The 𝑝-adic variation of the Gross–Kohnen–Zagier theorem

Longo, Matteo; Nicole, Marc-Hubert

doi:10.1515/forum-2018-0129

Cited by 5 publications

(1 citation statement)

References 36 publications

(58 reference statements)

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“…This variant further allowed them to obtain a similar relationship as in Kohnen's formula for central critical derivatives, with the role of the Fourier coefficient a |D| (g) being played by the first derivative of the |D|-th Fourier coefficient of a p-adic family of half-integral forms. Also in this line, the p-adic variation of the Gross-Kohnen-Zagier theorem, including the existence of Λ-adic families of Jacobi forms, is studied in [LN19a,LN19b]. In a different direction and with a different flavour, there is also the work of Ono-Skinner [OS98], studying the divisibility of Fourier coefficients of half-integral weight modular forms by looking at the residual Galois representations of integral weight modular forms in correspondence with them.…”

Section: Introductionmentioning

confidence: 99%

p-adic families of d-th Shintani liftings

Casazza,

de Vera-Piquero

2019

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In this note we give a detailed construction of a Λ-adic d-th Shintani lifting. We derive a Λ-adic version of Kohnen's formula relating Fourier coefficients of half-integral weight modular forms and special values of twisted L-series. As a by-product we obtain a mild generalization of such classical formula. Contents 1. Introduction 1 2. d-th Shintani lifting 5 3. d-th Shintani liftings of p-stabilized newforms 11 4. Hida theory and modular symbols 19 5. The Λ-adic d-th Shintani lifting 25 References 34

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Section: Introductionmentioning

confidence: 99%

p-adic families of d-th Shintani liftings

Casazza,

de Vera-Piquero

2019

Preprint

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Generalized Heegner cycles on Mumford curves

Longo

Pati

2020

Math. Z.

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We study generalised Heegner cycles, originally introduced by Bertolini-Darmon-Prasanna for modular curves in [BDP13], in the context of Mumford curves. The main result of this paper relates generalized Heegner cycles with the two variable anticyclotomic p-adic L-function attached to a Coleman family f∞ and an imaginary quadratic field K, constructed in [BD07] and [Sev14]. While in [BD07] and [Sev14] only the restriction to the central critical line of this 2 variable p-adic L-function is considered, our generalised Heegner cycles allow us to study the restriction of this function to non-central critical lines. The main result expresses the derivative along the weight variable of this anticyclotomic p-adic L-function restricted to non necesserely central critical lines as a combination of the image of generalized Heegner cycles under a p-adic Abel-Jacobi map. In studying generalised Heegner cycles in the context of Mumford curves, we also obtain an extension of a result of Masdeu [Mas12] for the (one variable) anticyclotomic p-adic L-function of a modular form f and K at non-central critical integers.

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