Slopes of overconvergent 2-adic modular forms

2003/11/20 by Kevin Buzzard, Buzzard, Kevin, Frank Calegari +1
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/0311364

17 pages

arxiv created 2003/11/20 · openalex publication_date 2003/11/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The slope of a p-adic overconvergent eigenform of weight k is the p-adic valuation of its U_p eigenvalue. We find the slope of all 2-adic finite slope overconvergent eigenforms of tame level 1 and weight 0. As a consequence we prove that any finite slope 2-adic overconvergent eigenform of tame level 1 and weight 0 has coefficients in Q₂. These results provide evidence towards conjectures of the first author that predict the slopes of classical p-adic modular forms under a mild ``Gamma₀(N)''-regular hypothesis on p.

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