Abstract
We introduce and study finite slope nearly overconvergent (elliptic) modular forms. We give an application of this notion to the construction of the Rankin-Selberg p-adic L-function on the product of two eigencurves..
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- 1.
It will be clear to the reader that it could be generalized to any Shimura variety of PEL type.
- 2.
His definition follows a suggestion of P. Deligne.
- 3.
The first half of this note was also presented in my lecture given at H. Hida’s 60th birthday conference.
- 4.
Those facts are mainly due to Shimura
- 5.
We will see a similar fact in the p-adic case. See Proposition 6.
- 6.
We leave it as an exercise to check that this operator corresponds to the classical Maass-Shimura operator via the isomorphism of Proposition 1.
- 7.
In Pilloni (to appear), this sheaf is constructed in a purely geometric way and the existence of \(\mathcal{M}_{\mathfrak{U}}^{\rho }\) is deduced from it.
- 8.
This maximum is < ∞ since \(Q^{{\ast}}(0) \in A(\mathfrak{U})^{\times }\).
- 9.
When the level is not 1, one uses the theory of primitive forms which described the maximal semi-simple direct factor of \(T_{R,\mathfrak{V}} \otimes F(\mathfrak{V})\)
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Acknowledgements
The author would like to thank Giovanni Rosso and Chris Skinner for interesting conversations during the preparation of this work and Vincent Pilloni for pointing out an error in a previous version of this text. He would like also to thank Pierre Colmez who encouraged him to write this note. He is also grateful to the organizers of the conference Iwasawa 2012 held in Heidelberg for their invitation and for giving the opportunity to publish this paper in the proceedings of this conference. This work was also lectured during the Postech winter school in January 2013. The author would like to thank the organizers of this workshop for their invitation. Finally the author would like to thank the Florence Gould Foundation for its support when he was a Member at the Institute for Advanced Studies and when some part of this work was conceived.
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Urban, E. (2014). Nearly Overconvergent Modular Forms. In: Bouganis, T., Venjakob, O. (eds) Iwasawa Theory 2012. Contributions in Mathematical and Computational Sciences, vol 7. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55245-8_14
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