A determination of the points which are rational over Q of three modular curves

It is proven that the cusps are the only points which are rational over Q on X₀(N) for N = 53, 113 and 137.     

Fonctions rationnelles aux différences à valeurs entières dans les vecteurs de Witt

We transpose Kochen's results on integral valued rational functions over the p-adic numbers to difference rational functions over the Witt vectors with their Frobenius. To cite this article: L. Bélair, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Simple components of group algebras Q[G] central over real quadratic fields

Let k be a real quadratic field, and $\text{U}$ a central division quaternion algebra over k. In this paper sufficient conditions are given to insure that $\text{U}$ appears in a simple component of the group algebra Q[G] of some finite group G over the rational field Q. In particular, when k is assumed to be Q(√2) or Q(√5), the necessary and sufficient conditions for $\text{U}$ to appear in some Q[G] are given.     

Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry

Following the approach of Gromov and Witten, we construct invariants under deformation of real rational symplectic 4-manifolds. These invariants provide lower bounds for the number of real rational J-holomorphic curves in a given homology class passing through a given real configuration of points. To cite this article: J.-Y. Welschinger, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Rational frames of minimal twist along space curves under specified boundary conditions

An adapted orthonormal frame (f₁(ξ),f₂(ξ),f₃(ξ)) on a space curve r(ξ), ξ ∈ [ 0, 1 ] comprises the curve tangent \(\mathbf {f}_{1}(\xi ) =\mathbf {r}^{\prime }(\xi )/|\mathbf {r}^{\prime }(\xi )|\) and two unit vectors f₂(ξ),f₃(ξ) that span the normal plane. The variation of this frame is specified by its angular velocity Ω = Ω₁f₁ + Ω₂f₂ + Ω₃f₃, and the twist of the framed curve is the integral of the component Ω₁ with respect to arc length. A minimal twist frame (MTF) has the least possible twist value, subject to prescribed initial and final orientations f₂(0),f₃(0) and f₂(1),f₃(1) of the normal–plane vectors. Employing the Euler–Rodrigues frame (ERF) — a rational adapted frame defined on spatial Pythagorean–hodograph curves — as an intermediary, an exact expression for an MTF with Ω₁ = constant is derived. However, since this involves rather complicated transcendental terms, a construction of rational MTFs is proposed by the imposition of a rational rotation on the ERF normal–plane vectors. For spatial PH quintics, it is shown that rational MTFs compatible with the boundary conditions can be constructed, with only modest deviations of Ω₁ about the mean value, by a rational quartic normal–plane rotation of the ERF. If necessary, subdivision methods can be invoked to ensure that the rational MTF is free of inflections, or to more accurately approximate a constant Ω₁. The procedure is summarized by an algorithm outline, and illustrated by a representative selection of computed examples.     

On the spiral-likeness of rational functions

The object of this paper is to extend some results concerning the univalence, starlikeness, and convexity of rational functions recently obtained by Reade, Silverman, and Todorov. The domain of variability of log{f(z)/z} for a fixedz and for such functionsf ranging over the class of λ-spirallike functions of order α are also determined.     

Approximation simultanée d'un nombre et de son carré

In 1969, H. Davenport and W.M. Schmidt established a measure of the simultaneous approximation for a real number ξ and its square by rational numbers with the same denominator, assuming only that ξ is not rational nor quadratic over $\text{Q}$. Here, we show by an example, that this measure is optimal. We also indicate several properties of the numbers for which this measure is optimal, in particular with respect to approximation by algebraic integers of degree at most three. To cite this article: D. Roy, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Special values of L-functions of elliptic curves over Q and their base change to real quadratic fields

For an elliptic curve E over Q, and a real quadratic extension F of Q, satisfying suitable hypotheses, we study the algebraic part of certain twisted L-values for E/F. The Birch and Swinnerton-Dyer conjecture predicts that these L-values are squares of rational numbers. We show that this question is related to the ratio of Petersson inner products of a quaternionic form on a definite quaternion algebra over Q and its base change to F.     

