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