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

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.