Related papers: Canonical heights and division polynomials
Given a sextic CM field $K$, we give an explicit method for finding all genus 3 hyperelliptic curves defined over $\mathbb{C}$ whose Jacobians are simple and have complex multiplication by the maximal order of this field, via an…
Let $K$ be a field of characteristic different from $2$, $\bar{K}$ its algebraic closure. Let $n \ge 3$ be an odd integer. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without repeated roots. Let us consider…
We study the universal family of odd hyperelliptic curves of genus $g \geq 1$ over $\mathbb{Q}$. We relate the heights of $\mathbb{Q}$-points of Jacobians of curves in this family to the reduction theory of the representation of…
Explicit monoid structure is provided for the class of canonical subfield preserving polynomials over finite fields. Some classical results and asymptotic estimates will follow as corollaries.
Let A be the jacobian variety of a hyperelliptic curve defined over a number field k. We provide a decomposition formula for the Faltings height of A and for the N\'eron-Tate height of k-rational points on A. We formulate a question of…
We present an efficient endomorphism for the Jacobian of a curve $C$ of genus 2 (hyperelliptic) for divisors having a Non disjoint support. This extends the work of Costello and Lauter in [12] who calculated explicit formulae for divisor…
Canonical differential calculus is defined for finitely generated abelian group with an involution existing consistently. Two such canonical calculi are found out. Fermionic representation for canonical calculus is defined based on…
We develop a notion of a `canonical $\mathcal{C}$-henselian valuation' for a class $\mathcal{C}$ of field extensions, generalizing the construction of the canonical henselian valuation of a field. We use this to show that the $p$-adic…
We construct jacobians of plane quartics without complex multiplication, using Del Pezzo surfaces of degree 2.
We discuss for the first time canonical differential equations for hyperelliptic Feynman integrals. We study hyperelliptic Lauricella functions that include in particular the maximal cut of the two-loop non-planar double box, which is known…
We study the interplay between canonical heights and endomorphisms of an abelian variety $A$ over a number field $k$. In particular we show that whenever the ring of endomorphisms defined over $k$ is strictly larger than $\Z$ there will be…
We provide an asymptotic estimate for the number of rational points of bounded height on a non-singular conic over the rationals. The estimate is uniform in the coefficients of the underlying quadratic form.
We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. From this algorithm, we derive a new…
In his previous papers the author proved that in characteristic different from 2 the jacobian J(C) of a hyperelliptic curve C: y^2=f(x) has only trivial endomorphisms over an algebraic closure K_a of the ground field K if the Galois group…
We propose a geometric explanation for the observation that generic quadratic polynomials over split quaternions may have up to six different factorizations while generic polynomials over Hamiltonian quaternions only have two. Split…
The traditional method of teaching canonical transformations involves the introduction of generating functions of various types. This method obscures the underlying structure of the Hamiltonian least-action principle, and can make a…
We use a global version of Heath-Brown's $p-$adic determinant method developed by Salberger to give upper bounds for the number of rational points of height at most $B$ on non-singular cubic curves defined over $\mathbb{Q}$. The bounds are…
The height of an algebraic number $\alpha$ is a measure of how arithmetically complicated $\alpha$ is. We say $\alpha$ is totally $p$-adic if the minimal polynomial of $\alpha$ splits completely over the field $\mathbb{Q}_p$ of $p$-adic…
We give some bounds on the numbers of rational points on abelian varieties and jacobians varieties over finite fields. The main result is that we determine the maximum and minimum number of rational points on jacobians varieties of…
In this paper, we give some counting results on integer polynomials of fixed degree and bounded height whose distinct non-zero roots are multiplicatively dependent. These include sharp lower bounds, upper bounds and asymptotic formulas for…