On the slopes of the $U_5$ operator acting on overconvergent modular forms

Kilford, L. J. P

Journal de théorie des nombres de Bordeaux, Volume 20 (2008) no. 1, pp. 165-182.

We show that the slopes of the $U_5$ operator acting on 5-adic overconvergent modular forms of weight $k$ with primitive Dirichlet character $\chi$ of conductor 25 are given by either

$$\left\{\frac{1}{4}·⌊\frac{8i}{5}⌋:i\in \mathbb{N}\right\}\text{or}\left\{\frac{1}{4}·⌊\frac{8i+4}{5}⌋:i\in \mathbb{N}\right\},$$

depending on $k$ and $\chi$.

We also prove that the space of classical cusp forms of weight $k$ and character $\chi$ has a basis of eigenforms for the Hecke operators $T_p$ and $U_5$ which is defined over ${\mathbf{Q}}_5(\sqrt[4]{5},\sqrt{3})$.

Nous démontrons que les pentes de l’opérateur $U_5$ agissant sur 5-adique formes modulaires surconvergentes de poids $k$ avec caractère de Dirichlet primitif $\chi$ de conducteur 25 sont

$$\left\{\frac{1}{4}·⌊\frac{8i}{5}⌋:i\in \mathbb{N}\right\}\text{ou}\left\{\frac{1}{4}·⌊\frac{8i+4}{5}⌋:i\in \mathbb{N}\right\}.$$

Nous prouvons aussi que l’espace de forms parabolique de poids $k$ et caractère $\chi$ a une base des formes propres pour les opérateurs de Hecke $T_p$ et $U_5$ définie sur ${\mathbf{Q}}_5(\sqrt[4]{5},\sqrt{3})$.

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