The pro-$l$ outer Galois actions associated to modular curves of prime power level

Hoshi, Yuichiro; Iijima, Yu

Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 3, pp. 781-799.

Soit $l$ un nombre premier. Dans cet article, nous étudions la pro-$l$ action galoisienne extérieure associée à une courbe modulaire de niveau une puissance de $l$. En particulier, nous discutons de la question de savoir si cette action se factorise à travers d’un pro-$l$ quotient du groupe de Galois absolu d’un certain corps de nombres. Comme application, nous établissons aussi une relation entre les variétés Jacobiennes de courbes modulaires de niveau puissance d’un nombre premier et l’ensemble défini par Rasmussen et Tamagawa.

Let $l$ be a prime number. In the present paper, we study the pro-$l$ outer Galois action associated to a modular curve of level a power of $l$. In particular, we discuss the issue of whether or not the pro-$l$ outer Galois action factors through a pro-$l$ quotient of the absolute Galois group of a certain number field. Moreover, as an application, we also obtain a result concerning the relationship between the Jacobian varieties of modular curves of prime power level and a set defined by Rasmussen and Tamagawa.

Classification : 14H30, 11G18
Mots-clés : modular curve, pro-$l$ outer Galois action

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