ABC and the Hasse principle for quadratic twists of hyperelliptic curves

Conditionally on the ABC conjecture, we apply work of Granville to show that a hyperelliptic curve $C_{/\mathbb{Q}}$ of genus at least three has infinitely many quadratic twists that violate the Hasse Principle iff it has no $\mathbb{Q}$-rational hyperelliptic branch points.     

Projectively normal embeddings of an algebraic curve with a net

Let $X$ be a smooth irreducible algebraic curve of genus $g$. The projective normality of a complete embedding of $X$ is determined by only its quadratic normality in case the embedding is of degree at least $g$. This means that the complete embedding fails to be projectively normal if and only if it admits an effective divisor which fails to impose independent conditions on quadrics in the embedded projective space. Thus if $X$ admits a net, then it is interesting to compare the conditions for the projective normality of an embedding of $X$ with properties of conic sections of the plane curve given by the net. For such a curve $X$ with a net, we show that the projectively normal embeddings are closely related to properties of conic sections.     

On unramified normal coverings of hyperelliptic curves

It is well known that the number of unramified normal coverings of an irreducible complex algebraic curve C with a group of covering transformations isomorphic to Z₂⊕Z₂ is (2^4g−3⋅2^2g+2)/6. Assume that C is hyperelliptic, say . Horiouchi has given the explicit algebraic equations of the subset of those covers which turn out to be hyperelliptic themselves. There are of this particular type. In this article, we provide algebraic equations for the remaining ones.     

Uniform bound for the effective Bogomolov conjecture

We obtain a uniform bound for the effective Bogomolov conjecture, which depends only on the genus g of the curve. The bound grows as $O(g^{?3})$ as g tends to infinity.     

Version explicite du théorème de Beilinson pour la courbe modulaire X1(N)

We state an explicit version of Beilinson's theorem for the modular curve $X_1(N)$. We deduce from it, for any elliptic curve E of prime conductor N, a formula giving $L(E,2)$ in terms of the twisted values $L(E,\chi ,1)$, where χ is a character modulo N. We illustrate this result and its consequences in the case of the elliptic curve $E=X_1(11)$. To cite this article: F. Brunault, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Complete weight enumerators of a class of linear codes with two weights

In this paper, a class of $p$-ary linear codes with two weights is constructed by using the properties of cyclotomic classes of ${\mathbb{F}}_{p^2}^1$. The complete weight enumerators of these linear codes are also determined. In some cases, they are optimal and can be employed to obtain secret sharing schemes with interesting access structures and asymptotically optimal systematic authentication codes.