Companion forms over totally real fields
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- Gee, T. manuscripta math. (2008) 125: 1. doi:10.1007/s00229-007-0128-9
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We show that if F is a totally real field in which p splits completely and f is a mod p Hilbert modular form with parallel weight 2 < k < p, which is ordinary at all primes dividing p and has tamely ramified Galois representation at all primes dividing p, then there is a “companion form” of parallel weight k′ := p + 1 − k. This work generalises results of Gross and Coleman–Voloch for modular forms over Q.