RATIONAL POINTS ON THREE SUPERELLIPTIC CURVES

Rational points on Erdős–Selfridge superelliptic curves

Given k > 2, we show that there are at most finitely many rational numbers x and y 6= 0 and integers ` > 2 (with (k, `) 6= (2, 2)) for which x(x+ 1) · · · (x+ k − 1) = y. In particular, if we assume that ` is prime, then all such triples (x, y, `) satisfy either y = 0 or ` < exp(3k).

Rational points on curves

Rational points on curves

2 Faltings’ theorem 15 2.1 Prelude: the Shafarevich problem . . . . . . . . . . . . . . . . 15 2.2 First reduction: the Kodaira–Parshin trick . . . . . . . . . . . 17 2.3 Second reduction: passing to the jacobian . . . . . . . . . . . 19 2.4 Third reduction: passing to isogeny classes . . . . . . . . . . . 19 2.5 Fourth reduction: from isogeny classes to `-adic representations 21 2.6 The isogen...

Rational Points on Curves

This is an extended version of an invited lecture I gave at the Journées Arithmétiques in St. Étienne in July 2009. We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective) geometrically integral curve C over Q. The focus is on practical aspects of this problem in the case that the genus of C is at least 2, and therefore the set of rat...

Higher-order Weierstrass Weights of Branch Points on Superelliptic Curves

In this paper we consider the problem of calculating the higherorder Weierstrass weight of the branch points of a superelliptic curve C. For any q > 1, we give an exact formula for the q-weight of an affine branch point. We also find a formula for the q-weight of a point at infinity in the case where n and d are relatively prime. With these formulas, for any fixed n, we obtain an asymptotic for...