The modified ghost conjecture for 2-adic slopes

Let NN be a positive integer, let p=2p=2 be a Γ0(N)\Gamma_0(N)-regular prime, and let GκG^\circ_\kappa be the modified ghost series incorporating the fractional repeated slopes from the spaces Sk(Γ0(N)Γ1(8),η8±)S_k(\Gamma_0(N)\cap\Gamma_1(8),\eta_8^{\pm}). Let PκP_\kappa be the corresponding Fredholm series, and write NP()\operatorname{NP}(-) for the Newton polygon.

Modified ghost conjecture. For each κW\kappa\in\mathcal W,

NP(Gκ)=NP(Pκ).\operatorname{NP}(G^\circ_\kappa)=\operatorname{NP}(P_\kappa).

The modification is designed to account for non-integral repeated slopes that the original ghost series does not detect at p=2p=2 and higher tame level. The claim is presented as a proposed salvage of the ghost conjecture and is not resolved in the source.

Sources & referencesView supporting material

Primary source

John Bergdall and Robert Pollack, “Slopes of modular forms and the ghost conjecture (unabridged version)”, arXiv:1607.04658 (2016).

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