Twists of Superelliptic Curves Without Rational Points

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 Atkin-lehner Twists of Modular Curves

These are the (more detailed) notes accompanying a talk that I am to give at the University of Pennsylvania on July 21, 2006. The topic is rational points on Atkin-Lehner twists of the modular curves X0(N). Apart from being an interesting Diophantine problem in its own right, there is an ulterior motive: Q-rational points correspond to “elliptic Q-curves” and thus to projective Galois represent...

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...

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...