On the Artin formalism for triple product $p$-adic $L$-functions: Chow--Heegner points vs. Heegner points

TL;DR

This paper explores the factorization of triple product $p$-adic $L$-functions using Artin formalism, focusing on explicit reciprocity laws and comparisons of Chow--Heegner and Heegner points in Hida families.

Contribution

It provides an expository analysis of the steps needed to prove a factorization formula for triple product $p$-adic $L$-functions, emphasizing reciprocity laws and point comparisons.

Findings

Establishes explicit reciprocity laws linking cycles to $p$-adic $L$-functions.

Compares Chow--Heegner points and twisted Heegner points in Hida families.

Lays groundwork for factorization formulas guided by Artin formalism.

Abstract

Our main objective in this paper (which is expository for the most part) is to study the necessary steps to prove a factorization formula for a certain triple product $p$-adic $L$-function guided by the Artin formalism. The key ingredients are: a) the explicit reciprocity laws governing the relationship of diagonal cycles and generalized Heegner cycles to $p$-adic $L$-functions; b) a careful comparison of Chow--Heegner points and twisted Heegner points in Hida families, via formulae of Gross--Zagier type.