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- W2800723456 abstract "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 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." @default.
- W2800723456 created "2018-05-17" @default.
- W2800723456 creator A5037931106 @default.
- W2800723456 date "2018-05-09" @default.
- W2800723456 modified "2023-09-27" @default.
- W2800723456 title "A New p-adic Maass-Shimura Operator and Supersingular Rankin-Selberg p-adic L-functions" @default.
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