Rational points with large denominator on Erdos--Selfridge superelliptic curves

2021 ◽  
Vol 99 (3-4) ◽  
pp. 317-329
Author(s):  
N. Saradha
Keyword(s):  
Rational Points ◽  
Superelliptic Curves

VARIANTS OF ERDŐS–SELFRIDGE SUPERELLIPTIC CURVES AND THEIR RATIONAL POINTS

Mathematika ◽  
2018 ◽  
Vol 64 (2) ◽  
pp. 380-386 ◽  
Author(s):  
Pranabesh Das ◽  
Shanta Laishram ◽  
N. Saradha
Keyword(s):  
Rational Points ◽  
Superelliptic Curves

POWERS FROM PRODUCTS OF CONSECUTIVE TERMS IN ARITHMETIC PROGRESSION

2006 ◽  
Vol 92 (2) ◽  
pp. 273-306 ◽  
Author(s):  
M. A. BENNETT ◽  
N. BRUIN ◽  
K. GYÖRY ◽  
L. HAJDU
Keyword(s):  
Positive Integer ◽  
Arithmetic Progression ◽  
Diophantine Equations ◽  
Galois Representations ◽  
Rational Points ◽  
Arithmetic Progressions ◽  
Superelliptic Curves

We show that if $k$ is a positive integer, then there are, under certain technical hypotheses, only finitely many coprime positive $k$-term arithmetic progressions whose product is a perfect power. If $4 \leq k \leq 11$, we obtain the more precise conclusion that there are, in fact, no such progressions. Our proofs exploit the modularity of Galois representations corresponding to certain Frey curves, together with a variety of results, classical and modern, on solvability of ternary Diophantine equations. As a straightforward corollary of our work, we sharpen and generalize a theorem of Sander on rational points on superelliptic curves.

RATIONAL POINTS ON SOME HYPER- AND SUPERELLIPTIC CURVES

2011 ◽  
Vol 07 (01) ◽  
pp. 115-132
Author(s):  
ANDREW BREMNER ◽  
MACIEJ ULAS
Keyword(s):  
Lower Bounds ◽  
Rational Points ◽  
High Genus ◽  
Superelliptic Curves

We construct families of certain hyper- and superelliptic curves that contain a (small) number of rational points. This leads to lower bounds for the ranks of Jacobians of certain high genus curves.

scholarly journals Rational points on Erdős–Selfridge superelliptic curves

2016 ◽  
Vol 152 (11) ◽  
pp. 2249-2254 ◽  
Author(s):  
Michael A. Bennett ◽  
Samir Siksek
Keyword(s):  
Rational Numbers ◽  
Rational Points ◽  
Superelliptic Curves ◽  
Image Position

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 $\ell <\exp (3^{k})$.

scholarly journals RATIONAL POINTS ON THREE SUPERELLIPTIC CURVES

2011 ◽  
Vol 85 (1) ◽  
pp. 105-113 ◽  
Author(s):  
ZHONGYAN SHEN ◽  
TIANXIN CAI
Keyword(s):  
Rational Points ◽  
Superelliptic Curves ◽  
Image Position

AbstractIn this paper, we obtain all rational points (x,y) on the superelliptic curves

scholarly journals Counting and equidistribution in quaternionic Heisenberg groups

2021 ◽  
pp. 1-38
Author(s):  
JOUNI PARKKONEN ◽  
FRÉDÉRIC PAULIN
Keyword(s):  
Hyperbolic Geometry ◽  
Quaternion Algebra ◽  
Rational Points ◽  
Hyperbolic Spaces ◽  
Heisenberg Groups ◽  
Arithmetic Groups ◽  
Hermitian Forms ◽  
Dimension 2 ◽  
Counting Formula ◽  
The Relationship

Abstract We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

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