SLOPES OF MODULAR FORMS AND REDUCIBLE GALOIS REPRESENTATIONS, AN OVERSIGHT IN THE GHOST CONJECTURE

John Bergdall; Robert Pollack

doi:10.1090/bproc/136

SLOPES OF MODULAR FORMS AND REDUCIBLE GALOIS REPRESENTATIONS, AN OVERSIGHT IN THE GHOST CONJECTURE

John Bergdall
, Robert Pollack

Research output: Contribution to journal › Article › peer-review

Abstract

The ghost conjecture, formulated by this article’s authors, predicts the list of p-adic valuations of the non-zero a_p-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.

Original language	English (US)
Pages (from-to)	432-444
Number of pages	13
Journal	Proceedings of the American Mathematical Society, Series B
Volume	9
DOIs	https://doi.org/10.1090/bproc/136
State	Published - 2022
Externally published	Yes

ASJC Scopus subject areas

Analysis
Algebra and Number Theory
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.1090/bproc/136

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title = "SLOPES OF MODULAR FORMS AND REDUCIBLE GALOIS REPRESENTATIONS, AN OVERSIGHT IN THE GHOST CONJECTURE",

abstract = "The ghost conjecture, formulated by this article{\textquoteright}s authors, predicts the list of p-adic valuations of the non-zero ap-eigenvalues ({\textquotedblleft}slopes{\textquotedblright}) for overconvergent p-adic modular eigenforms in terms of the Newton polygon of an easy-to-describe power series (the {\textquotedblleft}ghost series{\textquotedblright}). 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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SLOPES OF MODULAR FORMS AND REDUCIBLE GALOIS REPRESENTATIONS, AN OVERSIGHT IN THE GHOST CONJECTURE

Abstract

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