Generalized triple product $p$-adic $L$-functions and rational points on elliptic curves

Abstract

We generalize and simplify the constructions of Darmon–Rotger (2014) and Hsieh (2021) of an unbalanced triple product $p$-adic $L$-function attached to a triple $(\mathbf{f},\mathbf{g},\mathbf{h})$ of $p$-adic families of modular forms, allowing more flexibility for the choice of $\mathbf{g}$ and $\mathbf{h}$.
Assuming that $\mathbf{g}$ and $\mathbf{h}$ are families of theta series of infinite $p$-slope, we prove a factorization of (an improvement of) such $p$-adic $L$-function in terms of two anticyclotomic $p$-adic $L$-functions. As a corollary, when $\mathbf{f}$ specializes in weight $2$ to the newform attached to an elliptic curve $E$ over $\mathbb{Q}$ with multiplicative reduction at $p$, we relate certain Heegner points on $E$ to certain $p$-adic partial derivatives of the triple product $p$-adic $L$-function evaluated at the critical triple of weights $(2,1,1)$.