Weights in Serre's conjecture for Hilbert modular forms: The ramified case

Michael M. Schein

doi:10.1007/s11856-008-1035-9

Weights in Serre's conjecture for Hilbert modular forms: The ramified case

Michael M. Schein

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

29 Scopus citations

Abstract

Let F be a totally real field and p ≥ 3 a prime. If ρ : Gal(F̄/F) → GL₂ (|double strok F sign̄_p) 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 ℚ(√5 ) to provide evidence in support of the conjecture.

Original language	English
Pages (from-to)	369-391
Number of pages	23
Journal	Israel Journal of Mathematics
Volume	166
DOIs	https://doi.org/10.1007/s11856-008-1035-9
State	Published - Aug 2008
Externally published	Yes

Bibliographical note

Funding Information:
∗ The author thanks the NSF for a Graduate Research Fellowship that supported him during part of this work. Received October 16, 2006 and in revised form February 14, 2007

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10.1007/s11856-008-1035-9

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title = "Weights in Serre's conjecture for Hilbert modular forms: The ramified case",

abstract = "Let F be a totally real field and p ≥ 3 a prime. If ρ : Gal({\=F}/F) → GL2 (|double strok F sig{\=n}p) 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 Demb{\'e}l{\'e}'s computations of Hilbert modular forms over ℚ(√5 ) to provide evidence in support of the conjecture.",

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AB - Let F be a totally real field and p ≥ 3 a prime. If ρ : Gal(F̄/F) → GL2 (|double strok F sign̄p) 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 ℚ(√5 ) to provide evidence in support of the conjecture.

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Weights in Serre's conjecture for Hilbert modular forms: The ramified case

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