Where the Slopes Are

Abstract

Fix a prime number p and choose, once and for all, an embedding of the algebraic closure of Q into Qp. Let k and N be integers, and suppose N is not divisible by p. If f is a modular form of weight k, level N, and trivial character which is an eigenform for the p-th Hecke operator Tp, we define the slope of f to be the p-adic valuation of the eigenvalue of Tp. This paper reports on computations that suggest that there is quite a lot of structure to the set of slopes for eigenforms of varying weight k. In particular, we find that the slopes are often smaller than expected, that they are almost always integers, that there is evidence of a connection between fractional slopes and slopes which are "bigger than usual", and that there are some hints of a connection to the theory of theta-cycles.

Cited by in corpus (8)

• Slopes of modular forms and the ghost conjecture
• Slopes of eigencurves over boundary disks
• Slopes of modular forms and the ghost conjecture, II
• Slopes of modular forms and the ghost conjecture (unabridged version)
• Limiting measures of supersingularities
• Bounding slopes of $p$-adic modular forms
• An explicit computation of the Hecke operator and the ghost conjecture
• On Frobenius semisimplicity in Hida families