Nearly Overconvergent Modular Forms

  • Eric UrbanEmail author
Conference paper
Part of the Contributions in Mathematical and Computational Sciences book series (CMCS, volume 7)


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..


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.



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.


  1. Andreatta, F., Iovita, A., Pilloni, V.: Families of siegel modular forms. Ann. Math. (to appear)Google Scholar
  2. Bellaiche, J.: Critical p-adic L-functions. Invent. Math. 189(1), 1–60 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Berthelot, P., Ogus, A.: Notes on Crystalline Cohomology. Princeton University Press/University of Tokyo Press, Princeton/Tokyo, vi+243pp. (1978)Google Scholar
  4. Coleman, R.: Overconvergent and classical modular forms. Invent. Math. 124, 215–241 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Coleman, R.: p-adic Banach spaces and families of modular forms. Invent. Math. 127(3), 417–479 (1997)Google Scholar
  6. 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)Google Scholar
  7. Coleman, R., Gouvêa, F., Fernando Q., Jochnowitz, N.: E 2, Θ and overconvergence. Internat. Math. Res. Notices 1, 23-41 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 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)MathSciNetzbMATHGoogle Scholar
  9. Harris, M.: Arithmetic vector bundles and automorphic forms on Shimura varieties. I. Invent. Math. 82, 151–189 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Harris, M.: Arithmetic vector bundles and automorphic forms on Shimura varieties. II. Compos. Math. tome 60, 323–378 (1986)MathSciNetzbMATHGoogle Scholar
  11. Hida, H.: A p-adic measure attached to the zeta functions associated with two elliptic modular forms. I. Invent. Math. 79, 159–195 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Hida, H.: Elementary Theory of L-Functions and Eisenstein Series. London Mathematical Society Student Texts, vol. 26. Cambridge University Press, Cambridge/New York (1993)Google Scholar
  14. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Katz, N.: p-adic properties of modular schemes and modular forms. In: International Summer School on Modular Functions, Antwerp (1972)zbMATHGoogle Scholar
  16. Nappari, M.: Holomorphic forms canonically attached to nearly holomorphic automorphic forms. Thesis, Brandeis University (1992)Google Scholar
  17. 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)Google Scholar
  18. 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)Google Scholar
  19. Shimura, G.: The special values of the zeta functions associated with cusp forms. Commun. Pure Appl. Math. 29(6), 783–804 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Skinner, C.M., Urban, E.: p-adic families of nearly holomorphic forms and applications (in preparation)Google Scholar
  21. 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)Google Scholar
  22. 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)Google Scholar
  23. Wiles, A.: On ordinary λ-adic representations associated to modular forms. Invent. Math. 94(3), 529–573 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Harron, R., Xiao, L.: Gauss-Manin connections for p-adic families of nearly overconvergent modular forms. arXiv:1308.1732Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA

Personalised recommendations