Related papers: p-adic properties of division polynomials and elli…

We show that the vector of period ratios of a cubic surface is rational over $Q(\omega)$, where $\omega = \exp(2\pi i/3)$ if and only if the associate abelian variety is isogeneous to a product of Fermat elliptic curves. We also show how to…

It is known, that for every elliptic curve over Q there exists a quadratic extension in which the rank does not go up. For a large class of elliptic curves, the same is known with the rank replaced by the 2-Selmer group. We show, however,…

Given an elliptic curve E/Q and a prime p at which E has good reduction, let e_p be the exponent of the group E_p(F_p) of F_p-rational points on the reduction of E modulo p. Under the Generalized Riemann Hypothesis (GRH) for the Dedekind…

We derive necessary and sufficient conditions for periodic and for elliptic periodic trajectories of billiards within an ellipse in the Minkowski plane in terms of an underlining elliptic curve. We provide several examples of periodic and…

We show that for an elliptic curve E defined over a number field K, the group E(A) of points of E over the adele ring A of K is a topological group that can be analyzed in terms of the Galois representation associated to the torsion points…

Let k be an algebraically closed field of characteristic p. Let X(p^e;N) be the curve parameterizing elliptic curves with full level N structure (where p does not divide N) and full level p^e Igusa structure. By modular curve, we mean a…

We study the existence of primes and of primitive divisors in classical divisibility sequences defined over function fields. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields…

We are concerned with power series in 1/T over a finite field of 3 elements $\F_3$. In a previous article, Alain Lasjaunias investigated the existence of particular power series of elements algebraic over $\F_3[T]$, having all partial…

Let $E/\bbq$ be an elliptic curve defined over $\bbq$ with conductor $N$ and $\gq$ the absolute Galois group of an algebraic closure $\bar{\bbq}$ of $\bbq$. We prove that for every $\sigma\in \gq$, the Mordell-Weil group $E(\oqs)$ of $E$…

We study the \Q_p-unipotent Albanese map for curves over local fields with residue characteristic different from p and show that it has finite image. As a corollary, we deduce a simple `pi_1'-proof of Siegel's theorem for rank 1 elliptic…

We study Hill's differential equation with potential expressed by elliptic functions which arises in some problems of physics and mathematics. Analytical method can be applied to study the local properties of the potential in asymptotic…

Let $K$ be a complete discretely valued field of rank one, with residue field $\Q_p$. It is well known that period equals index in $\Br(K)$. We prove that when $p=2$ there exist noncyclic $K$-division algebras of every $2$-power degree…

Let $E$ be an elliptic curve over the finite field $\mathbb{F}_p$, and $P \in E(\mathbb{F}_p)$ be an $\mathbb{F}_p$-rational point. We study the sums \[ S_{\chi,P}(N,h) = \sum_{n=1}^N \chi(\psi_n(P)) \chi(\psi_{n+h}(P)), \] where…

Let $E$ be an elliptic curve over a quartic field $K$. By the Mordell-Weil theorem, $E(K)$ is a finitely generated group. We determine all the possibilities for the torsion group $E(K)_{tor}$ where $K$ ranges over all quartic fields $K$ and…

Fix a prime $p$. We prove that the set of sentences true in all but finitely many finite extensions of $\mathbb{Q}_p$ is undecidable in the language of valued fields with a cross-section. The proof goes via reduction to characteristic $p$,…

Consider an algebraic function like $F(x) = \sqrt{x^3 - 1}$. If $p \in \mathbb{Q}$ is a rational number, how many iterates of $p$ under $F$ can also be rational? The dynamics of algebraic functions may be formalized in the language of…

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…

Let $E$ be an elliptic curve defined over $\mathbb Q$ and $\widetilde{E}_p$ denote the reduction of $E$ modulo a prime $p$ of good reduction for $E$. The divisibility of $|\widetilde{E}_{p}(\mathbb{F}_p)|$ by an integer $m\ge 2$ for a set…

Let $C$ be an algebraic curve embedded transversally in a power $E^N$ of an elliptic curve $E$. In this article we produce a good explicit bound for the height of all the algebraic points on $C$ contained in the union of all proper…

Considering an arbitrary pair of distinct and non constant polynomials, $a$ and $b$ in $\mathbb{F}_2[t]$, we build a continued fraction in $\mathbb{F}_2((1/t))$ whose partial quotients are only equal to $a$ or $b$. In a previous work of the…