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..
Keywords
- Modular Form
- Irreducible Component
- Eisenstein Series
- Admissible Pair
- Fredholm Determinant
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 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 \(T_{R,\mathfrak{V}} \otimes F(\mathfrak{V})\)
References
Andreatta, F., Iovita, A., Pilloni, V.: Families of siegel modular forms. Ann. Math. (to appear)
Bellaiche, J.: Critical p-adic L-functions. Invent. Math. 189(1), 1–60 (2012)
Berthelot, P., Ogus, A.: Notes on Crystalline Cohomology. Princeton University Press/University of Tokyo Press, Princeton/Tokyo, vi+243pp. (1978)
Coleman, R.: Overconvergent and classical modular forms. Invent. Math. 124, 215–241 (1996)
Coleman, R.: p-adic Banach spaces and families of modular forms. Invent. Math. 127(3), 417–479 (1997)
Coleman, R., Mazur, B.: The eigencurve. In: Galois Representations in Arithmetic Algebraic Geometry (Durham, 1996). Volume 254 of London Mathematical Society Lecture Note Series, pp. 1–113. Cambridge University Press, Cambridge (1998)
Coleman, R., Gouvêa, F., Fernando Q., Jochnowitz, N.: E 2, Θ and overconvergence. Internat. Math. Res. Notices 1, 23-41 (1995)
Darmon, H., Rotger, V.: Diagonal cycles and Euler systems I: a p-adic Gross-Zagier formula. Ann. Sci. Éc. Norm. Supér. 47(4), 779–832 (2014)
Harris, M.: Arithmetic vector bundles and automorphic forms on Shimura varieties. I. Invent. Math. 82, 151–189 (1985)
Harris, M.: Arithmetic vector bundles and automorphic forms on Shimura varieties. II. Compos. Math. tome 60, 323–378 (1986)
Hida, H.: A p-adic measure attached to the zeta functions associated with two elliptic modular forms. I. Invent. Math. 79, 159–195 (1985)
Hida, H.: A p-adic measure attached to the zeta functions associated with two elliptic modular forms. II. Annales de l’Institut Fourier tome 38(3), 1–83 (1988)
Hida, H.: Elementary Theory of L-Functions and Eisenstein Series. London Mathematical Society Student Texts, vol. 26. Cambridge University Press, Cambridge/New York (1993)
Katz, N.: Nilpotent connections and the monodromy theorem: applications of a result of Turrittin. Publications mathématiques de l’Institut des Hautes Études Scientifique 39, 175–232 (1970)
Katz, N.: p-adic properties of modular schemes and modular forms. In: International Summer School on Modular Functions, Antwerp (1972)
Nappari, M.: Holomorphic forms canonically attached to nearly holomorphic automorphic forms. Thesis, Brandeis University (1992)
Pilloni, V.: Overconvergent modular forms. Ann. Inst. Fourier 63(1), 219–239 (2013) Pilloni, V.: Formes modulaires surconvergentes. Annales de l’Institut Fourier (to appear)
Shimura, G.: Introduction to the Arithmetic Theory of Automorphic Functions, Kanō Memorial Lectures, No. 1, Publications of the Mathematical Society of Japan, No. 11, xiv+267pp. Iwanami Shoten/Princeton University Press, Tokyo/Princeton (1971)
Shimura, G.: The special values of the zeta functions associated with cusp forms. Commun. Pure Appl. Math. 29(6), 783–804 (1976)
Skinner, C.M., Urban, E.: p-adic families of nearly holomorphic forms and applications (in preparation)
Tilouine, J.: Companion forms and classicity in the GL 2(Q)-case. In: Number Theory. Ramanujan Mathematical Society Lecture Notes Series, vol. 15, pp. 119–141. Ramanujan Mathematical Society, Mysore (2011)
Urban, E.: On the ranks of Selmer groups of elliptic curves over Q. In: Automorphic Representations and L-Functions. Proceedings of the International Colloquium held at the Tata Institute of Fundamental Research, Mumbai, pp. 651–680 (2013)
Wiles, A.: On ordinary λ-adic representations associated to modular forms. Invent. Math. 94(3), 529–573 (1988)
Harron, R., Xiao, L.: Gauss-Manin connections for p-adic families of nearly overconvergent modular forms. arXiv:1308.1732
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-642-55245-8_14
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-55244-1
Online ISBN: 978-3-642-55245-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)