p-adic adjoint L-functions for Hilbert modular forms

Let $F$ be a totally real field. Let $\pi$ be a cuspidal
cohomological automorphic representation for $\mathrm{GL}_2/F$. Let
$L(s, \mathrm{Ad}^0, \pi)$ denote the adjoint $L$-function associated
to $\pi$. The special values of this $L$-function and its relation to
congruence primes have been studied by Hida, Ghate and Dimitrov. Let
$p$ be an integer prime.  In this talk, I will discuss the
construction of a $p$-adic  adjoint $L$-function in neighbourhoods of
very decent points of the Hilbert eigenvariety.  As a consequence, we
relate the ramification locus of this eigenvariety to the zero set of
the $p$-adic $L$-functions. This was first established by Kim when
$F=\mathbb{Q}$. We follow Bellaiche's description of Kim's method,
generalizing it to arbitrary totally real number fields. This is joint
work with John Bergdall and Matteo Longo.