L'arithmétique des variétés algébri

L'arithmétique des variétés algébriques est un sujet très riche et vieux en théorie des nombres et la géométrie algébrique arithmétique. La théorie moderne est en fait a une approche très fantastique, dire l'idée de la Grothendieck sur la théorie du régime. Sur cette base, nombreuses conjectures énormes a été abordé dans le dernier demi-siècle, comme Conjecture de Weil, Conjecture de Mordell, la Conjecture de Shimura - Taniyama, le dernier théorème de Fermat et les deux ont fait énormément de résultats significatifs dans le programme de Langlands. En revanche, il y a encore beaucoup de problèmes qui est encore loin d'atteindre, même si dans la plus faible dimension, comme le BSD conjecture et nombreuses conjectures liées à la représentation de Galois géométrique et ainsi de suite.
Nous nous concentrons principalement sur le cas lorsque le champ de base est de caractéristiques positives et l'autre dimension, dans ce cas il y a une très subtile correspondance entre les deux colonnes géantes dites, le champ fonction algébrique et les courbes algébriques sur un corps fini, et de façon presque équivalente, la classification de l'extension algébrique du genre important de champ global est identique à la classification des courbes algébriques sur les corps finis , c'est-à-dire, dans de nombreuses situations que nous pouvons faire en fait la transformation directionnel deux entre ces deux colonnes.
Donc notre travail vise à la les régimes qui sont de type fini sur le corps fini k dont les composantes irréductibles ont la dimension 1, nous prendrons en considération les phénomènes qui ne se produira lorsque la caractéristique est positive, c'est, nous avons les concepts de la p-rang de l'associé à la jacobienne des courbes algébriques dans ce cas. 
Une bonne question est de savoir comment faire la classification des courbes en ce qui concerne les p-rangs des variétés jacobienne associées, en particulier comment obtenir le nombre des courbes avec le spécifique p-rang et le champ de masse spécifique, comment le sous-ensemble de l'espace de modules des courbes avec structure de rang-p ressemble, c'est-à-dire ce qui est sa géométrie et comment le décrire.
Les deux cas extrêmes lorsque le p-rang est 0 et g, le genre de la courbe, que nous appelons l'affaire super-singulier et l'ordinaire.
Pour hyperelliptiques cas, Jeffrey D. Achter, Rachel Pries donner quelques résultats importants sur les modules des courbes hyperelliptiques avec le p-grade donné et le genre. Beaucoup d'autres auteurs ont également considéré ce problème comme Faber & Van der Geer, Darren Glass, etc. YUI donne quelques résultats sur la structure de p-rang des courbes hyperelliptics.

結果 (英文) 1:

The arithmetic of algebraic varieties is a rich subject and old in number theory and arithmetic algebraic geometry. Modern theory is in fact has a very fantastic approach, say the idea of the Grothendieck theory of the regime. On this basis, many enormous conjecture has been addressed in the last half century as Conjecture de Weil Conjecture of Mordell, the Conjecture of Shimura - Taniyama, Fermat's last theorem and the two made a lot of significant results in the Langlands program. On the other hand, there are still many problems that is still far from reach, even if in the lower dimension, as the BSD conjecture and numerous conjectures related to the geometric Galois representation and so on.We focus mainly on the case when the base field is of positive characteristics and the other dimension, in this case there is a very subtle correspondence between the two so-called giant columns, the algebraic function field and algebraic over a finite field, and almost equivalently curves, the classification of the important global field genus algebraic extension is identical to the classification of algebraic over finite curves , i.e., in many situations we can actually make two directional transformation between these two columns.Therefore our work is aimed to the regimes that are finished over the finite field k whose irreducible components have dimension 1, we will take into account the phenomena that occur when the feature is positive, is, we have the concepts of the p-rank of the Jacobian of algebraic curves in this case associated. A good question is how to make the classification of curves with regard to the p-rank of associated Jacobian varieties, in particular how obtain the number of curves with the specific rank p and the specific mass field, how the subset of the moduli space of curves with rank-p structure looks like, i.e. What is its geometry and how to describe it.Two extreme cases when the p-rank is 0 and g, the genus of the curve, that we call the supersingular case and the ordinary.For hyperelliptic case, Jeffrey D. Achter, Rachel Pries give some important results on hyperelliptic curves with the given p-rank modules and the kind. Many other authors have also considered this problem as Faber & Van der Geer, Darren Glass, etc. YUI gives some results on the structure of p-rank of the hyperelliptics curves.

