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.

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.

AN APPLICATION OF NESTERENKO'S DIMENSION ESTIMATE TO p-ADIC q-SERIES

2011 ◽  
Vol 07 (02) ◽  
pp. 431-447 ◽  
Author(s):  
PETER BUNDSCHUH ◽  
KEIJO VÄÄNÄNEN
Keyword(s):  
Lower Bounds ◽  
Difference Equations ◽  
Functional Equations ◽  
Rational Points ◽  
Vector Spaces ◽  
Entire Solutions ◽  
Dimension Estimate

Very recently, Nesterenko proved a p-adic analogue of his famous dimension estimate from 1985. The main aim of our present paper is to use this criterion to obtain lower bounds for the dimension of ℚ-vector spaces spanned by the values at certain rational points of p-adic solutions of a class of linear q-difference equations. For the application of Nesterenko's new estimate, we first need a p-adic analogue of Töpfer's results on entire solutions of such functional equations, and secondly, very precise evaluations of certain p-adic Schnirelman integrals.

scholarly journals On lower bounds for the Ihara constants  and 

2013 ◽  
Vol 149 (7) ◽  
pp. 1108-1128
Author(s):  
Iwan Duursma ◽  
Kit-Ho Mak
Keyword(s):  
Lower Bounds ◽  
Rational Points ◽  
Class Field ◽  
Class Field Tower

AbstractLet $ \mathcal{X} $ be a curve over ${ \mathbb{F} }_{q} $ and let $N( \mathcal{X} )$, $g( \mathcal{X} )$ be its number of rational points and genus respectively. The Ihara constant $A(q)$ is defined by $A(q)= {\mathrm{lim~sup} }_{g( \mathcal{X} )\rightarrow \infty } N( \mathcal{X} )/ g( \mathcal{X} )$. In this paper, we employ a variant of Serre’s class field tower method to obtain an improvement of the best known lower bounds on $A(2)$ and $A(3)$.

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