On elliptic Galois representations and genus-zero modular units

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

Journal de Théorie des Nombres de Bordeaux (2007)

  • Volume: 19, Issue: 1, page 141-164
  • ISSN: 1246-7405

Abstract

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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 .

How to cite

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Fernández, Julio, and Lario, Joan-C.. "On elliptic Galois representations and genus-zero modular units." Journal de Théorie des Nombres de Bordeaux 19.1 (2007): 141-164. <http://eudml.org/doc/249968>.

@article{Fernández2007,
abstract = {Given an odd prime  $p$  and a representation $\varrho $  of the absolute Galois group of a number field $k$ onto $\mathrm\{PGL\}_2(\mathbb\{F\}_p)$ with cyclotomic determinant, the moduli space of elliptic curves defined over $k$ with $p$-torsion giving rise to $\varrho $ 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: $\mathrm\{PGL\}_2(\mathbb\{F\}_p)\simeq \mathcal\{S\}_n$ 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 $\varrho $ and its principality, that is, the existence in its fixed field of an element $\alpha $ of degree $n$ over $k$  such that $\alpha $ and $\alpha ^2$ have both trace zero over $k$.},
affiliation = {Departament de Matemàtica Aplicada 4 Universitat Politècnica de Catalunya EPSEVG, av. Víctor Balaguer E-08800 Vilanova i la Geltrú; Departament de Matemàtica Aplicada 2 Universitat Politècnica de Catalunya Edifici Omega, Campus Nord E-08034 Barcelona},
author = {Fernández, Julio, Lario, Joan-C.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {elliptic Galois representations; genus-zero modular units},
language = {eng},
number = {1},
pages = {141-164},
publisher = {Université Bordeaux 1},
title = {On elliptic Galois representations and genus-zero modular units},
url = {http://eudml.org/doc/249968},
volume = {19},
year = {2007},
}

TY - JOUR
AU - Fernández, Julio
AU - Lario, Joan-C.
TI - On elliptic Galois representations and genus-zero modular units
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2007
PB - Université Bordeaux 1
VL - 19
IS - 1
SP - 141
EP - 164
AB - Given an odd prime  $p$  and a representation $\varrho $  of the absolute Galois group of a number field $k$ onto $\mathrm{PGL}_2(\mathbb{F}_p)$ with cyclotomic determinant, the moduli space of elliptic curves defined over $k$ with $p$-torsion giving rise to $\varrho $ 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: $\mathrm{PGL}_2(\mathbb{F}_p)\simeq \mathcal{S}_n$ 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 $\varrho $ and its principality, that is, the existence in its fixed field of an element $\alpha $ of degree $n$ over $k$  such that $\alpha $ and $\alpha ^2$ have both trace zero over $k$.
LA - eng
KW - elliptic Galois representations; genus-zero modular units
UR - http://eudml.org/doc/249968
ER -

References

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  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. Zbl1084.11025MR2131387
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  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. Zbl0837.11061MR1340079
  7. G. Ligozat, Courbes modulaires de niveau 11 . Modular functions of one variable V, 149–237. Lecture Notes in Math. 601, Springer, Berlin, 1977. Zbl0357.14006MR463118
  8. B. Mazur, Rational points on modular curves. Modular functions of one variable V, 107–148. Lecture Notes in Math. 601, Springer, Berlin, 1977. Zbl0357.14005MR450283
  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. Zbl0943.14009MR1696485
  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. Zbl0897.11019
  11. K. Y. Shih, On the construction of Galois extensions of function fields and number fields. Math. Ann. 207 (1994), 99–120. Zbl0279.12102MR332725
  12. G. Shimura, Introduction to the arithmetic theory of automorphic functions. Publications of the Mathematical Society of Japan 11. Iwanami Shoten Publishers, Tokyo, 1971. Zbl0221.10029MR314766

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