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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 $k$.

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