Unitary Friedberg--Jacquet periods and anticyclotomic p-adic L-functions

We extend the construction of the $p$-adic $L$-function interpolating unitary Friedberg--Jacquet periods in previous work of the author to include the $p$-adic variation of Maass--Shimura differential operators. In particular, we develop a theory of nearly overconvergent automorphic forms in higher degrees of coherent cohomology for unitary Shimura varieties generalising previous work for modular curves. The construction of this $p$-adic $L$-function can be viewed as a higher-dimensional generalisation of the work of Bertolini--Darmon--Prasanna and Castella--Hsieh, and the inclusion of this extra variable arising from the $p$-adic iteration of differential operators will play a key role in relating values of this $p$-adic $L$-function to $p$-adic regulators of special cycles on unitary Shimura varieties.