ℒ-Invariants, p-Adic Heights, and Factorization of p-Adic L-Functions

Abstract

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