?-Invariants, <i>p</i>-Adic Heights, and Factorization of <i>p</i>-Adic <i>L</i>-Functions

Abstract We continue with our study of the noncritical exceptional zeros Katz’ $p$-adic $L$-functions attached to a CM field $K$, following two threads. In 1st thread, we redefine (group-ring-valued) ${\mathcal {L}}$-invariant associated each ${\mathbb {Z}}_p$-extension $K_\Gamma $ $K$ in terms height pairings and interpolate them as varies universal (multivariate) group-ring-valued {L}}$-invariant. 2nd use results Rankin–Selberg at specializations self-products nearly ordinary families, via factorization statements establish. The theorems are extensions due Greenberg Palvannan.