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On $p$-adic adjoint $L$-functions for Bianchi cuspforms: the $p$-split case
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abstract
We construct a Hecke-equivariant pairing on the overconvergent cohomology of Bianchi threefolds. Applying the strategy of Kim and Bella\"iche, 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).
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Overconvergent Eichler-Shimura morphisms for $\mathrm{GSp}_4$
The full four-step Eichler-Shimura decomposition for GSp4 is interpolated along p-adic weight families, and near any nice-enough eigenvariety point it splits with Hodge-Tate-Sen weights (-3, κ2-2, κ1-1, κ1+κ2).
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