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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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.
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 …"