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

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 .f; g; h/ of p-adic families of modular forms, allowing more flexibility for the choice of g and h. Assuming that g and 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 f specializes in weight 2 to the newform attached to an elliptic curve E over 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/.