$\mathcal{L}$-invariants, $p$-adic heights and factorization of $p$-adic $L$-functions

Abstract

We continue with our study of the non-critical exceptional zeros of Katz' $p$-adic $L$-functions attached to a CM field $K$, following two threads. In the first thread, we redefine our (group-ring-valued) $\mathcal{L}$-invariant associated to 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 second thread, we use our results to study the exceptional zeros of the non-genuine Rankin--Selberg $p$-adic $L$-functions attached to 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.

48 pages, final version (to appear in IMRN). Journal version may be slightly different

References in corpus (3)

• Integral Eisenstein cocycles on GLn, I : Sczech's cocycle and p-adic L-functions of totally real fields
• Height pairings, exceptional zeros and Rubin's Formula: The multiplicative group
• On the non-critical exceptional zeros of Katz $p$-adic $L$-functions for CM fields