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p-adic families of \(\mathfrak d\)th Shintani liftings

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Abstract

In this note we give a detailed construction of a \(\Lambda \)-adic \(\mathfrak d\)th Shintani lifting. We obtain a \(\Lambda \)-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 derive a mild generalization of such classical formulae, and also point out a relation between Fourier coefficients of \(\Lambda \)-adic \(\mathfrak d\)th Shintani liftings and Stark–Heegner points.

Résumé

Dans cet article nous donnons une construction détaillée d’un \(\mathfrak d\)-ième relèvement de Shintani \(\Lambda \)-adique. Nous obtenons une version \(\Lambda \)-adique de la formule de Kohnen reliant les coefficients de Fourier des formes modulaires de poids semi-intégral et les valeurs spéciales de séries L tordues. Comme sous-produit, nous déduisons une généralisation de telles formules classiques, et nous soulignons également une relation entre les coefficients de Fourier \(\Lambda \)-adiques des \(\mathfrak d\)-ièmes relèvements de Shintani \(\Lambda \)-adiques et les points de Stark–Heegner.

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Notes

  1. Notice that this is stronger than our previous assumption \(\mathrm {gcd}(N_0,\mathfrak d)=1\).

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Acknowledgements

D.C. acknowledges financial support by Spanish Ministry of Economy and Competitiveness, through the Severo Ochoa programme for Centres of Excellence in R&D (SEV-2015-0554), and C.dVP. acknowledges financial support by a Junior Researcher Grant through the project AGAUR PDJ2012. Both authors are also grateful to ICMAT and UB for their warm hospitality, and to Prof. Dieulefait for partial support through the project MTM2015-66716-P. This project has also received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation programme (Grant agreement no 682152) and from the Government of Ireland through the Fellowship GOIPD/2019/877.

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Authors and Affiliations

  1. University College Dublin, Belfield, Dublin 4, Ireland

    Daniele Casazza

  2. Facultad de Ciencias, Universidad de Zaragoza, C/ Pedro Cerbuna 12, 50009, Zaragoza, Spain

    Carlos de Vera-Piquero

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  1. Daniele Casazza
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  2. Carlos de Vera-Piquero
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Correspondence to Carlos de Vera-Piquero.

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Casazza, D., Vera-Piquero, C.d. p-adic families of \(\mathfrak d\)th Shintani liftings. Ann. Math. Québec 46, 419–460 (2022). https://doi.org/10.1007/s40316-021-00182-6

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  • DOI: https://doi.org/10.1007/s40316-021-00182-6

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