Slopes of modular forms and reducible Galois representations: an oversight in the ghost conjecture

2021/10/15 by Bergdall, John, Pollack, Robert
#11F33 (11F85) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2110.07973

Abstract

The ghost conjecture, formulated by this article's authors, predicts the list of p-adic valuations of the non-zero p-th eigenvalues ("slopes") for overconvergent p-adic modular eigenforms in terms of the Newton polygon of an easy-to-describe power series (the "ghost series"). The prediction is restricted to eigenforms whose Galois representation modulo p is reducible on a decomposition group at p. It has been discovered, however, that the conjecture is not formulated correctly. Here we explain the issue and propose a salvage.

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