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Let $L / K$ be a solvable (or cyclic) Galois extension of totally real fields, and let $f$ be a Hilbert modular newform over $L$.

Suppose that, for every $\sigma \in Gal(L / K)$, the conjugate newform $f_\sigma$ is twist-equivalent to $f$, i.e. there exists a Hecke character $\chi_\sigma$ of $L$ such that $a_{\sigma(\mathfrak{p})}(f) = \chi_{\sigma}(\mathfrak{p}) a_\mathfrak{p}(f)$ for all but finitely many primes $\mathfrak{p}$ of $L$, where $a_{\mathfrak{p}}(f)$ is the Hecke eigenvalue at $\mathfrak{p}$.

Does it follow that $f$ is twist-equivalent to the base-change to $L$ of a Hilbert modular form over $K$?

I'm happy to assume that $f$ is non-CM, and that all weights of $f$ are $\ge 2$, if that helps.

(Note that Galois is acting on $L$ here, not on the coefficients of $f$ -- this is not the same setup as the Ribet--Momose theory of "inner twists".)

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  • $\begingroup$ So is $f_\sigma$ just obtained from $f$ by permuting the weights? In particular, if $f$ is a parallel weight form, $f_\sigma = f$? $\endgroup$
    – Kimball
    Commented Jun 24, 2016 at 7:11
  • $\begingroup$ No. If $f$ has parallel weight 2 and is associated to a modular elliptic curve $E / L$, then $f^\sigma$ is associated to $E^\sigma / L$ (the curve obtained by acting by $\sigma$ on the coefficients of a Weierstrass equation for $E$), and $E^\sigma$ is not usually isomorphic (or even isogenous) to $E$. $\endgroup$
    – David Loeffler
    Commented Jun 24, 2016 at 8:08

2 Answers 2

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Use Galois Representations. By Schur's Lemma, the projective representation extends to G_K. By Tate's theorem, this projective representation lifts to a genuine representation of G_K. The restriction of this representation to G_L is a twist of the original representation. Hence, after twisting, the HMF is invariant under Gal(L/K). Then use cyclic base change.

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I'm going to add an answer to my own question, not because there's anything wrong with user94346's answer, but because I found a paper in the literature treating exactly this question:

Erez Lapid and Jonathan Rogawski, On twists of cuspidal representations of GL(2). Forum Math. 10 (1998), no. 2, 175–197. DOI: 10.1515/form.10.2.175

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