On the Artin formalism for triple product $p$-adic $L$-functions: Super-factorization

TL;DR

The paper proves a CM-case instance of the Artin-formalism factorization for triple-product $p$-adic $L$-functions by developing a super-factorization mechanism: when the middle factor $\mathbf g$ has CM, the cubic $p$-adic $L$-function factors as a product of two Beilinson–Donovan–Perrin-Riou (BDP) type $p$-adic $L$-functions, up to an explicit algebraic scalar. The core analytic result expresses $\mathcal L_p^{\mathbf g}(\mathbf f\otimes\mathbf g\otimes\mathbf g^c)^2$ on a suitable weight region as $\mathscr C(\kappa)\cdot \mathcal L_p^{\rm ad}(\mathbf f\otimes{\rm ad}^0\mathbf g)\cdot {\rm Log}_{\omega_{\mathbf f}}({\rm BK}_{\mathbf f}^{\dagger})$, with $\mathscr C$ algebraic at crystalline points and tied to Katz-type $p$-adic $L$-values. The authors also establish the corresponding algebraic factorization via Selmer complexes and leading-term modules, providing a structural parallel between analytic and algebraic $p$-adic $L$-functions in the CM setting. The work integrates Beilinson–Kato and Beilinson–Flach reciprocity principles, BDV-type machinery, and Katz $p$-adic $L$-functions to realize a robust CM-compatible form of the $p$-adic Artin formalism for triple products with explicit control of constants, periods, and Euler factors.

Abstract

We prove the factorization conjecture for triple-product $p$-adic $L$-functions formulated in a companion article in the special case when two of the (three) factors have complex multiplication.

Theorems & Definitions (65)

• Theorem 1.1: Theorem \ref{['Thm:8=6+2CM']}
• Theorem 1.2: Theorem \ref{['thm_main_8_4_4_factorization_bis']} below
• Theorem 2.1: Kitagawa
• Theorem 2.2: Hida AIF_Hida88
• Theorem 2.3: Hsieh
• Conjecture 2.4
• Theorem 3.1
• proof
• Remark 3.2
• Corollary 3.3