We study cuspidal automorphic representations of unitary groups of $2n$ variables with $ϵ$-factor $-1$ and their central $L$-derivatives by constructing their arithmetic theta liftings, which are Chow cycles of codimension $n$ on Shimura varieties of dimension $2n-1$ of certain unitary groups. We give a precise conjecture for the arithmetic inner product formula, originated by Kudla, which relates the height pairing of these arithmetic theta liftings and the central $L$-derivatives of certain automorphic representations. We also prove an identity relating the archimedean local height pairing and derivatives of archimedean Whittaker functions of certain Eisenstein series, which we call an arithmetic local Siegel–Weil formula for archimedean places. This provides some evidence toward the conjectural arithmetic inner product formula.

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.