p-adic adjoint L-functions for Hilbert modular forms

Exposé dans le cadre du séminaire de théorie des nombres

Orateur : Baskar Balasubramanyam (IISER Pune, Inde)

Let F be a totally real field. Let π be a cuspidal cohomological automorphic representation for GL2/F. Let L(s,Ad0,π) denote the adjoint L-function associated to π. 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=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.