The pro-l outer Galois actions associated to modular curves of prime power level
Hoshi, Yuichiro1 ; Iijima, Yu2
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
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.

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DOI : 10.5802/jtnb.1049
Classification : 14H30, 11G18
Mots-clés : modular curve, pro-l outer Galois action
Hoshi, Yuichiro 1 ; Iijima, Yu 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 
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Hoshi, Yuichiro; Iijima, Yu. 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://www.numdam.org/articles/10.5802/jtnb.1049/

[1] Anderson, Greg; Ihara, Yasutaka Pro-l branched coverings of P1 and higher circular l-units, Ann. Math., Volume 128 (1988), pp. 271-293 | DOI | MR | Zbl

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

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

[4] Hoshi, Yuichiro 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 | MR | Zbl

[5] Hoshi, Yuichiro 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 | MR | Zbl

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

[7] Ihara, Yasutaka Some arithmetic aspects of Galois actions in the pro-p fundamental group of ℙ1-{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 | DOI | MR | Zbl

[8] Katz, Nicholas M. 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 | MR | Zbl

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

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

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

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

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

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

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

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