Soit $f$ un revêtement de la droite projective défini sur $\overline{\mathbb{Q}}$, de groupe de monodromie $G$. Soit $K$ le compositum des corps de rationalité des points de branchement $f$, et $M$ le corps des modules correspondants. Partant du lien entre corps des modules et espaces de Hurwitz, on étudie la géométrie et l’arithmétique de ces espaces et des espaces de configuration de points complétés pour évaluer la ramification dans $M/K$ des mauvaises places de $f$ qui ne divisent pas l’ordre de $G$, mais où les points de branchements...

Given the category $X$ of coherent sheaves over a weighted projective line $X=X(\lambda ,p)$ (of any representation type), the endomorphism ring $\Sigma =\left(𝒯\right)$ of an arbitrary tilting sheaf - which is by definition an almost concealed canonical algebra - is shown to satisfy a rank additivity property (Theorem 3.2). Moreover, this property extends to the representationinfinite quasi-tilted algebras of canonical type (Theorem 4.2). Finally, it is demonstrated that rank additivity does not generalize to the case of tilting complexes...

We construct a family of elliptic curves with six parameters, arising from a system of Diophantine equations, whose rank is at least five. To do so, we use the Brahmagupta formula for the area of cyclic quadrilaterals (p³,q³,r³,s³) not necessarily representing genuine geometric objects. It turns out that, as parameters of the curves, the integers p,q,r,s along with the extra integers u,v satisfy u⁶+v⁶+p⁶+q⁶ = 2(r⁶+s⁶), uv = pq, which, by previous work, has infinitely many integer solutions.

We report on a large-scale project to investigate the ranks of elliptic curves in a quadratic twist family, focussing on the congruent number curve. Our methods to exclude candidate curves include 2-Selmer, 4-Selmer, and 8-Selmer tests, the use of the Guinand-Weil explicit formula, and even 3-descent in a couple of cases. We find that rank 6 quadratic twists are reasonably common (though still quite difficult to find), while rank 7 twists seem much more rare. We also describe our inability to find...

A point $({\xi }_1,{\xi }_2)$ with coordinates in a subfield of $ℝ$ of transcendence degree one over $\mathbb{Q}$, with $1,{\xi }_1,{\xi }_2$ linearly independent over $\mathbb{Q}$, may have a uniform exponent of approximation by elements of ${\mathbb{Q}}^2$ that is strictly larger than the lower bound $1/2$ given by Dirichlet’s box principle. This appeared as a surprise, in connection to work of Davenport and Schmidt, for points of the parabola $\{(\xi ,{\xi }^2);\xi \in ℝ\}$. The goal of this paper is to show that this phenomenon extends to all real conics defined over $\mathbb{Q}$, and that the largest exponent of...

In this paper we consider rational Bézier curves with control points having rational coordinates and rational weights, and we give necessary and sufficient conditions for such a curve to have infinitely many points with integer coefficients. Furthermore, we give algorithms for the construction of these curves and the computation of theirs points with integer coefficients.

If $V_{d,\delta }$ denotes the variety of irreducible plane curves of degree $d$ with exactly $\delta$ nodes as singularities, Diaz and Harris (1986) have conjectured that $\mathrm{Pic}\left(V_{d,\delta }\right)$ is a torsion group. In this note we study rational equivalence on some families of singular plane curves and we prove, in particular, that $\mathrm{Pic}\left(V_{d,1}\right)$ is a finite group, so that the conjecture holds for $\delta =1$. Actually the order of $\mathrm{Pic}\left(V_{d,1}\right)$ is $6(d-2)d^2-3d+1)$, the group being cyclic if $d$ is odd and the product of ${\mathbb{Z}}_2$ and a cyclic group of order $3(d-2)(d^2-3d+1)$ if $d$ is even.