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.

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Last updated: 13 Sep 2024