• Corpus ID: 244896160

The average number of integral points on the congruent number curves

@inproceedings{Chan2021TheAN,
  title={The average number of integral points on the congruent number curves},
  author={Stephanie Chan},
  year={2021}
}
We show that the total number of non-torsion integral points on the curves ED : y = x − Dx, where D ranges over positive squarefree integers less than N , is ≪ N(logN). 
View PDF on arXiv
1 Citations
Background Citations
1

One Citation

Integral points on cubic twists of Mordell curves

Abstract. Fix a non-square integer k 6= 0. We show that the the number of curves EB : y 2 = x + kB containing an integral point, where B ranges over positive integers less than N , is bounded by ≪k…

References

SHOWING 1-10 OF 20 REFERENCES

Linear Growth for Certain Elliptic Fibrations

We prove that the number of rational points of bounded height on certain del Pezzo surfaces of degree 1 defined over Q grows linearly, as predicted by Manin's conjecture.

Reduction of Binary Cubic and Quartic Forms

A reduction theory is developed for binary forms (homogeneous polynomials) of degrees three and four with integer coefficients. The resulting coefficient bounds simplify and improve on those in the…

The second moment of the number of integral points on elliptic curves is bounded

In this paper, we show that the second moment of the number of integral points on elliptic curves over $\mathbb{Q}$ is bounded. In particular, we prove that, for any $0 < s < \log_2 5 = 2.3219…

A quantitative version of Siegel's theorem: integral points on elliptic curves and Catalan curves.

The two fundamental finiteness theorems in the arithmetic theory of elliptic curves are the Mordell-Weil theorem, which says that the group of rational points is finitely generated, and Siegel's…

On the equivalence of binary quartics

Integral points on the congruent number curve

We study integral points on the quadratic twists $\mathcal{E}_D:y^2=x^3-D^2x$ of the congruent number curve. We give upper bounds on the number of integral points in each coset of…

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

Introduction To Elliptic Curves And Modular Forms

The size of Selmer groups for the congruent number problem

where Np is the number of solutions of the congruence y ≡ x −D2x (mod p). Then LD(s) has an analytic continuation as an entire function on the complex plane. The conjecture of Birch and…

Elliptic Curves: Diophantine Analysis

I. General Algebraic Theory.- I. Elliptic Functions.- II. The Division Equation.- III. p-Adic Addition.- IV. Heights.- V. Kummer Theory.- V1. Integral Points.- II. Approximation of Logarithms.- VII.…