Nous rappelons que Manin décrit l’homologie singulière relative aux pointes de la courbe modulaire $X_0\left(N\right)$ comme un quotient du groupe ${\mathbf{Z}}^{\left({\mathbf{P}}^1\right(\mathbf{Z}/N\mathbf{Z}\left)\right)}$. En s’appuyant sur des techniques de fractions continues, nous donnons une expression indépendante de $N$ d’un relèvement de l’action des opérateurs de Hecke de $H_1(X_0\left(N\right),ptes,\mathbf{Z})$ sur ${\mathbf{Z}}^{\left({\mathbf{P}}^1\right(\mathbf{Z}/N\mathbf{Z}\left)\right)}$.

In this article, we formalize operations of points on an elliptic curve over GF(p). Elliptic curve cryptography [7], whose security is based on a difficulty of discrete logarithm problem of elliptic curves, is important for information security. We prove that the two operations of points: compellProjCo and addellProjCo are unary and binary operations of a point over the elliptic curve.

On peut définir la pente d'un mot écrit avec des 0 et des 1 comme le nombre de 1 divisé par le nombre de 0, et généraliser cette définition aux mots de longueur infinie. Considérant le lien entre les mots de Christoffel et les fractions continues, on se propose d'étudier le comportement de tels mots lorsqu'on additionne leurs pentes, ou qu'on les multiplie par un entier positif. Après un bref exposé des différentes notions liées aux mots de Christoffel, l'étude de la somme et de la multiplication...

The law of the iterated logarithm for discrepancies of lacunary sequences is studied. An optimal bound is given under a very mild Diophantine type condition.

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...