id: 05822210
dt: j
an: 2011a.00058
au: Rice, Adrian
ti: “To a factor près”: Cayley’s partial anticipation of the Weierstrass
$\wp$-function.
so: Am. Math. Mon. 117, No. 4, 291-302 (2010).
py: 2010
pu: Mathematical Association of America (MAA), Washington, DC
la: EN
cc: A30 I80
ut: invariants; Weierstrass $\wp$-function; elliptic functions; elliptic
integrals
ci: Zbl 0965.11025
li: doi:10.4169/000298910X480766
ab: In a nutshell, this article presents ‒ apart from biographical material
on Cayley and Weierstraß ‒ Cayley’s transformation of an integral
$\int \frac{dx}y$, where $y^2 = f_4(x)$ equals a quartic polynomial,
into an integral of Weierstraß 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. 1-2, 71‒87 (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.
rv: Franz Lemmermeyer (Jagstzell)