Ramification of the Hilbert Eigenvariety

Abstract

Andreatta–Iovita–Pilloni constructed eigenvarieties for cuspidal Hilbert modular forms. The eigenvariety has a natural map to the weight space, called the weight map. We compute the dimension of the tangent space of the fiber of the weight map using Galois deformation theory. Along with the classicality theorem due to Tian–Xiao, this enables us to characterize the classical points of the eigenvariety which are ramified over the weight space, in terms of the local splitting behavior of the associated Galois representation. In particular, a modular form with complex multiplication of critical slope gives rise to a ramification point on the eigenvariety. This is equivalent to saying that there is an over- convergent generalized Hecke eigenform with the same Hecke eigenvalues as the CM form, which is not a scalar multiple of it. In the case of elliptic modular forms, we give a description of the Fourier coefficients of the generalized Hecke eigenform in terms of certain Galois cohomology classes.