arXiv subjects

Nicolas Ratazzi

Publications and source records attributed to Nicolas Ratazzi.

11 recordsLinked to original sources

arXiv2016

Torsion pour les varietes abeliennes de type I et II

Let A be an abelian variety defined over a number field K, the number of torsion points rational over a finite extension L is bounded polynomially in terms of the degree [L : K]. When A is isogenous to a product of simple abelian varieties of type I or II in Albert classification and is "fully of Lefschetz type", i.e. whose Mumford-Tate group is the group of symplectic similitudes commuting with endomorphisms and which satisfy the Mumford-Tate conjecture, we compute the optimal exponent for this bound in terms of the dimensions of the abelian subvarieties of A and their rings of endomorphisms. The result is unconditional for a product of simple abelian varieties of type I or II with odd relative dimension. Extending work of Serre, Pink and Hall, we also prove that the Mumford-Tate conjecture is true for a few new cases for such abelian varieties.

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arXiv2014

Classe d'isogénie de variétés abéliennes pleinement de type GSp

Faltings in 1983 proved that a necessary and sufficient condition for two abelian varieties $A$ and $B$ to be isogenous over a number field $K$ is that the local factors of the L-series of $A$ and $B$ are equal for almost all primes of $K$ ; for each such prime this implies that $A$ and $B$ have the same number of points over the residue field. We show in this article that for abelian varieties faithfully of type GSp (a class containing the abelian varieties with endomorphism ring $\mathbb{Z}$ and of odd dimension) `having the same number of points' may be replaced by `the number of points have the same prime divisors' and still gives a sufficient condition for $A$ and $B$ to be $K$-isogenous. The proof is based on ideas of Serre \cite{serreim72} and Frey-Jarden \cite{FJ} and follows closely Hall-Perucca \cite{hallp} who proved the result for elliptic curves.

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arXiv2010

Points de torsion sur les varietes abeliennes de type GSp

Let $A$ be an abelian variety defined over a number field $K$, the number of torsion points rational over a finite extension $L$ is bounded polynomially in terms of the degree $[L:K]$. When $A$ is isogenous to a product of simple abelian varieties of $\GSp$ type, i.e. whose Mumford-Tate group is "generic" (isomorphic to the group of symplectic similitudes) and which satisfy the Mumford-Tate conjecture, we compute the optimal exponent for this bound in terms of the dimensions of the abelian subvarieties of $A$. The result is unconditional for a product of simple abelian varieties with endomorphism ring $\Z$ and dimension outside an explicit exceptional set $\mathcal{S}=\{4,10,16,32,...\}$. Furthermore, following a strategy of Serre, we also prove that if the Mumford-Tate conjecture is true for some abelian varieties of $\GSp$ type, it is then true for a product of such abelian varieties.

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arXiv2008

Torsion dans un produit de courbes elliptiques

Let $A$ be an abelian variety defined over a number field $K$, the number of torsion points rational over a finite extension $L$ is bounded polynomially in terms of the degree $[L:K]$. We formulate a question suggesting the optimal exponent for this bound in terms of the dimension of the Mumford-Tate groups of the abelian subvarieties of $A$; we study the behaviour under product and then give a positive answer to our question when $A$ is the product of elliptic curves.

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arXiv2008

Intersection de courbes et de sous-groupes, et problèmes de minoration de hauteur dans les variétés abéliennes C.M

We prove a special case of the following conjecture of Zilber-Pink generalising the Manin-Mumford conjecture : let $X$ be a curve inside an Abelian variety $A$ over $\bar{\Q}$, provided $X$ is not contained in a torsion subvariety, the intersection of $X$ with the union of all subgroup schemes of codimension at least 2 is finite ; we settle the case where $A$ is a power of a simple Abelian variety of C.M. type. This generalises the previous known result, due to Viada and Rémond-Viada (who was able to prove the conjecture for power of an elliptic curve with complex multiplication). The proof is based on the strategy of Rémond (following Bombieri, Masser and Zannier) with two new ingredients, one of them, being at the heart of this article : it is a lower bound for the Néron-Tate height of points on Abelian varieties $A/K$ of C.M. type in the spirit of Lehmer's problem. This lower bound is an analog of the similar result of Amoroso and David \cite{ad2003} on $\G_m^n$ and is a generalisation of the theorem of David and Hindry \cite{davidhindry} on the abelian Lehmer's problem. The proof is an adaptation of \cite{davidhindry} using in our abelian case the new ideas introduced in \cite{ad2003}. Furthermore, as in \cite{ad2003} and adapting in the abelian case their proof, we give another application of our result : a lower bound for the absolute minimum of a subvariety $V$ of $A$. Although lower bounds for this minimum were already known (decreasing multi-exponential function of the degree for Bombieri-Zannier), our methods enable us to prove, up to an $ε$ the optimal result that can be conjectured.

