Related papers: Canonical heights and division polynomials

Let $K$ be a global field of characteristic different from 2 and $u(x)\in K[x]$ be an irreducible polynomial of even degree $2g\ge 6$, whose Galois group over $K$ is either the full symmetric group $S_{2g}$ or the alternating group…

In order to achieve a Hamiltonian-based canonical derivation of the Hawking effect, one usually faces multiple hurdles. Firstly, the spacetime foliation using Schwarzschild time does not lead to hyper-surfaces which are always spacelike.…

We explicitly construct the Kummer variety associated to the Jacobian of a hyperelliptic curve of genus 3 that is defined over a field of characteristic not equal to 2 and has a Weierstra{\ss} point defined over the same field. We also…

Combining $2$-descent techniques with Riemann-Roch and B\'ezout's theorems, we give an upper bound on the number of rational points of bounded height on elliptic and hyperelliptic curves over function fields of characteristic $\neq 2$. We…

In this paper, we propose a domain decomposition method for multiscale second order elliptic partial differential equations with highly varying coefficients. The method is based on a discontinuous Galerkin formulation. We present both a…

We consider the problem of efficient computation in the Jacobian of a hyperelliptic curve of genus 3 defined over a field whose characteristic is not 2. For curves with a rational Weierstrass point, fast explicit formulas are well known and…

For $\theta$ a non-algebraic point on a quasi projective variety over a number field, I prove that $\theta$ has an approximation by a series of algebraic points of bounded height and degree which is essentially best possible. Applications…

We fix a counting function of multiplicities of algebraic points in a projective hypersurface over a number field, and take the sum over all algebraic points of bounded height and fixed degree. An upper bound for the sum with respect to…

Let $\pi : E\to B$ be an elliptic surface defined over a number field $K$, where $B$ is a smooth projective curve, and let $P: B \to E$ be a section defined over $K$ with canonical height $\hat{h}_E(P)\not=0$. In this article, we show that…

Building on work of Davenport and Schmidt, we mainly prove two results. The first one is a version of Gel'fond's transcendence criterion which provides a sufficient condition for a complex or $p$-adic number $\xi$ to be algebraic in terms…

We develop a strategy for bounding from above the height of rational points of modular curves with values in number fields, by functions which are polynomial in the curve's level. Our main technical tools come from effective Arakelov…

We prove that for any two integers $g\geq 4$ and $g'\leq 2g-1$, there exist abelian varieties over $\overline{\mathbb{Q}}$ which are not quotients of a Jacobian of dimension $g'$. Our method in fact proves that most Abelian varieties…

We give a formula for the component at p of the p-adic height pairing of a divisor of degree 0 on a hyperelliptic curve. We use this to give a Chabauty-like method for finding p-adic approximations to p-integral points on such curves when…

A polynomial over a ring is called decomposable if it is a composition of two nonlinear polynomials. In this paper, we obtain sharp lower and upper bounds for the number of decomposable polynomials with integer coefficients of fixed degree…

An important problem in computational arithmetic geometry is to find changes of coordinates to simplify a system of polynomial equations with rational coefficients. This is tackled by a combination of two techniques, called minimisation and…

We prove that the number of geometrically indecomposable representations of fixed dimension vector d of a canonical algebra C defined over a finite field Fq is given by a polynomial in q (depending on C and d). We prove a similar result for…

For a composite $n$ and an odd $c$ with $c$ not dividing $n$, the number of solutions to the equation $n+a\equiv b\mod c$ with $a,b$ quadratic residues modulus $c$ is calculated. We establish a direct relation with those modular solutions…

We show that a genus $2$ curve over a number field whose jacobian has complex multiplication will usually have stable bad reduction at some prime. We prove this by computing the Faltings height of the jacobian in two different ways. First,…

In this article, we present a method for computing rational points on hyperelliptic curves of genus~3 and isolated quadratic points on hyperelliptic curves of genus~2 and~3 whose Jacobians have rank~0. Our approach begins by computing the…

We derive a straightening-free algorithm that computes the canonical bases of any higher-level q-deformed Fock space.