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 ({\boldsymbol f},{\boldsymbol g},{\boldsymbol h}) of p -adic families of modular forms, allowing more flexibility for the choice of {\boldsymbol g} and {\boldsymbol h} .Assuming that {\boldsymbol g} and {\boldsymbol 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 {\boldsymbol 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) .