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 Y of a modular curve Y. We define a new notion of p-adic modular forms as sections of a certain sheaf OΔ of "nearly rigid functions" which transform under the action of subgroups of the Galois group Gal(Y/Y) by O×Δ-valued weight characters. This extends Katz's notion of p-adic modular forms as functions on the Igusa tower YIg; indeed we may recover Katz's theory by restricting to a natural Z×p-covering YIg of YIg, viewing YIg⊂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 OΔ 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 GL2×GL1 over imaginary quadratic fields K/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.