
05822210
j
2011a.00058
Rice, Adrian
``To a factor pr\`es'': Cayley's partial anticipation of the Weierstrass $\wp$function.
Am. Math. Mon. 117, No. 4, 291302 (2010).
2010
Mathematical Association of America (MAA), Washington, DC
EN
A30
I80
invariants
Weierstrass $\wp$function
elliptic functions
elliptic integrals
Zbl 0965.11025
doi:10.4169/000298910X480766
In a nutshell, this article presents  apart from biographical material on Cayley and Weierstra\ss{}  Cayley's transformation of an integral $\int \frac{dx}y$, where $y^2 = f_4(x)$ equals a quartic polynomial, into an integral of Weierstra\ss{} form $\int \frac{dx}{w}$, where $w^2 = f_3(x)$ equals a cubic polynomial. Neither the result nor Cayley's method of proof via invariants are as obscure as the author seems to believe: all of these classical results have found their way into modern presentations of the general $2$descent on elliptic curves [see e.g. {\it J. E. Cremona}, J. Symb. Comput. 31, No. 12, 7187 (2001; Zbl 0965.11025)]. For this reason, the article under review is recommended to number theorists who wonder where the invariants, covariants and syzygies they are using came from.
Franz Lemmermeyer (Jagstzell)