Let $f$ be a nonclassical finite slope overconvergent $p$ adic eigenform of weight 1, and let $\rho_f$ be the associated $p$-adic Galois representation of $Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$. Is $\rho_f|_{Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)}$ necessarily Hodge-Tate? When $f$ is classical or when the weight is larger than 1 this is always true, but in weight $1$ this can perhaps fail because the Sen operator may not be semisimple.

  • $\begingroup$ If it had Hodge-Tate weights all 0, wouldn’t the image of inertia be finite? So it would be de Rham. $\endgroup$ Jun 11, 2020 at 18:29


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