# 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} }

## One Citation

### The average number of integral points on the congruent number curves

- Mathematics
- 2021

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

### Computing Elliptic Curves Having Good Reduction Everywhere over Quadratic Fields

- Mathematics
- 2001

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…

### Elliptic curves with good reduction everywhere over quadratic fields and having rational $j$-invariant

- Mathematics
- 1981

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…

### When the sieve works II

- MathematicsJournal 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…

### Solving S-unit, Mordell, Thue, Thue-Mahler and generalized Ramanujan-Nagell equations via Shimura-Taniyama conjecture

- 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…

### Reduction of elliptic curves over certain real quadratic number fields

- MathematicsMath. Comput.
- 1999

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.

### ABC allows us to count squarefrees

- Mathematics
- 1998

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…

### Reduction of elliptic curves over imaginary quadratic number fields.

- Mathematics
- 1983

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…

### Computing All Elliptic Curves Over an Arbitrary Number Field with Prescribed Primes of Bad Reduction

- MathematicsExp. Math.
- 2019

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…

### UPPER BOUNDS FOR |L(1, χ)|

- 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…