. Sortie, famille F (2n,2n) 2n associéesassociéesà la courbe. {Cet algorithme donné egalement les thêta constantes de niveau n. Par ailleurs, en utilisant les formules de changement de bases

D. Cependant and . Le-premierprobì-eme, nous pouvonségalementpouvonségalement demander de calculer l'image d'un point par l'isogénie. Les variétés initiales et finales peuventêtrepeuventêtre données de différentes façons : 1, Thêta constantes de niveau n

. Dans-ce-chapitre, nous supposons données les thêta constantes d'unepremì ere variété et une description d'un noyau d'une isogénie Nous allons calculer alors les thêta constantes de ladeuxì eme variété. Par ailleurs nous expliquons comment calculer l'image d'un point. Nous nous plaçons en caractéristique différente de 2. Par ailleurs, nous nous intéressons au calcul de d-isogénies séparablesséparablesétant donnés leurs noyaux

. Dans-le-cadre-des-formules-de-thomae, C) ayant ses zéros et pôles fixés En effet, nous serions alors capable d'utiliser la formule du lemme 6.1.2 pour obtenir les puissances 2n-i` emes des thêta constantes de niveau n associéesassociéesà une courbe hyperelliptique de genre g quelconque Dans le cadre du calcul d'isogénies entre courbes hyperelliptiques de genre 2, nous avons expliqué (chapitre 7) comment calculer des (, )-isogénies. L'´ etape suivante est le calcul de (, 1)-isogénies (et plus généralement de (, 1, . . . , 1)-isogénies en dimension g) Pour calculer des -isogénies, nous avons considéré les tores complexes C g /? ? . Ces tores sont naturellement munis d'une forme de Riemann c'est` a-dire d'une polarisation. Les fonctions thêta sontégalementsontégalement intrinsèquement liéesliéesà des polarisations (non principales). Les -isogéniesisogéniesétudiées au chapitre 7 respectent ces polarisations, or ce n'est pas le cas des (, 1)-isogénies. Ceci constitue une obstruction pour trouver des formules permettant le calcul de ces isogénies avec les fonctions thêta, il serait intéressant de savoir construire une fraction rationnelle sur la variété Jac

. De-ce-fait, -isogénies sont utilisables ce qui limite le nombre de courbes hyperelliptiques sur lesquelles cette méthode est applicable On pourrait essayer d'exploiter le calcul de -isogénies entre variétés abéliennes représentées par les fonctions thêta qui a ´ eté présenté au chapitre 7. Par ailleurs, nous avons expliqué comment passer d'une courbe hyperelliptique sous forme de Weierstraß et des coordonnées de Mumford aux fonctions thêta. La brique manquante pour la résolution de ceprobì eme est donc la reconstruction d'une courbe de genre 3 non hyperelliptiquè a partir des thêta constantes associées, Deux probì emes apparaissent naturellement : ? reconstruire une courbe de genre 3 non hyperelliptiquè a partir des thêta constantes associées, ? savoir changer de coordonnées

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