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A New p-Adic Maass-Shimura Operator and Supersingular Rankin-Selberg p-Adic L-Functions
Başlık:
A New p-Adic Maass-Shimura Operator and Supersingular Rankin-Selberg p-Adic L-Functions
Yazar:
Kriz, Daniel J., author.
ISBN:
9780438048706
Yazar Ek Girişi:
Fiziksel Tanımlama:
1 electronic resource (197 pages)
Genel Not:
Source: Dissertation Abstracts International, Volume: 79-10(E), Section: B.
Advisors: Shou-Wu Zhang; Christopher Skinner Committee members: Nicholas Katz; Christopher Skinner; Shou-Wu Zhang.
Özet:
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 ODelta of "nearly rigid functions" which transform under the action of subgroups of the Galois group Gal(Y/Y) by Ox Delta-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 xp-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 ODelta 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 affinoid subdomain of Y, and can be viewed as an extension of the Serre-Tate coordinate on Y Ig. 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 x 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.
Notlar:
School code: 0181
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Yer Numarası | Demirbaş Numarası | Shelf Location | Lokasyon / Statüsü / İade Tarihi |
---|---|---|---|
XX(681829.1) | 681829-1001 | Proquest E-Tez Koleksiyonu | Arıyor... |
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