Rational points on Erdős–Selfridge superelliptic curves
نویسندگان
چکیده
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).
منابع مشابه
Rational points on Jacobians of hyperelliptic curves
We describe how to prove the Mordell-Weil theorem for Jacobians of hyperelliptic curves over Q and how to compute the rank and generators for the Mordell-Weil group.
متن کاملDescent and Covering Collections
We explain several approaches that allow to prove that a given curve over Q has no rational points.
متن کاملChabauty and the Mordell-Weil Sieve
These notes are based on lectures given at the “Arithmetic of Hyperelliptic Curves” workshop, Ohrid, Macedonia, 28 August–5 September 2014. They offer a brief (if somewhat imprecise) sketch of various methods for computing the set of rational points on a curve, focusing on Chabauty and the Mordell–Weil sieve.
متن کاملHomology of braid groups, the Burau representation, and Fq-points on superelliptic curves
The (reduced) Burau representation Vn of the braid group Bn is obtained from the action of Bn on the homology of an infinite cyclic cover of the disc with n punctures. In this paper, we calculate H∗(Bn;Vn). As an application, we show that the expected number of Fq-points on a random superelliptic curve is equal to q.
متن کاملHomology of braid groups, the Burau representation, and points on superelliptic curves over finite fields
The (reduced) Burau representation Vn of the braid group Bn is obtained from the action of Bn on the homology of an infinite cyclic cover of the disc with n punctures. In this paper, we calculate H∗(Bn;Vn). Our topological calculation has the following arithmetic interpretation (which also has different algebraic proofs): the expected number of points on a random superelliptic curve of a fixed ...
متن کامل