Slopes of 2-adic overconvergent modular forms with small level

2003/02/13 by L. J. P. Kilford, L J P Kilford, Kilford, L J P
Mathematics · #11F11 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11F11

paper · pdf · doi:10.48550/arxiv.math/0302153

arxiv created 2003/02/13 · openalex publication_date 2003/02/13 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let τ be the primitive Dirichlet character of conductor 4, let χ be the primitive even Dirichlet character of conductor 8 and let k be an integer. Then the U₂ operator acting on cuspidal overconvergent modular forms of weight 2k+1 and character τ has slopes in the arithmetic progression 2,4,...,2n,..., and the U₂ operator acting on cuspidal overconvergent modular forms of weight k and character χ⋅ τ^k has slopes in the arithmetic progression 1,2,...,n,.... We then show that the characteristic polynomials of the Hecke operators U₂ and T_p acting on the space of classical cusp forms of weight k and character either τ or χ⋅τ^k split completely over \qtwo.

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