Bounding slopes of p-adic modular forms

2007/05/24 by Lawren Smithline, Smithline, Lawren
Mathematics · #11G18 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #math.AG #msc:11G18

paper · pdf · doi:10.48550/arxiv.0705.3614

15 pages. June 2001 preprint

arxiv created 2007/05/24 · openalex publication_date 2007/05/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p be prime, N be a positive integer prime to p, and k be an integer. Let P_k(t) be the characteristic series for Atkin's U operator as an endomorphism of p-adic overconvergent modular forms of tame level N and weight k. Motivated by conjectures of Gouvea and Mazur, we strengthen Wan's congruence between coefficients of P_k and P_k' for k' close p-adically to k. For p-1 | 12, N = 1, k = 0, we compute a matrix for U whose entries are coefficients in the power series of a rational function of two variables. We apply this computation to show for p = 3 a parabola below the Newton polygon N₀ of P₀, which coincides with N₀ infinitely often. As a consequence, we find a polygonal curve above N₀. This tightest bound on N₀ yields the strongest congruences between coefficients of P₀ and P_k for k of large 3-adic valuation.

Citations

• Where the Slopes Are

Related

• On the U_p operator acting on p-adic overconvergent modular forms when X₀(p) has genus 1
• Slopes of overconvergent 2-adic modular forms
• Traces on Hecke algebras and families of p-adic modular forms
• A note on cusp forms as p-adic limits
• Slopes of 2-adic overconvergent modular forms with small level