On the Artin formalism for triple product $p$-adic $L$-functions
Kâzım Büyükboduk, Ryotaro Sakamoto
TL;DR
The paper addresses irregular factorization phenomena for triple product $p$-adic $L$-functions, formulating a $p$-adic Artin formalism that predicts a decomposition into products of auxiliary $p$-adic $L$-functions and Beilinson–Kato data even when interpolation ranges vanish. The authors develop a comprehensive framework based on Hida families, Greenberg Selmer complexes, and Perrin-Riou logarithms, introducing modules of leading terms as algebraic avatars of $p$-adic $L$-functions and proving an algebraic factorization theorem (Theorem 8.4.4) that ties ${\delta}(T_2^{\dagger},\Delta_{\mathbf g})$ to ${\delta}(M_2^{\dagger},\mathrm{tr}^{*}\Delta_{\mathbf g})$ and ${\rm Log}_{\omega_{\mathbf f}}({\rm BK}_{\mathbf f}^{\dagger})$. The work draws a bridge between diagonal (Gross–Kudla) cycles and Heegner cycles, guided by ETNC-like philosophy, and provides a strategic plan to prove the conjectural factorization via a comparison of conjectural Gross–Kudla and Gross–Zagier–Zhang formulae. It also clarifies the algebraic counterpart through determinants and leading-term modules, with a companion paper offering further evidence in the CM-analytic regime. Overall, the article lays a rigorous foundation for an irregular factorization picture of triple product $p$-adic $L$-functions and their algebraic avatars, with explicit structural and computational components.
Abstract
Our main objective in the present article is to study the factorization problem for triple-product $p$-adic $L$-functions, particularly in the scenarios when the defining properties of the $p$-adic $L$-functions involved have no bearing on this problem, although Artin formalism would suggest such a factorization. Our analysis, which is guided by the ETNC philosophy, recasts this problem as a comparison of diagonal cycles, Beilinson--Kato elements, and Heegner cycles.