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).