Automorphisms of real rational surfaces and weighted blow-up singularities

Let X be a singular real rational surface obtained from a smooth real rational surface by performing weighted blow-ups. Denote by Aut(X) the group of algebraic automorphisms of X into itself. Let n be a natural integer and let e = [e ₁, . . . , e _? ] be a partition of n. Denote by X ^e the set of ?-tuples (P ₁, . . . , P _? ) of disjoint nonsingular curvilinear subschemes of X of orders (e ₁, . . . , e _? ). We show that the group Aut(X) acts transitively on X ^e . This statement generalizes earlier work where the case of the trivial partition e = [1, . . . , 1] was treated under the supplementary condition that X is nonsingular. As an application we classify singular real rational surfaces obtained from nonsingular surfaces by performing weighted blow-ups.     

On the Łojasiewicz numbers

Let f be a holomorphic function of two complex variables with an isolated critical point at $\text{0∈}\text{C}{}^2$. We give some necessary conditions for a rational number to be the smallest θ>0 in the ?ojasiewicz inequality |gradf(z)|?C|z|^θ for z near $\text{0∈}\text{C}{}^2$. To cite this article: E. Garc??a Barroso, A. P?oski, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Parametrization of Abelian K-surfaces with quaternionic multiplication

We prove that the Abelian K-surfaces whose endomorphism algebra is a rational quaternion algebra are parametrized, up to isogeny, by the K-rational points of the quotient of certain Shimura curves by the group of their Atkin–Lehner involutions. To cite this article: X. Guitart, S. Molina, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

On the contact boundaries of normal surface singularities

The abstract boundary M of a normal complex-analytic surface singularity is canonically equipped with a contact structure. We show that if M is a rational homology sphere, then this contact structure is uniquely determined by the topological type of M. An essential tool is the notion of open book carrying a contact structure, defined by E. Giroux. To cite this article: C. Caubel, P. Popescu-Pampu, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

The iwasawa invariant μ in the composite of two Zl-extensions

Let k₀ be a finite extension field of the rational numbers, and assume k₀ has at least two Z_l-extensions. Assume that at least one Z_l-extension $\text{K}\text{k}{}_0$ has Iwasawa invariant μ = 0, and let L be the composite of K and some other Z_l-extension of k₀. In this paper we find an upper bound for the number of Z_l-extensions of k₀ contained in L with nonzero μ.     

Une surface d'Enriques sans point sur C((t))

We give an explicit example of an Enriques surface without a rational point over $\text{C}\text{((t))}$. To cite this article: G. Lafon, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Triviality of Xsplit(N)(Q) for certain congruence classes of N

We give a criterion to check if, given a prime number N, the only rational points of the modular curve X_split(N) are trivial (i.e., cusps or points furnished by complex multiplication). We then prove that this criterion is verified for large enough N satisfying some explicit congruences. To cite this article: P. Parent, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Réalisation ℓ-adique des motifs mixtes

We provide an integral ?-adic realization functor for the geometrical mixed motives of Voevodsky over a noetherian separated scheme and derive in some case a moderate realization functor. We prove that our realization functor is the same up to isomorphism as the one constructed by A. Huber for rational mixed motives over a ground field of characteristic zero. To cite this article: F. Ivorra, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Chow groups of surfaces with h2,0⩽1

We will investigate the geometry of rational equivalence classes of points on a surface S. We will show that if S is a general projective K3 surface then these equivalence classes are dense in the complex topology. We will also show that if S has the property that these equivalence classes are Zariski dense, then h^2,0(S)?1. To cite this article: C. Maclean, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Rigidity for equivariant K-theory

We extend the classical rigidity results for K-theory to the equivariant setting of linear algebraic group actions. These results concern rigidity for rational points, field extensions, and Hensel local rings. To cite this article: S. Yagunov, P.A. Østvær, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

On a finite rational criterion for the irreducibility of a matrix

A matrix A ∈ M _n (C) is said to be irreducible if the only orthoprojectors that commute with A are the zero and unit matrices. A finite rational criterion for irreducibility is proposed. The criteria for verification of this property that can be found in the literature are neither finite nor rational.