Gee, Toby2009-05-132008Gee, Toby. 2008. Companion forms over totally real fields. Manuscripta Mathematica 125(1): 1-41.0025-2611http://nrs.harvard.edu/urn-3:HUL.InstRepos:2943908We 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 \(Q\).en-USMazurs principleShimura curvesGalois representationsHilbert modular-formsJournal Article10.1007/s00229-007-0128-9