Éric Gaudron

Textes mathématiques

1. 23 Isogénies et places réelles avec Gaël Rémond — 2024
   It has long been known that isogeny theorems between two elliptic curves over a number field are easier to obtain and yield better numerical constants when the number field admits a real embedding. We explain this phenomenon and extend it to arbitrary abelian varieties. Indeed we have associated in a previous work to any abelian variety $A$ another canonically defined abelian variety which governs isogenies from $A$ and here we show that, in the case of a real embedding, this second abelian variety splits as a product of two natural abelian subvarieties, up to $2$-torsion. This allows us to apply the period theorem to varieties whose dimension is halved compared to the general case and produce constants which are roughly reduced by a square root. We give explicit statements to illustrate this and also include new bounds in the elliptic case.
2. 22 Adelic approximation on spheres Combinatorics and Number Theory, 2024
   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 $\alpha\in E\otimes_{K}K_{v}$ such that $q(\alpha)=1$. 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 $(\upsilon,\phi)\in E\times K$, with $\max{(\Vert\upsilon\Vert_{E,v},\vert\phi\vert_{v})}\le T$ and $\max{(\Vert\upsilon\Vert_{E,w},\vert\phi\vert_{w})}$ controlled for any place $w$, satisfying $q(\upsilon)=\phi^{2}\ne 0$ and $\vert q(\alpha\phi-\upsilon)\vert_{v}\le c\vert\phi\vert_{v}/T$. This remains true for certain 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 Rémond (2017).
3. 21 Nombre de petits points sur une variété abélienne avec Gaël Rémond — Journal de l'Institut de Mathématiques de Jussieu, 2025
   Given a polarised abelian variety over a number field, we provide totally explicit upper bounds for the cardinality of the rational points whose Néron-Tate height is less than a small threshold. These imply new estimates for the number of torsion points as well as the minimal height of a non-torsion point. Our bounds involve the Faltings height and dimension of the abelian variety together with the degrees of the polarisation and the number field but we also get a stronger statement where we use certain successive minima associated to the period lattice at a fixed archimedean place, in the spirit of a result of David for elliptic curves.
4. 20 Nouveaux théorèmes d'isogénies avec Gaël Rémond — Mémoires de la SMF 176, 2023
   Given a finitely generated field extension $K$ of the rational numbers and an abelian variety $C$ over $K$, we consider the class of all abelian varieties over $K$ which are isogenous (over $K$) to an abelian subvariety of a power of $C$. We show that there is a single, naturally constructed abelian variety $C^\flat$ in the class whose ring of endomorphisms controls all isogenies in the class. Precisely, this means that if $d$ is the discriminant of this ring then for any pair of isogenous abelian varieties in the class there exists an isogeny between them whose kernel has exponent at most $d$. Furthermore we prove that, for any element $A$ in the class, the same number $d$ governs several invariants attached to $A$ such as the smallest degree of a polarisation on $A$, the discriminant of its ring of endomorphisms or the size of the invariant part of its geometric Brauer group. All these are bounded only in terms of $d$ and the dimension of $A$. In the case where $K$ is a number field we can go further and show that the period theorem applies to $C^\flat$ in a natural way and gives an explicit bound for $d$ in terms of the degree of $K$, the dimension of $C^\flat$ and the stable Faltings height of $C$. This in turn yields explicit upper bounds for all the previous quantities related to isogenies, polarisations, endomorphisms, Brauer groups which significantly improve known results.
5. 19 Some explicit computations in Arakelov geometry of abelian varieties J. Ramanujan Math. Soc., 2019
   Given a polarized complex abelian variety $(\mathsf{A},\mathsf{L})$, a Gromov lemma makes a comparison between the sup and $\mathrm{L}^{2}$ norms of a global section of $\mathsf{L}$. We give here an explicit bound which depends on the dimension, degree and injectivity diameter of $(\mathsf{A},\mathsf{L})$. It rests on a more general estimate for the jet of a global section of $\mathsf{L}$. As an application we deduce some estimates of the maximal slope of the tangent and cotangent spaces of a polarized abelian variety defined over a number field. These results are effective versions of previous works by Masser and Wüstholz on one hand and Bost on the other. They also improve some similar statements established by Graftieaux in 2000.
6. 18 Minima and slopes of rigid adelic spaces Lecture Notes in Mathematics, 2021
   In this lecture, we present an abstract of the theory of rigid adelic spaces over an algebraic extension of $\mathbb{Q}$, developed in a previous article with G. Rémond (2017). We define the Harder-Narasimhan filtration, the slopes and several types of minima associated to such spaces. This formalism generalizes the Minkowski geometry of numbers for ellipsoids, the twisted height theory by Roy and Thunder as well as the slope theory of Hermitian vector bundles by Bost.
