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

TL;DR

This paper investigates the factorization of triple-product $p$-adic $L$-functions, focusing on cases where traditional properties do not influence the problem, using a framework inspired by the ETNC philosophy.

Contribution

It introduces a new approach to the factorization problem by comparing various cycles and elements guided by the ETNC philosophy.

Findings

Recasts the factorization problem as a comparison of cycles and elements.

Provides insights into the relationship between $p$-adic $L$-functions and algebraic cycles.

Suggests new avenues for understanding Artin formalism in the context of $p$-adic $L$-functions.

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.