On the characteristic power series of the U operator

Fernando Q. Gouvêa; Barry Mazur

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On the slopes of the $U_{5}$ operator acting on overconvergent modular forms

L. J. P Kilford (2008)

Journal de Théorie des Nombres de Bordeaux

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We show that the slopes of the $U_{5}$ operator acting on 5-adic overconvergent modular forms of weight $k$ with primitive Dirichlet character $χ$ of conductor 25 are given by either $\{\frac{1}{4} \cdot ⌊\frac{8 i}{5}⌋ : i \in ℕ\} or \{\frac{1}{4} \cdot ⌊\frac{8 i + 4}{5}⌋ : i \in ℕ\},$ depending on $k$ and $χ$ . We also prove that the space of classical cusp forms of weight $k$ and character $χ$ has a basis of eigenforms for the Hecke operators $T_{p}$ and $U_{5}$ which is defined over $Q_{5} (\sqrt[4]{5}, \sqrt{3})$ .