7. 17 bis Lower bound for the Néron-Tate height Oberwolfach report, 2016
8. 17 Logarithmes des points rationnels des variétés abéliennes avec Vincent Bosser — Canadian J. of Math., 2019
   Let $u$ be a logarithm of an algebraic point $p$ of an abelian variety defined over a number field. Let $A_{u}$ be the smallest abelian subvariety whose tangent space at the origin contains $u$. We give a bound for the geometrical degree of $A_{u}$ which generalizes the famous period theorem by Masser and Wüstholz. From this statement, we also deduce some lower bounds for the norm of $u$ and for the Néron-Tate height of $p$. The main new feature of these results is that the bounds are fully explicit in terms of the classical invariants of the problem, among which the dimension and the Faltings height of the abelian variety. The proofs rest on tools from Gel'fond-Baker theory of linear forms in logarithms and from Arakelov slope theory.
9. 16 Torsion des variétés abéliennes CM avec Gaël Rémond — Proc. AMS, 2018
   In this note, we improve on a result of Silverberg giving an upper bound for the order of a rational torsion point on a CM abelian variety over a number field in terms of the degree of the field and the dimension of the variety. The proof uses the main theorem of complex multiplication and class field theory.
10. 15 bis Adelic quadratic spaces Oberwolfach report, 2016
11. 15 Espaces adéliques quadratiques avec Gaël Rémond — Math. Proc. Cambridge Philos. Soc., 2017
    We study quadratic forms defined on an adelic vector space over an algebraic extension $K$ of the rationals. Under the sole condition that a Siegel lemma holds over $K$, we provide height bounds for several objects naturally associated to the quadratic form, such as an isotropic subspace, a basis of isotropic vectors (when it exists) or an orthogonal basis. Our bounds involve the heights of the form and of the ambient space. In several cases, we show that the exponents of these heights are best possible. The results improve and extend previously known statements for number fields and the field of algebraic numbers.
12. 14 Corps de Siegel avec Gaël Rémond — Crelle, 2017
    We define a Siegel field to be a subfield $K$ of the algebraic numbers over which a Siegel lemma holds. Known examples include number fields (classical geometry of numbers) and the field of algebraic numbers itself (Roy-Thunder and Zhang). We investigate the situation for other fields. We provide new examples such as real algebraic numbers or Hilbert class field towers, as well as counterexamples. We also relate the Siegel condition to other properties of $K$, for example the Northcott property introduced by Bombieri-Zannier. For this we define several series of successive minima for an adelic vector space over $K$, which give rise to diverse numerical invariants of $K$ (akin to Hermite's constant). When $K$ is the field of algebraic numbers, we determine their exact values.
13. 13 Polarisations et isogénies avec Gaël Rémond — Duke Math. J., 2014
    Let $A$ be an abelian variety over a number field $k$. We prove that $A$ admits a polarisation of degree explicitly bounded in terms of the Faltings height of $A$, its dimension and the degree of $k$. The fact that this could be done effectively was known only in special cases, thanks to the work of Masser and Wüstholz. We also provide sharpened, explicit versions of their isogeny and factorisation estimates, as well as of a result of Bertrand about almost complements of abelian subvarieties. One crucial tool is our recent period theorem and the proofs proceed through the detailed study of lattices of morphisms of abelian varieties, endowed with euclidean metrics deduced from suitable Rosati involutions.
14. 12 Minorations simultanées de formes linéaires de logarithmes de nombres algébriques Bulletin de la S.M.F., 2014
    This work falls within the theory of linear forms in logarithms over a commutative linear algebraic 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 integrate into the proof the modern tools of adelic slope theory (auxiliary section method), using a new small values Siegel's lemma.
15. 11 Théorème des périodes et degrés minimaux d'isogénies avec Gaël Rémond — Comment. Math. Helv., 2014
    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.
16. 10 Minima, pentes et algèbre tensorielle avec Gaël Rémond — Israel Journal of Math., 2013
    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.
17. 9 Lemmes de Siegel d'évitement avec Gaël Rémond — Acta Arith., 2012
    Siegel's lemma is the generic name of a series of statements yielding a small nonzero element in a lattice or more generally an adelic vector bundle. We study the problem with a condition of avoidment where the small vector is sought outside a given algebraic subset. We review the known results and prove stronger and simpler versions.
