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

Reçu le : 2017-02-28
Révisé le : 2017-10-30
Accepté le : 2017-11-28
Publié le : 2019-03-28
DOI : https://doi.org/10.5802/jtnb.1049
Classification : 14H30,  11G18
Mots clés: modular curve, pro-l outer Galois action
@article{JTNB_2018__30_3_781_0,
     author = {Yuichiro Hoshi and Yu Iijima},
     title = {The pro-$l$ outer Galois actions associated to modular curves of prime power level},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {3},
     year = {2018},
     pages = {781-799},
     doi = {10.5802/jtnb.1049},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_2018__30_3_781_0/}
}
Yuichiro Hoshi; Yu Iijima. The pro-$l$ outer Galois actions associated to modular curves of prime power level. Journal de Théorie des Nombres de Bordeaux, Tome 30 (2018) no. 3, pp. 781-799. doi : 10.5802/jtnb.1049. https://jtnb.centre-mersenne.org/item/JTNB_2018__30_3_781_0/

[1] Greg Anderson; Yasutaka Ihara Pro-l branched coverings of P 1 and higher circular l-units, Ann. Math., Volume 128 (1988), pp. 271-293 | Zbl 0692.14018

[2] Michael P. Anderson Exactness properties of profinite completion functors, Topology, Volume 13 (1974), pp. 229-239 | Zbl 0324.20041

[3] Fred Diamond; Jerry Shurman A first course in modular forms, Graduate Texts in Mathematics, Volume 228, Springer, 2005, xv+436 pages | Zbl 1062.11022

[4] Yuichiro Hoshi On monodromically full points of configuration spaces of hyperbolic curves, The Arithmetic of Fundamental Groups - PIA 2010 (Contributions in Mathematical and Computational Sciences) Volume 2, Springer, 2010, pp. 167-207 | Zbl 1317.14065

[5] Yuichiro Hoshi On the kernels of the pro-l outer Galois representations associated to hyperbolic curves over number fields, Osaka J. Math., Volume 52 (2015) no. 3, pp. 647-675 | Zbl 06502589

[6] Yuichiro Hoshi; Yu Iijima A pro-l version of the congruence subgroup problem for mapping class groups of genus one, J. Algebra, Volume 520 (2019), pp. 1-31 | Zbl 06993577

[7] Yasutaka Ihara Some arithmetic aspects of Galois actions in the pro-p fundamental group of 1 -{0,1,}, Arithmetic fundamental groups and noncommutative algebra (Berkeley, CA, 1999) (Proceedings of Symposia in Pure Mathematics) Volume 70, American Mathematical Society, 2002, pp. 247-273 | Zbl 1065.14025

[8] Nicholas M. Katz p-adic properties of modular schemes and modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) (Lecture Notes in Mathematics) Volume 350, Springer, 1973, pp. 69-190 | Zbl 0271.10033

[9] Nicholas M. Katz; Barry Mazur Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, Volume 108, Princeton University Press, 1985, xiv+514 pages | Zbl 0576.14026

[10] Barry Mazur Modular curves and the Eisenstein ideal, Publ. Math., Inst. Hautes Étud. Sci., Volume 47 (1977), pp. 33-186 | Zbl 0394.14008

[11] Shinichi Mochizuki; Akio Tamagawa The algebraic and anabelian geometry of configuration spaces, Hokkaido Math. J., Volume 37 (2008) no. 1, pp. 75-131 | Zbl 1143.14306

[12] Matthew Papanikolas; Christopher Rasmussen On the torsion of Jacobians of principal modular curves of level 3 n , Arch. Math., Volume 88 (2007) no. 1, pp. 19-28 | Zbl 1125.11034

[13] Despina T. Prapavessi On the Jacobian of the Klein curve, Proc. Am. Math. Soc., Volume 122 (1994) no. 4, pp. 971-978 | Zbl 0823.14016

[14] Christopher Rasmussen; Akio Tamagawa A finiteness conjecture on abelian varieties with constrained prime power torsion, Math. Res. Lett., Volume 15 (2008) no. 6, pp. 1223-1231 | Zbl 1182.11027

[15] William Stein The Modular Forms Database (http://wstein.org/Tables)