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).
We prove a companion-forms theorem for mod (l) Hilbert modular forms. This work generalises results of Gross [Gr] and Coleman and Voloch [CV] for modular forms over (Q) and gives a new proof of their results in many cases. The methods used are completely different to previous work in this area and rely on modularity lifting theorems and the general theory of deformations of Galois representations.