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arXiv2007

Borne sur la torsion dans les variétés abéliennes de type C.M

Let A be an abelian variety of dimension g defined over a number field K. We study the size of the torsion group A(F)_{tors} where F/K is a finite extension and more precisely we study the possible exponent γin the inequality Card(A(F)_{tors})<< [F:K]^γ when F is any extension of K. In the C.M. case we give an exact formula for the best possible exponent in terms of the characters of the Mumford-Tate group--a torus in this case--and discuss briefly the general case. Finally we give applications of this result in direction of a conjecture of Rémond generalising the Manin-Mumford conjecture.

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arXiv2007

Lemmes de multiplicites et constante de Seshadri

We establish an improvement of Philippon's zero estimates primarily in the multiplicity setting. The improvement is made possible by a more geometric approach and in particular the use of Seshadri constants.

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arXiv2004

Theoreme de Dobrowolski-Laurent pour les extensions abeliennes sur une courbe elliptique a multiplication complexe

Let E/K be an elliptic curve with complex multiplication and let $K^{ab}$ be the Abelian closure of $K$. We prove in this article that there exists a constant $c(E/K)$ such that : for all point $P\in E(\bar{K})-E_{tors}$, we have \[\hat{h}(P)\geq\frac{c(E/K)}{D}(\frac{\log \log 5D}{\log 2D})^{13},\] where $D=[K^{ab}(P):K^{ab}]$. This result extends to the case of elliptic curve s with complex multiplication the previous resultof Amoroso-Zannier \cite{AZ} on the analogous problem on the multiplicative group $\mathbb{G}_m$, and generalizes to the case of extensions of degree D the result of Baker \cite{baker} on the lower bound of the Néron-Tate height of the points defined over an Abelian extension of an elliptic curve with complex multiplication. This result also enables us to simplify the proof of a theorem of Viada \cite{viada}.

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arXiv2004

Deux remarques sur le probleme de Lehmer sur les varietes abeliennes

Let $A/K$ be an abelian variety over a number field $K$. We prove in this article that a good lower bound (in terms of the degree $[K(P):K]$) for the Néron-Tate height of the points $P$ of infinite order modulo every strict abelian subvarieties of $A$ implies a good lower bound for the height of all the non-torsion points of $A$. In particular when $A$ is of C.M. type, a theorem of David and Hindry enables us to deduce, up to ``log'' factors, an optimal lower bound for the height of the non-torsion points of $A$. In the C.M. type case, this improves the previous result of Masser \cite{lettre}. Using the same theorem of David and Hindry we prove in the second part an optimal lower bound, up to ``log'' factors, for the product of the Néron-Tate height of $n$ End$(A)$-linearly independant non-torsion points of a C.M. type abelian variety.

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arXiv2004

Densité de points et minoration de hauteur

We obtain a lower bound for the normalised height of a non-torsion subvariety $V$ of a C.M. abelian variety. This lower bound is optimal in terms of the geometric degree of $V$, up to a power of a ``log''. We thus extend the results of F. Amoroso and S. David on the same problem on a multiplicative group $\mathbb{G}_m^n$. We prove furthermore that the optimal lower bound (conjectured by S. David and P. Philippon) is a corollary of the conjecture of S. David and M. Hindry on the abelian Lehmer's problem. We deduce these results from a density theorem on the non-torsion points of $V$.

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arXiv2003

Problème de Lehmer pour les hypersurfaces de variétés abéliennes de type C.M

We obtain a lower bound for the normalised height of a non-torsion hypersurface $V$ of a C.M. abelian variety $A$ which is a refinement of a precedent result. This lower bound is optimal in terms of the geometric degree of $V$, up to an absolute power of a ``log'' (independant of the dimension of $A$). We thus extend the results of F. Amoroso and S. David on the same problem on a multiplicative group $\mathbb{G}_m^n$. When $A$ is an elliptic curve and $V=\bar{P}$ is the set of conjugates of a non torsion $\bar{k}$-point, we reobtain the result of M. Laurent on the elliptic Lehmer's problem.

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