Kevin Buzzard

A few years ago Gouvea and Mazur made some conjectures about the eigenvalues of Hecke operators acting on spaces of modular forms. These conjectures have "essentially" been proven by Coleman. One can generalise these conjectures to the case of Hilbert modular forms (definitions will be given!) and then ask if they are true. Coleman's methods might work but pushing them through in this case will probably be rather technical. I will avoid this problem completely by presenting a low-level argument which again "essentially" proves the conjectures in many cases.