On elliptic Galois representations and genus-zero modular units
Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 141-164.

Given an odd prime  p  and a representation ϱ  of the absolute Galois group of a number field k onto PGL 2 (𝔽 p ) with cyclotomic determinant, the moduli space of elliptic curves defined over k with p-torsion giving rise to ϱ consists of two twists of the modular curve X(p). We make here explicit the only genus-zero cases p=3 and p=5, which are also the only symmetric cases: PGL 2 (𝔽 p )𝒮 n for n=4 or n=5, respectively. This is done by studying the corresponding twisted Galois actions on the function field of the curve, for which a description in terms of modular units is given. As a consequence of this twisting process, we recover an equivalence between the ellipticity of ϱ and its principality, that is, the existence in its fixed field of an element α of degree n over k  such that α and α 2 have both trace zero over k.

Etant donnés un nombre premier p impair et une représentation ϱ du groupe de Galois absolu d’un corps de nombres k sur PGL 2 (𝔽 p ) avec déterminant cyclotomique, l’espace des modules des courbes elliptiques définies sur k et dont la p-torsion donne lieu à ϱ  est composé de deux tordues galoisiennes de la courbe modulaire X(p). On explicite ici les seuls cas de genre zéro, p=3 et p=5, qui sont aussi les seuls cas symétriques : PGL 2 (𝔽 p )𝒮 n pour n=4 ou n=5, respectivement. Dans ce but, on étudie les actions galoisiennes correspondantes aux deux tordues sur le corps de fonctions de la courbe, duquel on donne une description au moyen d’unités modulaires. Comme conséquence, on retrouve une équivalence entre l’ellipticité de ϱ et sa pincipalité, c’est-à-dire l’existence dans son corps fixe d’un élément α de degré n sur k tel que α and α 2 ont tous les deux trace zéro sur k.

Received:
Published online:
DOI: 10.5802/jtnb.578
Julio Fernández 1; Joan-C. Lario 2

1 Departament de Matemàtica Aplicada 4 Universitat Politècnica de Catalunya EPSEVG, av. Víctor Balaguer E-08800 Vilanova i la Geltrú
2 Departament de Matemàtica Aplicada 2 Universitat Politècnica de Catalunya Edifici Omega, Campus Nord E-08034 Barcelona
@article{JTNB_2007__19_1_141_0,
     author = {Julio Fern\'andez and Joan-C. Lario},
     title = {On elliptic {Galois} representations and genus-zero modular units},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {141--164},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {1},
     year = {2007},
     doi = {10.5802/jtnb.578},
     zbl = {pre05186979},
     mrnumber = {2332058},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.578/}
}
TY  - JOUR
TI  - On elliptic Galois representations and genus-zero modular units
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2007
DA  - 2007///
SP  - 141
EP  - 164
VL  - 19
IS  - 1
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.578/
UR  - https://zbmath.org/?q=an%3Apre05186979
UR  - https://www.ams.org/mathscinet-getitem?mr=2332058
UR  - https://doi.org/10.5802/jtnb.578
DO  - 10.5802/jtnb.578
LA  - en
ID  - JTNB_2007__19_1_141_0
ER  - 
%0 Journal Article
%T On elliptic Galois representations and genus-zero modular units
%J Journal de Théorie des Nombres de Bordeaux
%D 2007
%P 141-164
%V 19
%N 1
%I Université Bordeaux 1
%U https://doi.org/10.5802/jtnb.578
%R 10.5802/jtnb.578
%G en
%F JTNB_2007__19_1_141_0
Julio Fernández; Joan-C. Lario. On elliptic Galois representations and genus-zero modular units. Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 141-164. doi : 10.5802/jtnb.578. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.578/

[1] R.W. Carter, Simple groups of Lie type. Pure and Applied Mathematics 28. John Wiley & Sons, London-New York-Sydney, 1972. | MR: 407163 | Zbl: 0248.20015

[2] J. Fernández, Elliptic realization of Galois representations. PhD thesis, Universitat Politècnica de Catalunya, 2003.

[3] J. Fernández, J-C. Lario, A. Rio, On twists of the modular curves X(p). Bull. London Math. Soc. 37 (2005), 342–350. | MR: 2131387 | Zbl: 1084.11025

[4] J. González, Equations of hyperelliptic modular curves. Ann. Inst. Fourier (Grenoble) 41 (1991), 779–795. | Numdam | MR: 1150566 | Zbl: 0758.14010

[5] D. S. Kubert, S. Lang, Modular units. Grundlehren der Mathematischen Wissenschaften 244. Springer-Verlag, New York, 1981. | MR: 648603 | Zbl: 0492.12002

[6] J-C. Lario, A. Rio, An octahedral-elliptic type equality in Br 2 (k). C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), 39–44. | MR: 1340079 | Zbl: 0837.11061

[7] G. Ligozat, Courbes modulaires de niveau 11. Modular functions of one variable V, 149–237. Lecture Notes in Math. 601, Springer, Berlin, 1977. | MR: 463118 | Zbl: 0357.14006

[8] B. Mazur, Rational points on modular curves. Modular functions of one variable V, 107–148. Lecture Notes in Math. 601, Springer, Berlin, 1977. | MR: 450283 | Zbl: 0357.14005

[9] B. Mazur, Open problems regarding rational points on curves and varieties. Galois representations in arithmetic algebraic geometry (Durham, 1996), 239–265. London Math. Soc. Lecture Note Ser. 254. Cambridge Univ. Press, 1998. | MR: 1696485 | Zbl: 0943.14009

[10] D. E. Rohrlich, Modular curves, Hecke correspondence, and L-functions. Modular forms and Fermat’s last theorem (Boston, 1995), 41–100. Springer, New York, 1997. | Zbl: 0897.11019

[11] K. Y. Shih, On the construction of Galois extensions of function fields and number fields. Math. Ann. 207 (1994), 99–120. | MR: 332725 | Zbl: 0279.12102

[12] G. Shimura, Introduction to the arithmetic theory of automorphic functions. Publications of the Mathematical Society of Japan 11. Iwanami Shoten Publishers, Tokyo, 1971. | MR: 314766 | Zbl: 0221.10029

Cited by Sources: