Arithmetic theta lifting and $L$-derivatives for unitary groups, II

Abstract

We prove the arithmetic inner product formula conjectured in the first paper of this series for $n=1$, that is, for the group $U{\left(1,1\right)}_F$ unconditionally. The formula relates central $L$-derivatives of weight-$2$ holomorphic cuspidal automorphic representations of $U{\left(1,1\right)}_F$ with $ϵ$-factor $-1$ with the Néron–Tate height pairing of special cycles on Shimura curves of unitary groups. In particular, we treat all kinds of ramification in a uniform way. This generalizes the arithmetic inner product formula obtained by Kudla, Rapoport, and Yang, which holds for certain cusp eigenforms of $PGL{\left(2\right)}_{\mathbb{Q}}$ of square-free level.