Slopes of modular forms and reducible Galois representations: an oversight in the ghost conjecture
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Pollack, Robert
Bergdall, John
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R. Pollack, J. Bergdall. "Slopes of modular forms and reducible Galois representations: an oversight in the ghost conjecture." Transactions of the American Mathematical Society, https://arxiv.org/abs/2110.07973v2
Abstract
The ghost conjecture, formulated by this article’s authors, predicts
the list of p-adic valuations of the non-zero ap-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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This work is distributed under a Creative Commons Attribution 4.0 license.