Related papers: Integral points on the congruent number curve

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 $O( N(\log N)^{-1/4+\epsilon})$. The proof involves a…

We study integral points on the quadratic twists $E_D : y^2 = x^3+D^2Ax+D^3B$ of a fixed elliptic curve $E : y^2 = x^3+Ax+B$ over $\overline{Q}$. For sufficiently large squarefree positive integers $D$, we prove that the number of integral…

We prove that, when elliptic curves $E/\mathbb{Q}$ are ordered by height, the average number of integral points $\#|E(\mathbb{Z})|$ is bounded, and in fact is less than $66$ (and at most $\frac{8}{9}$ on the minimalist conjecture). By…

Fix a non-square integer $k\neq 0$. We show that the number of curves $E_B:y^2=x^3+kB^2$ containing an integral point, where $B$ ranges over positive integers less than $N$, is bounded by $O_k(N(\log N)^{-\frac{1}{2}+\epsilon})$. In…

For a given elliptic curve E in short Weierstrass form, we show that almost all quadratic twists E_D have no integral points, as D ranges over square-free integers ordered by size. Our result is conditional on a weak form of the Hall-Lang…

Let $B$ be a sixth-power-free integer and $P$ be a non-torsion point on the Mordell curve $E_B:y^2=x^3+B$. In this paper, we study integral multiples $[n]P$ of $P$. Among other results, we show that $P$ has at most three integral multiples…

The subject matter of this work is the set of integral points(i.e. points with both coordinates integers) on the graphs of rational functions of the form f(x)=(x^2+bx+c)/(x+a), with a,b,c,being integers.Following the introduction, we…

If $E$ is a minimal elliptic curve defined over $\ZZ$, we obtain a bound $C$, depending only on the global Tamagawa number of $E$, such that for any point $P\in E(\QQ)$, $nP$ is integral for at most one value of $n>C$. As a corollary, we…

For positive rank $r$ elliptic curves $E(\mathbb{Q})$, we employ ideal class pairings $$ E(\mathbb{Q})\times E_{-D}(\mathbb{Q}) \rightarrow \mathrm{CL}(-D), $$ for quadratic twists $E_{-D}(\mathbb{Q})$ with a suitable ``small $y$-height''…

We show that the average number of integral points on elliptic curves, counted modulo the natural involution on a punctured elliptic curve, is bounded from above by $2.1 \times 10^8$. To prove it, we design a descent map, whose prototype…

We study the quadratic integral points-that is, (S-)integral points defined over any extension of degree two of the base field-on a curve defined in P_3 by a system of two Pell equations. Such points belong to three families explicitly…

On the coordinate plane, the slopes $a$ and $b$ of two straight lines and the slope $c$ of one of their angle bisectors satisfy the equation $(a-c)^2(b^2+1) = (b-c)^2(a^2+1).$ Recently, an explicit formula for nontrivial integral solutions…

Let $E: y^2=x(x-a^2)(x+b^2)$ be an elliptic curve with full $2$-torsion group, where $a$ and $b$ are coprime integers and $2(a^2+b^2)$ is a square. Assume that the $2$-Selmer group of $E$ has rank two. We characterize all quadratic twists…

The congruent number elliptic curves are defined by $E_d: y^2=x^3-d^2x$, where $d\in \mathbb{N}.$ We give a simple proof of a formula for $L(\mathrm{Sym}^2(E_d),3)$ in terms of the determinant of the elliptic trilogarithm evaluated at some…

We study the $2$-Selmer ranks of elliptic curves. We prove that for an arbitrary elliptic curve $E$ over an arbitrary number field $K$, if the set $A_E$ of 2-Selmer ranks of quadratic twists of $E$ contains an integer $c$, it contains all…

We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques ([BP], [HBR]) and methods based on quasiorthogonality in the Mordell-Weil lattice…

Clemm and Trebat-Leder (2014) proved that the number of quadratic number fields with absolute discriminant bounded by $x$ over which there exist elliptic curves with good reduction everywhere and rational $j$-invariant is $\gg…

We give a completely explicit upper bound for integral points on (standard) affine models of hyperelliptic curves, provided we know at least one rational point and a Mordell-Weil basis of the Jacobian. We also explain a powerful refinement…

The Thue-Siegel method is used to obtain an upper bound for the number of primitive integral solutions to a family of quartic Thue's inequalities. This will provide an upper bound for the number of integer points on a family of elliptic…

14H52 : Elliptic curves Let E be the elliptic curve given by a Mordell equation y^2=x^3-A where A is an integer. For certain A, we use Stoll's formula to compute a lower bound for the proportion of square-free integers D up to X such that…