Slopes of 2-adic overconvergent modular forms with small level

Let \(\tau\) be the primitive Dirichlet character of conductor 4, let \(\chi\) be the primitive even Dirichlet character of conductor 8 and let \(k\) be an integer. Then the \(U_2\) operator acting on cuspidal overconvergent modular forms of weight \(2k+1\) and character \(\tau\) has slopes in the a...

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Bibliographic Details
Published in:arXiv.org 2003-02
Main Author: Kilford, L J P
Format: Article
Language:English
Subjects:
Analytic functions
Arithmetic
Conductors
Dirichlet problem
Mathematical analysis
Polynomials
Slopes
Weight
Online Access:Get full text
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Summary:Let \(\tau\) be the primitive Dirichlet character of conductor 4, let \(\chi\) be the primitive even Dirichlet character of conductor 8 and let \(k\) be an integer. Then the \(U_2\) operator acting on cuspidal overconvergent modular forms of weight \(2k+1\) and character \(\tau\) has slopes in the arithmetic progression \({2,4,...,2n,...}\), and the \(U_2\) operator acting on cuspidal overconvergent modular forms of weight \(k\) and character \(\chi \cdot \tau^k\) has slopes in the arithmetic progression \({1,2,...,n,...}\). We then show that the characteristic polynomials of the Hecke operators \(U_2\) and \(T_p\) acting on the space of classical cusp forms of weight \(k\) and character either \(\tau\) or \(\chi\cdot\tau^k\) split completely over \(\qtwo\).
ISSN:2331-8422