# Quadratic Fields Admitting Elliptic Curves with Rational j-Invariant and Good Reduction Everywhere

@article{Matschke2021QuadraticFA,
title={Quadratic Fields Admitting Elliptic Curves with Rational j-Invariant and Good Reduction Everywhere},
author={Benjamin Matschke and Abhijit S. Mudigonda},
journal={Journal of Number Theory},
year={2021}
}
• Published 17 March 2021
• Mathematics
• Journal of Number Theory
1 Citations

We show that the total number of non-torsion integral points on the elliptic curves $\mathcal{E}_D:y^2=x^3-D^2x$, where $D$ ranges over positive squarefree integers less than $N$, is $O( N(\log ## References SHOWING 1-10 OF 50 REFERENCES has good reduction at every finite place of the ring of integers of$Q(\sqrt{29})\$ . Other examples of such elliptic curves are found by several authors (see for example [17] and [5]). The aim of
The problem of determining elliptic curves over complex quadratic fields having good reduction everywhere has been discussed by Stroeker in [2] and the present author in 1 ]. In 1 ], such curves were
• Mathematics
Journal für die reine und angewandte Mathematik (Crelles Journal)
• 2020
Abstract For a set of primes 𝒫 {\mathcal{P}} , let Ψ ⁢ ( x ; 𝒫 ) {\Psi(x;\mathcal{P})} be the number of positive integers n ≤ x {n\leq x} all of whose prime factors lie in 𝒫 {\mathcal{P}} . In
• Mathematics
• 2016
In the first part we construct algorithms which we apply to solve S-unit, Mordell, cubic Thue, cubic Thue-Mahler and generalized Ramanujan-Nagell equations. As a byproduct we obtain alternative
It is shown that an elliptic curve having good reduction everywhere over a real quadratic field has a 2-rational point under certain hypotheses (primarily on class numbers of related fields) and small fields satisfying the hypotheses are found.
We show several consequences of the abc-conjecture for questions in analytic number theory which were of interest to Paul Erd} os: For any given polynomial f(x) 2 Zx], we deduce, from the
It is shown that an elliptic curve defined over a complex quadratic field K, having good reduction at all primes, does not have a global minimal (Weierstrass) model. As a consequence of a theorem of
ABSTRACT In this article, we study the problem of how to determine all elliptic curves defined over an arbitrary number field K with good reduction outside a given finite set of primes S of K by
• Philosophy, Mathematics
• 2001
Given a non-principal Dirichlet character χ (mod q), an important problem in number theory is to obtain good estimates for the size of L(1, χ). The best bounds known give that q−ǫ ≪ǫ |L(1, χ)| ≪ log