A New $p$-adic Maass-Shimura operator and Supersingular Rankin-Selberg $p$-adic $L$-functions

We give a construction of a new $p$-adic Maass-Shimura operator defined on an affinoid subdomain of the preperfectoid $p$-adic universal cover $\mathcal{Y}$ of a modular curve $Y$. We define a new notion of $p$-adic modular forms as sections of a certain sheaf $\mathcal{O}_{\Delta}$ of "nearly rigid functions" which transform under the action of subgroups of the Galois group $\mathrm{Gal}(\mathcal{Y}/Y)$ by $\mathcal{O}_{\Delta}^{\times}$-valued weight characters. This extends Katz's notion of $p$-adic modular forms as functions on the Igusa tower $Y^{\mathrm{Ig}}$; indeed we may recover Katz's theory by restricting to a natural $\mathbb{Z}_p^{\times}$-covering $\mathcal{Y}^{\mathrm{Ig}}$ of $Y^{\mathrm{Ig}}$, viewing $\mathcal{Y}^{\mathrm{Ig}} \subset \mathcal{Y}$ as a sublocus. Our $p$-adic Maass-Shimura operator sends $p$-adic modular forms of weight $k$ to forms of weight $k + 2$. Its construction comes from a relative Hodge decomposition with coefficients in $\mathcal{O}_{\Delta}$ defined using Hodge-Tate and Hodge-de Rham periods arising from Scholze's Hodge-Tate period map and the relative $p$-adic de Rham comparison theorem. By studying the effect of powers of the $p$-adic Maass-Shimura operator on modular forms, we construct a $p$-adic continuous function which satisfies an "approximate" interpolation property with respect to the the algebraic parts of central critical $L$-values of anticyclotomic Rankin-Selberg families on $GL_2 \times GL_1$ over imaginary quadratic fields $K/\mathbb{Q}$, including the "supersingular" case where $p$ is not split in $K$. Finally we establish a new $p$-adic Waldspurger formula which, in the case of a newform, relates the formal logarithm of a Heegner point to a special value of the $p$-adic $L$-function.