On elliptic Galois representations and genus-zero modular units

Fernández, Julio; Lario, Joan-C.

Journal de théorie des nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 141-164.

Given an odd prime  $p$  and a representation $\varrho$  of the absolute Galois group of a number field $k$ onto ${\mathrm{PGL}}_2\left({𝔽}_p\right)$ 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 $X\left(p\right)$. We make here explicit the only genus-zero cases $p=3$ and $p=5$, which are also the only symmetric cases: ${\mathrm{PGL}}_2\left({𝔽}_p\right)\simeq {𝒮}_n$ 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 $k$.

Etant donnés un nombre premier $p$ impair et une représentation $\varrho$ du groupe de Galois absolu d’un corps de nombres $k$ sur ${\mathrm{PGL}}_2\left({𝔽}_p\right)$ avec déterminant cyclotomique, l’espace des modules des courbes elliptiques définies sur $k$ et dont la $p$-torsion donne lieu à $\varrho$  est composé de deux tordues galoisiennes de la courbe modulaire $X\left(p\right)$. On explicite ici les seuls cas de genre zéro, $p=3$ et $p=5$, qui sont aussi les seuls cas symétriques : ${\mathrm{PGL}}_2\left({𝔽}_p\right)\simeq {𝒮}_n$ 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 $\varrho$ et sa pincipalité, c’est-à-dire l’existence dans son corps fixe d’un élément $\alpha$ de degré $n$ sur $k$ tel que $\alpha$ and ${\alpha }^2$ ont tous les deux trace zéro sur $k$.

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