arXiv subjects

Éric Gaudron

Publications and source records attributed to Éric Gaudron.

6 recordsLinked to original sources

arXiv2024

Adelic approximation on spheres

We establish an adelic version of Dirichlet's approximation theorem on spheres. Let $K$ be a number field, $E$ be a rigid adelic space over $K$ and $q\colon E\to K$ be a quadratic form. Let $v$ be a place of $K$ and $α\in E\otimes_{K}K_{v}$ such that $q(α)=1$. We produce an explicit constant $c$ having the following property. If there exists $x\in E$ such that $q(x)=1$ then, for any $T>c$, there exists $(\upupsilon,\upphi)\in E\times K$, with $\max{(\Vert\upupsilon\Vert_{E,v},\vert\upphi\vert_{v})}\le T$ and $\max{(\Vert\upupsilon\Vert_{E,w},\vert\upphi\vert_{w})}$ controlled for any place $w$, satisfying $q(\upupsilon)=\upphi^{2}\ne 0$ and $\vert q(α\upphi-\upupsilon)\vert_{v}\le c\vert\upphi\vert_{v}/T$. This remains true for some infinite algebraic extensions as well as for a compact set of places of $K$. Our statements generalize and improve on earlier results by Kleinbock \\& Merrill (2015) and Moshchevitin (2017). The proofs rely on the quadratic Siegel's lemma in a rigid adelic space obtained by the author and R{é}mond (2017).

math.NT↗

arXiv2012

Minorations simultanées de formes linéaires de logarithmes de nombres algébriques

This work falls within the theory of linear forms in logarithms over a commutative linear group defined over a number field. We give lower bounds for simultaneous linear forms in logarithms of algebraic numbers, treating both the archimedean and $p$-adic cases. The proof includes Baker's method, Hirata's reduction, Chudnovsky's process of variable change. The novelty is that we integrated into the proof the modern tools of adelic slope theory, using also a new small values Siegel's lemma.

math.NT↗

arXiv2011

Minima, pentes et algèbre tensorielle

Slopes of an adelic vector bundle exhibit a behaviour akin to successive minima. Comparisons between the two amount to a Siegel lemma. Here we use Zhang's version for absolute minima over the algebraic numbers. We prove a Minkowski-Hlawka theorem in this context. We also study the tensor product of two hermitian bundles bounding both its absolute minimum and maximal slope, thus improving an estimate of Chen. We further include similar inequalities for exterior and symmetric powers, in terms of some lcm of multinomial coefficients.

math.NT↗

arXiv2011

Théorème des périodes et degrés minimaux d'isogénies

We give a new, sharpened version of the period theorem of Masser and Wüstholz, which is moreover totally explicit. We also present a new formulation involving all archimedean places. We then derive new bounds for elliptic isogenies, improving those of Pellarin. The small numerical constants obtained allow an application to Serre's uniformity problem in the split Cartan case, thanks to the work of Bilu, Parent and Rebolledo.

math.NT↗

arXiv2008

Pentes des fibrés vectoriels adéliques sur un corps global

At the end of the twentieth century, J.-B. Bost developped a slope theory of hermitian vector bundles over number fields. A new method of diophantine approximation, the so-called slope method, has emerged from his research. Our article proposes a generalisation to adelic vector bundles over global fields. The norms at the archimedean places are no longer supposed to be hermitian. The link with adelic successive minima is also mentioned. To get these results, we use several concepts from the geometry of finite dimensional Banach spaces.

math.NT↗

arXiv2008

Étude du cas rationnel de la théorie des formes linéaires de logarithmes. (French) [Study of the rational case of the theory of linear forms in logarithms]

We establish new measures of linear independence of logarithms on commutative algebraic groups in the so-called \emph{rational case}. More precisely, let k be a number field and v_{0} be an arbitrary place of k. Let G be a commutative algebraic group defined over k and H be a connected algebraic subgroup of G. Denote by Lie(H) its Lie algebra at the origin. Let u\in Lie(G(C_{v_{0}})) a logarithm of a point p\in G(k). Assuming (essentially) that p is not a torsion point modulo proper connected algebraic subgroups of G, we obtain lower bounds for the distance from u to Lie(H)\otimes_{k} C_{v_{0}}. For the most part, they generalize the measures already known when G is a linear group. The main feature of these results is to provide a better dependence in the height Log a of p, removing a polynomial term in LogLog a. The proof relies on sharp estimates of sizes of formal subschemes associated to H (in the sense of J.-B. Bost) obtained from a lemma by M. Raynaud as well as an absolute Siegel lemma and, in the ultrametric case, a recent interpolation lemma by D. Roy.

math.NT↗