Anticyclotomic p-adic L-function of Central Critical Rankin–Selberg L-value

Let M be an imaginary quadratic field, f a Hecke eigen-cusp form on ${\mathrm GL}_2(\mathbb Q)\setminus {\mathrm GL}_2(\mathbb A)$, and $\hat \pi_{\mathrm f}$ the unitary base-change to M of automorphic representation πf associated to f. Take a unitary arithmetic Hecke character χ of $M^\times\setminus M^\times_\mathbb A$ inducing the inverse of the central character of πf. The celebrated formula of Waldspurger relates the square of $L\chi\ ({\mathrm f}):=\int _{M^\times\setminus M^\times_\mathbb A}{\mathrm f}(t)\chi(t)d^\times t$ to the central critical value $L(\frac {1}{2},\ \hat \pi_{\mathrm f}\ \otimes\ \chi)$. We construct a new p-adic L-function that interpolates Lχ(f) over arithmetic χ's for a cusp form f in the spirit of the landmark result of Katz where he did that for the Eisenstein series.