結果 (英文) 2:

The arithmetic of algebraic varieties is a subject very rich old number theory and arithmetic algebraic geometry. The modern theory is actually a very fantastic approach, saying the idea of Grothendieck theory of the regime. On this basis, many huge speculation has been addressed in the last half-century, as Conjecture Weil Conjecture Mordell, Conjecture of Shimura - Taniyama on Fermat's Last Theorem and both have done a lot of significant results in the program Langlands. However, there are still many problems that are still far from achieving, even if in smaller scale, such as the BSD conjecture and many conjectures related to the representation of the Galois geometry and so on.
We mainly focus on the case when the base field is of positive characteristics, and the other dimension, in this case there is a very subtle correspondence between the two giant columns say, the algebraic function field and algebraic curves over a finite field, and so almost equal, the classification of algebraic extension of the large kind of global field is identical to the classification of algebraic curves over finite fields, that is to say, in many situations we can do actually directional transformation Both of these two columns.
So our work is to diets that are finitely generated over the finite k whose irreducible components have dimension 1, we will consider the phenomena that occur when the characteristic is positive, c east, we have the concepts of p-rank associated with the Jacobian of algebraic curves in this case. 
A good question is how to make the classification of curves regarding the p-ranks of Jacobian varieties associated , in particular how to obtain the number of curves with specific p-rank and the density field, how the subset of the modules of space curves with rank p-like structure, that is to say what is its geometry and how to describe it.
The two extreme cases when the p-rank is 0 and g, the kind of the curve, we call super-singular case and the ordinary.
For hyperelliptic case, Jeffrey D . Achter, Rachel Pries give some important results on the modules of hyperelliptic curves with the given p-grade and gender. Many other authors have also considered this problem as Faber & Van der Geer, Darren Glass, etc. YUI gives some results on the p-rank structure hyperelliptics curves.

結果 (英文) 3:

the arithmetic of algebraic varieties is very rich and old number theory and algebraic geometry arithmétique. la modern theory is in fact a very fantastic, the idea of the theory of grothendieck régime. sur this conjecture, many huge has been addressed in the past half century, as the conjecture of weil. mordell conjecture.the conjecture of shimura, taniyama, fermat's last theorem, and the two have done a lot of significant results in the langlands. en program. however, there are still a lot of problems, which is far from reach, even if in the lower dimension, such as the bsd conjecture and speculation related to the representation of the galois. and so on and so forth.we focus primarily on the case when the base field is of positive features and the other dimension, in this case, there is a very subtle correspondence between the two giant columns, the algebraic curves, algebraic function in a body, and the end, and so is almost the same.the classification of algebraic extension of important type of scope is identical to the classification of algebraic curves in the body finis , that is to say, in many situations, we can do the conversion between the two directional two columns.our work is in the plans that are of type over on the body over k whose irreducible components have dimension 1, we shall take account of the events that will occur when the characteristic is positive, that is, we have the concepts of the p status of the associated with the jacobienne algebraic curves in this cas. .a good question is how to do the classification of curves with respect to p rows associated jacobienne varieties, in particular how to obtain the number of lines with specific p - rank and the mass field, specifically, is the subset of the space of moduli of curves with level structure p looks like.that is to say, it is how to describe the geometry. the two extreme cases when p is 0 and g level, the genus of the curve, which we call the super - singular and regular. for hyperelliptiques, jeffrey d. keep rachel pray give some important results. the modules of the hyperelliptiques p curves with rank and gender. beaucoup other authors have also considered this issue to be expressed & van der geer, darren glass, etc. yui gives some results on the structure of p - hyperelliptics level curves.