Pak-Hin Lee | Searcharxiv

arXiv2024

On $p$-adic adjoint $L$-functions for Bianchi cuspforms: the $p$-split case

We construct a Hecke-equivariant pairing on the overconvergent cohomology of Bianchi threefolds. Applying the strategy of Kim and Bellaïche, we use this pairing to construct $p$-adic adjoint $L$-functions for Bianchi cuspforms and show that it detects the ramification locus of the cuspidal Bianchi eigenvariety over the weight space. Combining results of Barrera Salazar--Williams, we show a non-vanishing result of this $p$-adic adjoint $L$-function at certain points. Finally, we obtain a formula relating this pairing with the adjoint $L$-values of the corresponding cuspidal Bianchi eigenforms (of level 1).

math.NT↗

arXiv2021

A p-adic L-function for non-critical adjoint L-values

Let $K$ be an imaginary quadratic field, with associated quadratic character $α$. We construct an analytic $p$-adic $L$-function interpolating the twisted adjoint $L$-values $L(1, \mathrm{ad}(f) \otimes α)$ as $f$ varies in a Hida family; these special values are non-critical in the sense of Deligne. Our approach is based on Greenberg--Stevens' idea of $Λ$-adic modular symbols, which considers cohomology with values in a space of $p$-adic measures.

math.NT↗