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Galois representations of superelliptic curves
Part of
- Discontinuous groups and automorphic forms
- Arithmetic algebraic geometry
Ariel Pacetti, Angel Villanueva
Journal:

Glasgow Mathematical Journal / Volume 65 / Issue 2 / May 2023

Published online by Cambridge University Press:

24 November 2022, pp. 356-382

Print publication:

May 2023
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A superelliptic curve over a discrete valuation ring $\mathscr{O}$ of residual characteristic p is a curve given by an equation $\mathscr{C}\;:\; y^n=\,f(x)$, with $\textrm{Disc}(\,f)\neq 0$. The purpose of this article is to describe the Galois representation attached to such a curve under the hypothesis that f(x) has all its roots in the fraction field of $\mathscr{O}$ and that $p \nmid n$. Our results are inspired on the algorithm given in Bouw and WewersGlasg (Math. J. 59(1) (2017), 77–108.) but our description is given in terms of a cluster picture as defined in Dokchitser et al. (Algebraic curves and their applications, Contemporary Mathematics, vol. 724 (American Mathematical Society, Providence, RI, 2019), 73–135.).

Rational points on Erdős–Selfridge superelliptic curves
Part of
- Discontinuous groups and automorphic forms
- Diophantine equations
Michael A. Bennett, Samir Siksek
Journal:

Compositio Mathematica / Volume 152 / Issue 11 / November 2016

Published online by Cambridge University Press:

14 July 2016, pp. 2249-2254

Print publication:

November 2016
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Given $k\geqslant 2$ , we show that there are at most finitely many rational numbers $x$ and $y\neq 0$ and integers $\ell \geqslant 2$ (with $(k,\ell )\neq (2,2)$ ) for which
$$\begin{eqnarray}x(x+1)\cdots (x+k-1)=y^{\ell }.\end{eqnarray}$$
In particular, if we assume that $\ell$ is prime, then all such triples $(x,y,\ell )$ satisfy either $y=0$ or $\ell <\exp (3^{k})$ .