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Julio Fernández; Joan-C. Lario
On elliptic Galois representations and genus-zero modular units
Journal de théorie des nombres de Bordeaux, 19 no. 1 (2007), p. 141-164, doi: 10.5802/jtnb.578
Article PDF | Reviews MR 2332058 | Zbl pre05186979

Résumé - 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 $k$.

Bibliography

[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. Cedram |  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 $\mathrm{{B}r}_2(k)$. 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