Related papers: The Circle Method for Quadrics over Function Field…

This article presents and compares four approaches for computing the rotation of a point about an axis by an angle in $\mathbb{R}^3$. We illustrate these methods by computing, by hand, the rotation of point $P=(1,0,1)^T$ about axis…

We give some methods for computing equations for certain Shimura curves, natural maps between them, and special points on them. We then illustrate these methods by working out several examples in varying degrees of detail. For instance, we…

An $n$-cap in $k$-dimensional projective space is a set of $n$ points so that no three lie on a line. In this note, we provide an algorithm to count the number of $n$-caps in $\mathbb{P}^3(\mathbb{F}_q)$, which follows from our recent paper…

In this paper we give an upper bound on the number of rational points on an irreducible curve $C$ of degree $\delta$ defined over a finite field $\mathbb{F}_q$ lying on a Frobenius classical surface $S$ embedded in $\mathbb{P}^3$. This…

Quadratic Chabauty is a $p$-adic method for determining rational points on curves. Local heights are arithmetic invariants used in the quadratic Chabauty method. We present an algorithm to compute these local heights for hyperelliptic…

In this article we give an algorithm for the computation of the number of rational points on the Jacobian variety of a generic ordinary hyperelliptic curve defined over a finite field of cardinality $q$ with time complexity $O(n^{2+o(1)})$…

In this paper we prove a formula for the number of rational points of bounded height relative to all the generators of the cone of effective divisor for a toric variety over a number field.

We give an explicit description of the F_{q^i}-rational points on the Fermat curve u^{q-1}+v^{q-1}+w^{q-1}=0 for each i=1,2,3. As a consequence, we observe that for any such point (u,v,w), the product uvw is a cube in F_{q^i}. We also…

Let $K$ be a global field of positive characteristic. We give an asymptotic formula for the number of $K$-points of bounded height on the Hilbert scheme $\text{Hilb}^2\mathbb{P}^2$ and show that by eliminating an exceptional thin set, the…

I provide a systematic construction of points (defined over number fields) on Legendre elliptic curves over $\mathbb{Q}$: for any odd integer $n\geq 3$ my method constructs $n$ points on the Legendre curve and I show that rank of the…

In his paper from 1996 on quadratic forms Heath-Brown developed a version of the circle method to count points in the intersection of an unbounded quadric with a lattice of short period, if each point is given a weight, and approximated…

We consider the question of determining the maximum number of $\mathbb{F}_q$-rational points that can lie on a hypersurface of a given degree in a weighted projective space over the finite field $\mathbb{F}_q$, or in other words, the…

We prove analogues of several well-known results concerning rational morphisms between quadrics for the class of so-called quasilinear $p$-hypersurfaces. These hypersurfaces are nowhere smooth over the base field, so many of the geometric…

Let $\mathbb F_{q^n}$ denote the finite field with $q^n$ elements. In this paper we determine the number of $\mathbb F_{q^n}$-rational points of the affine Artin-Schreier curve given by $y^q-y = x(x^{q^i}-x)-\lambda$ and of the…

Using the circle method, we count integer points on complete intersections in biprojective space in boxes of different side length, provided the number of variables is large enough depending on the degree of the defining equations and…

We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension $2$.…

We describe a method to show a plane quartic over a number field has no rational points. The method can be adapted to show that a curve does not have divisors of degree 1 or 2 and can be generalized to arbitrary smooth projective curves.…

In this note we give exact formulas (and asymptotics) for the number of rational points of bounded height on weighted projective stacks over global function fields.

We give an algorithm which requires no integer factorization for computing the canonical height of a point in $\mathbb{P}^1(\mathbb{Q})$ relative to a morphism $\phi: \mathbb{P}_{\mathbb{Q}}^1 \rightarrow \mathbb{P}_{\mathbb{Q}}^1$ of…

We construct an algorithm for solving the following problem: given a number field $K$, a positive integer $N$, and a positive real number $B$, determine all points in $\mathbb P^N(K)$ having relative height at most $B$. A theoretical…