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Title: A New p-adic Maass-Shimura Operator and Supersingular Rankin-Selberg p-adic L-functions
Authors: Kriz, Daniel
Advisors: Zhang, Shou-Wu
Skinner, Christopher
Contributors: Mathematics Department
Keywords: arithmetic geometry
number theory
p-adic L-functions
p-adic modular forms
Subjects: Mathematics
Issue Date: 2018
Publisher: Princeton, NJ : Princeton University
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
Alternate format: The Mudd Manuscript Library retains one bound copy of each dissertation. Search for these copies in the library's main catalog:
Type of Material: Academic dissertations (Ph.D.)
Language: en
Appears in Collections:Mathematics

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