Stein and Watkins conjectured that for a certain family of elliptic curves E, the X₀(N)-optimal curve and the X₁(N)-optimal curve of the isogeny class 𝓒 containing E of conductor N differ by a 3-isogeny. In this paper, we show that this conjecture is true.

For i = 0,1, let $E_i$ be the $X_i\left(N\right)$-optimal curve of an isogeny class of elliptic curves defined over ℚ of conductor N. Stein and Watkins conjectured that E₀ and E₁ differ by a 5-isogeny if and only if E₀ = X₀(11) and E₁ = X₁(11). In this paper, we show that this conjecture is true if N is square-free and is not divisible by 5. On the other hand, Hadano conjectured that for an elliptic curve E defined over ℚ with a rational point P of order 5, the 5-isogenous curve E’ := E/⟨P⟩ has a rational point of order...

If the counting function N(x) of integers of a Beurling generalized number system satisfies both ${\int }_1^{\infty }{x}^{-2}|N\left(x\right)-Ax|dx<\infty$ and $x^{-1}\left(logx\right)(N\left(x\right)-Ax)=O\left(1\right)$, then the counting function π(x) of the primes of this system is known to satisfy the Chebyshev bound π(x) ≪ x/logx. Let f(x) increase to infinity arbitrarily slowly. We give a construction showing that ${\int }_1^{\infty }|N\left(x\right)-Ax|x^{-2}dx<\infty$ and $x^{-1}\left(logx\right)(N\left(x\right)-Ax)=O\left(f\left(x\right)\right)$ do not imply the Chebyshev bound.

We consider digit expansions ${\sum }_{j=0}^{\ell -1}{\Phi }^j\left(d_j\right)$ with an endomorphism $\Phi$ of an Abelian group. In such a numeral system, the $w$-NAF condition (each block of $w$ consecutive digits contains at most one nonzero) is shown to minimise the Hamming weight over all expansions with the same digit set if and only if it fulfills the subadditivity condition (the sum of every two expansions of weight $1$ admits an optimal $w$-NAF).This result is then applied to imaginary quadratic bases, which are used for scalar multiplication in elliptic...

Soit $K$ un corps $p$-adique, $G$ son groupe de Galois absolu et $v$ la valuation sur ${\mathbf{C}}_p$. Dans sa démonstration du théorème d’Ax-Sen-Tate, Ax montre que si pour un $A\in \mathbf{R}$, $x\in {\mathbf{C}}_p$ vérifie $v(\sigma x-x)\ge A$ pour tout $\sigma \in G$, alors il existe $y\in K$ tel que $v(x-y)\ge A-c$, avec $c=p/{(p-1)}^2$. Ax se pose la question de l’optimalité de la constante $c$, que nous étudions ici. En utilisant l’extension de $K$ engendrée par les racines $p^m$-es d’une uniformisante fixée de $K$, nous déterminons la constante optimale, ainsi que des informations supplémentaires sur les $x\in {\mathbf{C}}_p$ tels que $v(\sigma x-x)\ge A$ pour...

This paper is devoted to a systematic study of a class of binary trees encoding the structure of rational numbers both from arithmetic and dynamical point of view. The paper is divided into three parts. The first one is mainly expository and consists in a critical review of rather standard topics such as Stern-Brocot and Farey trees and their connections with continued fraction expansion and the question mark function. In the second part we introduce two classes of (invertible and non-invertible)...