Abstract
Let F be a totally real field and p ≥ 3 a prime. If ρ :
is continuous, semisimple, totally odd, and tamely ramified at all places of F dividing p, then we formulate a conjecture specifying the weights for which ρ is modular. This extends the conjecture of Diamond, Buzzard, and Jarvis, which required p to be unramified in F. We also prove a theorem that verifies one half of the conjecture in many cases and use Dembélé’s computations of Hilbert modular forms over \(\mathbb{Q}(\sqrt 5 )\) to provide evidence in support of the conjecture.
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The author thanks the NSF for a Graduate Research Fellowship that supported him during part of this work.
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Schein, M.M. Weights in Serre’s conjecture for Hilbert modular forms: The ramified case. Isr. J. Math. 166, 369–391 (2008). https://doi.org/10.1007/s11856-008-1035-9
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DOI: https://doi.org/10.1007/s11856-008-1035-9