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

Abstract

We give a construction of a new p-adic Maass-Shimura operator defined on an affi-

noid 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 O∆ of

“nearly rigid functions” which transform under the action of subgroups of the Galois

group Gal(Y/Y ) by O×∆-valued weight characters. This extends Katz’s notion of

p-adic modular forms as functions on the Igusa tower YIg transforming under the action

of the Galois group Gal(YIg/Y ord), where Yord ⊂ Y denotes the ordinary locus,

by a certain weight character; 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 O∆ 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. In

particular, the Hodge-de Rham period gives rise to a coordinate qdR on a large affi-

noid subdomain of Y, and can be viewed as an extension of the Serre-Tate coordinate

on YIg. By studying the effect of powers of the p-adic Maass-Shimura operator on

modular forms expressed in qdR-coordinates, 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. This gives a new one-variable anticyclotomic p-adic

L-function, resolving questions, dating back to work of Katz from the 70’s, regarding

the interpolation of such L-values, and extends work in the ordinary case done

by Katz, Bertolini-Darmon-Prasanna and Liu-Zhang-Zhang. 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