Over several centuries, many mathematicians dedicated themselves in studying the arithmetic of modular forms. One of the most beautiful results was due to R. Coleman and B. Mazur: they discovered that certain types of modular forms (i.e., finite-slope eigenforms) can be patched into a geometric object –the so-called eigencurve. The geometric properties of the eigencurve are mysterious and often encode interesting arithmetic information. For example, in his thesis, W. Kim constructed a Hecke-equivariant pairing on the cuspidal eigenvariety and showed that this pairing detects the ramification locus of the cuspidal eigencurve over the weight space and has an explicit relation with the adjoint L-function of cuspidal eigenforms.

In this talk, based on joint work with Pak-Hin Lee, I will report a generalisation of the aforementioned results. More precisely, I will discuss the construction of a Hecke-equivariant pairing on the cuspidal Bianchi eigenvariety. This pairing shares similar properties with Kim’s pairing: it also detects the ramification locus of the cuspidal Bianchi eigenvariety over the (parallel) weight space and has explicit relation with the adjoint L-function of cuspidal Bianchi eigenforms. If time permits, I will discuss an interesting question derived from these p-adic adjoint L-functions.