Linear Hodge-Newton decomposition and its applications

TL;DR

This paper presents a new proof of a key lemma related to Hodge-Newton decomposition in nonarchimedean settings and extends it to archimedean cases, with applications to modular forms.

Contribution

It offers a novel proof of a lemma on slopes of overconvergent modular forms and introduces an archimedean analogue of the Hodge-Newton decomposition.

Findings

New proof of the lemma using nonarchimedean linear Hodge-Newton decomposition

Archimedean analogue of the Hodge-Newton lemma

Applications to slopes of overconvergent modular forms

Abstract

Firstly, we provide a different proof of an important lemma in Buzzard and Calegari's work on slopes of overconvergent 2-adic modular forms via nonarchimedean linear Hodge-Newton decomposition. The lemma shows that two equivalent matrices with coefficients in the ring of integers in an archimedean field have the same Newton polygon under suitable conditions. Secondly, we give an archimedean analogue of the above lemma.