  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 $\mathrm{GL}_2$ 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 $p$-adic $L$-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.

Number Theory