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New proof shows equivalence of theta and Fontaine operators in overconvergent modular eigenforms, indicating classical status of forms.
Inspired by [Pan26], we give a new proof that for an overconvergent modular eigenform f of weight 1 + k with k ∈ ℤ ≥ 1 , assuming that its associated Galois representation ρ f : Gal ℚ → GL 2 ( ℚ ¯ p ) is irreducible, then f is classical if and only if the associated Galois representation ρ f is de Rham at p . For the proof, we prove that theta operator θ k coincides with Fontaine operator in a suitable sense.
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Yuanyang Jiang (2026) studied this question.
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