From: elkies@ramanujan.harvard.edu (Noam Elkies)
Newsgroups: sci.math.research
Subject: quintic question
Date: 17 Jan 1995 20:26:27 GMT
Summary: reference sought
Keywords: algebraic geometry enumeration
The following is surely known to someone, but it somehow escaped the
notice of the local alg.geom. gurus. Can anybody who has seen this
provide a reference?
Proposition: Let k be an algebraically closed field of characteristic 0,
and let P(t) be a polynomial of degree 5 over k with distinct roots.
Then there are exactly 80 polynomials y(t) of degree 3 such that
y^2 - P is the cube of a quadratic polynomial.
Proof sketch: Let C be the hyperelliptic curve u^2=P(t). Then
on C we have y^2-P = (y+u)(y-u). If that's a cube then the
div(y+u)=3D for some divisor D on C. Use Riemann-Roch to show
that the map y |--> [D] is a bijection from polynomials y
satisfying our condition to nontrivial 3-torsion elements of
the Jacobian Jac(C). Concluding by observing that the number
of such elements is 3^4 - 1 = 80.
Remark: It follows that there are 3*80=240 pairs (x,y) of polynomials
of degree 2,3 such that y^2 = x^3 + P(t). These may be identified
with the 240 roots of the E8 lattice, which arises from the Neron-
Severi group of the rational surface Y^2 = X^3 + P(T), or equivalently
(a la Shioda) as the Mordell-Weil lattice of the elliptic curve
y^2 = x^3 + P(t) over C(t). Our map y |--> [D] can then be
regarded as a manifestation of a 3-descent for this curve.
But the proof sketched above ought to have been known long before
this theory was developed; even Riemann-Roch is a fancier tool
than is really needed for this purpose.
--Noam D. Elkies (elkies@ramanujan.harvard.edu)
Dept. of Mathematics, Harvard University