18. 8 Habilitation (2009) : Géométrie des nombres adélique et formes linéaires de logarithmes dans un groupe algébrique commutatif
19. 7 Géométrie des nombres adélique et lemmes de Siegel généralisés Manuscripta Math., 2009
    A Siegel's lemma provides an explicit upper bound for a non-zero vector of minimal height in a finite dimensional vector space over a number field. This article explains how to obtain Siegel's lemmas for which the minimal vectors do not belong to a finite union of vector subspaces (Siegel's lemmas with conditions). The proofs mix classical results of adelic geometry of numbers and an adelic variant of a theorem of Henk about the number of lattice points of a centrally symmetric convex body in terms of the successive minima of the body.
20. 6 Pentes des fibrés vectoriels adéliques sur un corps global Rendiconti di Padova, 2008
    At the end of the twentieth century, J.-B. Bost developed 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.
21. 5 Étude du cas rationnel de la théorie des formes linéaires de logarithmes Journal of Number Theory, 2007
    We establish new measures of linear independence of logarithms on commutative algebraic groups in the so-called 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 $\operatorname{Lie}(H)$ its Lie algebra at the origin. Let $u\in\operatorname{Lie}(G(\mathbf{C}_{v_{0}}))$ be 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 $\operatorname{Lie}(H)\otimes_{k}\mathbf{C}_{v_{0}}$. 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 $\log\log a$. The proof relies on sharp estimates of sizes of formal subschemes associated to $H$ (in the sense of Bost) obtained from a lemma by Raynaud as well as an absolute Siegel lemma and, in the ultrametric case, a recent interpolation lemma by Roy.
22. 4 Formes linéaires de logarithmes effectives sur les variétés abéliennes Annales de l'É.N.S., 2006
    We prove new measures of linear independence of logarithms on an abelian variety defined over $\overline{\mathbf{Q}}$, which are totally explicit in function of the invariants of the abelian variety (dimension, Faltings height, degree of a polarization). Besides, except an extra-hypothesis on the algebraic point considered and a weaker numerical constant, we improve on earlier results (in particular David's lower bound). We also introduce into the main theorem an algebraic subgroup that leads to a great variety of different lower bounds. An important feature of the proof is the implementation of the slope method of Bost and some results of Arakelov geometry naturally associated with it.
23. 3 Mesures d'indépendance linéaire de logarithmes dans un groupe algébrique commutatif Invent. math., 2005
    This work falls within the theory of linear forms in logarithms over a connected and commutative algebraic group, defined over the field of algebraic numbers $\overline{\mathbb{Q}}$. Let $G$ be such a group. Let $W$ be a hyperplane of the tangent space at the origin of $G$, defined over $\overline{\mathbb{Q}}$, and $u$ be a complex point of this tangent space, such that the image of $u$ by the exponential map of the Lie group $G(\mathbb{C})$ is an algebraic point. Then we obtain a lower bound for the distance between $u$ and $W\otimes\mathbb{C}$, which improves the results known before and which is, in particular, the best possible for the height of the hyperplane $W$. The proof rests on Baker's method and Hirata's reduction as well as a new arithmetic argument (Chudnovsky's process of variable change) which enables us to give a precise estimate of the ultrametric norms of some algebraic numbers built during the proof.
24. 2 Approximation diophantienne sur les courbes elliptiques à multiplication complexe avec Mohammed Ably — C.R.A.S., 2003
    Let $\mathcal{E}$ be an elliptic curve with complex multiplication, defined over $\overline{\mathbf{Q}}$. We consider linear forms on $\mathrm{Lie}(\mathcal{E}^{n})$ with coefficients in the CM field of $\mathcal{E}$. Within this framework, we present a new measure of linear independence for elliptic logarithms in $(\log b)(\log a)^{n}$. Like recent advances in this domain (works by Ably, David, Hirata-Kohno), our result is best possible in terms of the height of the linear forms ($\log b$) while providing a better estimate in the height of algebraic points considered ($\log a$), removing a term in $\log\log a$.
25. 1 Mesure d'indépendance linéaire de logarithmes dans un groupe algébrique commutatif C.R.A.S., 2001
    We obtain some new results in the theory of linear forms in logarithms on a commutative algebraic group, defined over $\overline{\mathbb{Q}}$. We generalize recent progress of S. David and N. Hirata. In particular, we achieve an optimal linear independence measure in the height of the linear form and a measure more precise than Hirata's for parameters related to the logarithms. The proof rests on Baker's method and a new argument of arithmetical nature.
26. 0 Thèse de doctorat (2001) : Mesure d'indépendance linéaire de logarithmes dans un groupe algébrique commutatif