The pro-l outer Galois actions associated to modular curves of prime power level
Journal de Théorie des Nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 781-799.

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.

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1049
Classification: 14H30,  11G18
Keywords: modular curve, pro-l outer Galois action
Yuichiro Hoshi 1; Yu Iijima 2

1 Research Institute for Mathematical Sciences Kyoto University Kyoto 606-8502, Japan
2 Department of Mathematics Graduate School of Science Hiroshima University 1-3-1 Kagamiyama Higashi-Hiroshima 739-8526, Japan
@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},
     pages = {781--799},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {3},
     year = {2018},
     doi = {10.5802/jtnb.1049},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1049/}
}
TY  - JOUR
TI  - The pro-$l$ outer Galois actions associated to modular curves of prime power level
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2018
DA  - 2018///
SP  - 781
EP  - 799
VL  - 30
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1049/
UR  - https://doi.org/10.5802/jtnb.1049
DO  - 10.5802/jtnb.1049
LA  - en
ID  - JTNB_2018__30_3_781_0
ER  - 
%0 Journal Article
%T The pro-$l$ outer Galois actions associated to modular curves of prime power level
%J Journal de Théorie des Nombres de Bordeaux
%D 2018
%P 781-799
%V 30
%N 3
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.1049
%R 10.5802/jtnb.1049
%G en
%F 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, Volume 30 (2018) no. 3, pp. 781-799. doi : 10.5802/jtnb.1049. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1049/

[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)

Cited by Sources: