p-adic L-functions of Hilbert cusp forms and the trivial zero conjecture

We prove a strong form of the trivial zero conjecture at the central point for the p-adic L-function of a non-critically refined self-dual cohomological cuspidal automorphic representation of GL2 over a totally real field, which is Iwahori spherical at places above p. In the case of a simple zero we adapt the approach of Greenberg and Stevens, based on the functional equation for the p-adic L-function of a nearly finite slope family and on improved padic L-functions that we construct using automorphic symbols and overconvergent cohomology. For higher order zeros we develop a conceptually new approach studying the variation of the root number in partial families and establishing the vanishing of many Taylor coefficients of the p-adic L-function of the family.