Related papers: A conjecture of Yves Andre

We describe a model-theoretic setting for the study of Shimura varieties, and study the interaction between model theory and arithmetic geometry in this setting. In particular, we show that the model-theoretic statement of a certain…

We present a proof of a conjecture proposed by T. Yano about the generic $b$-exponents of irreducible plane curve singularities.

We prove conjectures of Breuil and Breuil-Dembele (C. Breuil, "Sur un probleme de compatibilite local-global modulo p pour GL(2)"), including a generalisation from the principal series to the cuspidal case, subject to a mild global…

Let s be a special point on a Shimura variety, and x a pre-image of s in a fixed fundamental set of the associated Hermitian symmetric domain. We prove that the height of x is polynomially bounded with respect to the discriminant of the…

We prove the $S=T$ conjecture proposed by Xiao--Zhu in \cite{2017arXiv170705700X}, making use of Scholze's theory of diamonds and v-stacks and Fargues--Scholze's geometric Satake equivalence. Following \cite{2018arXiv180205299X}, we deduce…

Let $X$ be a smooth projective curve of genus $\geq2$ over a number field. A natural variant of Grothendieck's Section Conjecture postulates that every section of the fundamental exact sequence for $X$ which everywhere locally comes from a…

In their study of the Yamabe problem in the presence of isometry group, Hebey and Vaugon announced a conjecture. This conjecture generalizes Aubin's conjecture, which has already been proven and is sufficient to solve the Yamabe problem. In…

We give an inductive proof that the generalized Severi varieties -- the varieties which parametrize (irreducible) plane curves of given degree and genus, with a fixed tangency profile to a given line at several general fixed points and…

We study the mod $p$-points of the Kisin--Pappas integral models of Shimura varieties of Hodge type with parahoric level. We show that if the group is quasi-split, then every isogeny class contains the reduction of a CM point, proving a…

Eisenbud and Harris conjectured in 1982 that an algebraic curve of high genus lies on a surface of low degree (which they proved for curves of very large degree). They observed constraints on the Hilbert function of a general hyperplane…

We prove a conjecture of Nakajima (for type A the result was announced by Ginzburg- Vasserot) giving a geometric realization, via quiver varieties, of the Yangian of type ADE (and more in general of the Yangian associated to every symmetric…

In this paper we prove the equidistribution of bounded sequences of special subvarieties in a general mixed Shimura varieties, a notion adapted from the pure case treated by Clozel, Ullmo, and Yafaev in the study of the Andre-Oort…

This is the text of my lecture (in french) at the Bourbaki Seminar (november 2003) on the proof by Claire Voisin of the Green conjecture for a generic curve. This conjecture predicts the structure of the minimal resolution of the ideal of a…

We prove a version of the Kawamata-Morrison ample cone conjecture for projective irreducible holomorphic symplectic manifolds deformation equivalent to either the Hilbert scheme of n points on a K3 surface, or a generalized Kummer variety.

It is conjectured that there exist finitely many isomorphism classes of simple endomorphism algebras of abelian varieties of GL_2-type over \Q of bounded dimension. We explore this conjecture when particularized to quaternion endomorphism…

The Geometric Shafarevich Conjecture and the Theorem of de Franchis state the finiteness of the number of certain holomorphic objects on closed or punctured Riemann surfaces. The analog of these kind of theorems for Riemann surfaces of…

We obtain examples of smooth projective varieties over $\mathbb{C}$ that violate the integral Hodge conjecture and for which the total Chow group is of finite rank. Moreover, we show that there exist such examples defined over number…

A Generalized Hyperelliptic Variety (GHV) is the quotient of an abelian variety by a free action of a finite group which does not contain any translation. These varieties are natural generalizations of bi-elliptic surfaces. In this paper we…

We propose a model-theoretic structure for Shimura varieties and give necessary and sufficient conditions to obtain categoricity. We show that these conditions are directly related to important conjectures in number theory coming from…

Let $S$ be a Shimura variety with reflex field $E$. We prove that the action of $\operatorname{Gal}(\overline{\mathbb{Q}}/E)$ on $S$ maps special points to special points and special subvarieties to special subvarieties. Furthermore, the…