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Documents  11G35 | enregistrements trouvés : 4

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Rational points on smooth projective curves of genus $g \ge 2$ over number fields are in finite number thanks to a theorem of Faltings from 1983. The same result was known over function fields of positive characteristic since 1966 thanks to a theorem of Samuel. The aim of the talk is to give a bound as uniform as possible on this number for curves defined over such fields. In a first part we will report on a result by Rémond concerning the number field case and on a way to strengthen it assuming a height conjecture. During the second part we will focus on function fields of positive characteristic and describe a new result obtained in a joined work with Pacheco. Rational points on smooth projective curves of genus $g \ge 2$ over number fields are in finite number thanks to a theorem of Faltings from 1983. The same result was known over function fields of positive characteristic since 1966 thanks to a theorem of Samuel. The aim of the talk is to give a bound as uniform as possible on this number for curves defined over such fields. In a first part we will report on a result by Rémond concerning the ...

14G05 ; 11G35

Multi angle  Differential descent obstructions
Voloch, José Felipe (Auteur de la Conférence) | CIRM (Editeur )

We will discuss a new obstruction to the existence of rational and integral points on algebraic varieties over function fields obtained by considering covers described by differential equations.

11G35 ; 14G17

Multi angle  Exponential motives
Fresán, Javier (Auteur de la Conférence) | CIRM (Editeur )

I will sketch the construction - following ideas of Kontsevich and Nori - of a Tannakian category of exponential motives over a subfield of the complex numbers. It is a universal cohomology theory for pairs of varieties and regular functions, whose de Rham and Betti realizations are given by twisted de Rham and rapid decay cohomology respectively. The upshot is that one can attach to any such pair a motivic Galois group which conjecturally generalizes the Mumford-Tate group of a Hodge structure and, over number fields, governs all algebraic relations between exponential periods. This is a joint work with Peter Jossen (ETH). I will sketch the construction - following ideas of Kontsevich and Nori - of a Tannakian category of exponential motives over a subfield of the complex numbers. It is a universal cohomology theory for pairs of varieties and regular functions, whose de Rham and Betti realizations are given by twisted de Rham and rapid decay cohomology respectively. The upshot is that one can attach to any such pair a motivic Galois group which conjecturally ...

11R58 ; 14G25 ; 11F80 ; 14C15 ; 11E72 ; 14D07 ; 11G35

Given two algebraic curves $X$, $Y$ over a finite field we might want to know if there is a rational map from $Y$ to $X$. This has been looked at from a number of perspectives and we will look at it from the point of view of diophantine geometry by viewing the set of maps as $X(K)$ where $K$ is the function field of $Y$. We will review some of the known obstructions to the existence of rational points on curves over global fields, apply them to this situation and present some results and conjectures that arise. Given two algebraic curves $X$, $Y$ over a finite field we might want to know if there is a rational map from $Y$ to $X$. This has been looked at from a number of perspectives and we will look at it from the point of view of diophantine geometry by viewing the set of maps as $X(K)$ where $K$ is the function field of $Y$. We will review some of the known obstructions to the existence of rational points on curves over global fields, apply them to ...

11G20 ; 11G35 ; 14G05

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