at 11:08
In your first link, the letter $n$ denotes the number of punctures/points, not $p$. –  YangMills May 12 '12 at 20:34
Thanks YangMills!! Corrected. –  Papiro May 12 '12 at 21:04
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By looking at Coolidge's "Algebraic Plane Curves" Ch. VII, one may guess that $p$ stands for Plücker. You should have a look at the reference cited by Coolidge in his footnote to the first page of Ch. VII with title "Plücker's equations and Klein's equation" where the notion of genus is presented. The footnote says "For an historical account, see Berzolari, p. 343". The citation is to:

Berzolari, `Allgemeine Theorie des höheren ebenen algebraischen Kurven', in Enzyklopädie der Math. Wissenschaften, vol. iii, Part $2^1$, Leipzig, 1906, 99.

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Berzolari's article is available at gdz.sub.uni-goettingen.de/en/dms/load/img/… –  JeffE May 13 '12 at 18:44
yes the people at GDZ did a great job a digitalizing pretty much everything written in German by 19th century mathematicians. –  Abdelmalek Abdesselam May 13 '12 at 18:49
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