Anticyclotomic p-adic L-functions and Ichino's formula

We give a new construction of a $p$-adic $L$-function $\mathcal{L}(f,\mathrm{\Xi })$, for $f$ a holomorphic newform and $\mathrm{\Xi }$ an anticyclotomic family of Hecke characters of $\mathbb{Q}(\sqrt{-d})$. The construction uses Ichino’s triple product formula to express the central values of $L(f,\xi ,s)$ in terms of Petersson inner products, and then uses results of Hida to interpolate them. The resulting construction is well-suited for studying what happens when $f$ is replaced by a modular form congruent to it modulo $p$, and has future applications in the case where $f$ is residually reducible.