Related papers: Integral points on the congruent number curve

We establish sharp lower and upper bounds for the number of integral points near dilations of a space curve with nowhere vanishing torsion.

Let $b\ge 2$ be a given integer. In this paper, we show that there only finitely many positive integers $d$ which are not squares, such that the Pell equation $X^2-dY^2=1$ has two positive integer solutions $(X,Y)$ with the property that…

We provide an estimate for the number of nontrivial integer points on the Pellian surface $t^2 - du^2 = 1$ in a bounded region. We give a lower bound on the size of fundamental solutions for almost all $d$ in a certain class, based on a…

We consider genus $g$ hyperelliptic curves over $\mathbb{Q}$ with a rational Weierstrass point, ordered by height. If $d < g$ is odd, we prove, under an assumption, that there exists $B_d$ such that a positive proportion of these curves…

Let C be an affine, plane, algebraic curve of degree d with integer coefficients. In 1989, Bombieri and Pila showed that if one takes a box with sides of length N then C can obtain no more than O_{d,\epsilon}(N^{1/d+\epsilon}) integer…

We estimate the maximal number of integral points which can be on a convex arc in the plane with given length, minimal radius of curvature and initial slope.

The Pell equation $x^2 - Dy^2 = 1$ with non-square $D > 1$ has infinitely many integer solutions, yet most research has centered on the asymptotic behavior of fundamental units as $D$ varies. By contrast, the exact distribution of solutions…

Let $e_1,e_2,e_3$ be nonzero integers satisfying $e_1+e_2+e_3=0$. Let $(a,b,c)$ be a primitive triple of odd integers satisfying $e_1a^2+e_2b^2+e_3c^2=0$. Denote by $E: y^2=x(x-e_1)(x+e_2)$ and $\mathcal E: y^2=x(x-e_1a^2)(x+e_2b^2)$.…

Let $K$ be a number field, $S$ a finite set of places. For $\mathbb{G}_m$ or an elliptic curve $E$ defined over $K$, we establish uniformity results on the number of $S$-integral torsion points relative to a non-torsion point $\beta$, as…

We prove matching upper and lower bounds for the average of the 6-torsion of class groups of quadratic fields. Furthermore, we count the number of integer solutions on an affine quartic threefold.

For an integer $d\geq 2$ which is not a square, we show that there is at most one value of the positive integer $X$ participating in the Pell equation $X^2-dY^2=\pm 1$ which is a Tribonacci number, with a few exceptions that we completely…

In this note we study the distribution of real inflection points among the ovals of a real non-singular hyperbolic curve of even degree. Using Hilbert's method we show that for any integers $d$ and $r$ such that $4\leq r \leq 2d^2-2d$,…

We use twisted stable maps to answer the following question. Let E\subset P^2 be a smooth cubic. How many rational degree d curves pass through a general points of E, have b specified tangencies with E and c unspecified tangencies, and pass…

We study the following question: given a set P of 3d-2 points and an immersed curve G in the real plane R^2, all in general position, how many real rational plane curves of degree d pass through these points and are tangent to this curve.…

For nonzero rational D, which may be taken to be a squarefree integer, let E_D be the elliptic curve Dy^2=x^3-x over Q arising in the "congruent number" problem. It is known that the L-function of E_D has sign -1, and thus odd analytic…

Fix an elliptic curve $E$ over a number field $F$ and an integer $n$ which is a power of $3$. We study the growth of the Mordell--Weil rank of $E$ after base change to the fields $K_d = F(\sqrt[2n]{d})$. If $E$ admits a $3$-isogeny, then we…

In this article, we combine Bhargava's geometry-of-numbers methods with the dynamical point-counting methods of Eskin--McMullen and Benoist--Oh to develop a new technique for counting integral points on symmetric varieties lying within…

Let $C$ be a smooth projective curve defined over the field of complex numbers. Let $E$ be a vector bundle on $C$, and fix an integer $d\geqslant 1$. Let $\mc Q:={\rm Quot}(E,d)$ be the Quot Scheme which parameterizes all torsion quotients…

We generalize Siegel's theorem on integral points on affine curves to integral points of bounded degree, giving a complete characterization of affine curves with infinitely many integral points of degree d or less over some number field.…

We obtain a recursive formula for the number of rational degree $d$ curves in $\mathbb{CP}^2$ that pass through $3d+1-m$ generic points and that have an $m$-fold singular point. The special case of counting curves with a triple point was…