In the 1980s, Wiles showed the Galois representation over a Hida family to be reducible when restricted to a decomposition group at p. This result is the basis for the study of variation of Selmer groups of modular forms in the family. In joint work with K. S. Kedlaya and L. Xiao, we prove the analogous result over the eigencurve, using a strong finiteness result for Galois cohomology of rigid analytic families of (phi,Gamma)-modules over the Robba ring. Applications to Iwasawa theory are then possible by our previous work. (A similar result has been found recently by R. Liu, using a significant strengthening of Kisin's method of interpolation of crystalline periods.)