p-adic adjoint L-functions for Hilbert modular forms — Baskar Balasubramanyam (IISER Pune)

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.