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Annales mathématiques du Québec

, Volume 40, Issue 1, pp 55–82 | Cite as

Differential operators, pullbacks, and families of automorphic forms on unitary groups

  • Ellen Elizabeth Eischen
Article
  • 104 Downloads

Abstract

This paper has two main parts. First, we construct certain differential operators, which generalize operators studied by G. Shimura. Then, as an application of some of these differential operators, we construct certain p-adic families of automorphic forms. Building on the author’s earlier work, these differential operators map automorphic forms on a unitary group of signature (nn) to (vector-valued) automorphic forms on the product \(U^\varphi \times U^{-\varphi }\) of two unitary groups, where \(U^\varphi \) denotes the unitary group associated to a Hermitian form \(\varphi \) of arbitrary signature on an n-dimensional vector space. These differential operators have both a p-adic and a \(C^{\infty }\) incarnation. In the scalar-weight, \(C^{\infty }\)-case, these operators agree with ones studied by Shimura. In the final section of the paper, we also discuss some generalizations to other groups and settings. The results from this paper apply to the author’s paper-in-preparation with J. Fintzen, E. Mantovan, and I. Varma and to her ongoing joint project with M. Harris, J. -S. Li, and C. Skinner; they also relate to her recent paper with X. Wan.

Keywords

p-adic automorphic forms Differential operators Shimura varieties 

Résumé

Cet article se divise principalement en deux parties. Tout d’abord. nous contrui-sons des opérateurs différentiels généralisant les opérateurs étudiés par G. Shimura. Puis, nous appliquons certains de ces opérateurs différentiels pour construire quelques familles p-adiques de formes automorphes. Il s’ensuit des travaux antérieurs de l’auteur que ces opérateurs différentiels envoient les formes automorphes sur le groupe unitaire de signature (nn) sur des formes automorphes (à valeurs vectoriels) sur le produit de deux groupes unitaires, \(U^{\varphi }\times U^{-\varphi }\), où \(U^{\varphi }\) désigne le groupe unitaire associé à une forme hermitienne de signature arbitraire sur un espace vectoriel de dimension n. Ces opérateurs différentiels ont à la fois une incarnation p-adique et une incarnation \(C^{\infty }\). Dans le cas \(C^{\infty }\) et lorsque le poids est scalaire, ces opérateurs correspondent à ceux étudiés par Shimura. Dans la dernière partie de l’article, nous abordons quelques généralisations à d’autres groupes et situations. Les résultats de cet article s’appliquent à l’article de J. Fintzen, E. Mantovan, I. Verma et l’auteur, actuellement en préparation, et au project qu’elle mène conjointement avec M. Harris, J.-S. Li et C. Skinner; ils sont également en lien avec son article récent avec X. Wan.

Mathematics Subject Classification

11F33 11F37 11F46 11F55 11F85 11F03 

Notes

Acknowledgments

I am grateful to Harris, Skinner, and Urban for helpful conversations and suggestions while I was working on this paper. I am also very thankful to Hida and K.-W. Lan for answering my questions about q-expansions.

References

  1. 1.
    Caraiani, A., Eischen, E., Fintzen, J., Mantovan, E., Varma, I.: \(p\)-expansion principles on unitary Shimura varieties, p. 36 (2015). Accepted for publication in Directions in Number Theory: Proceedings for the 2014 WIN3 workshop. http://arxiv.org/pdf/1411.4350.pdf
  2. 2.
    Deligne, P.: Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques. In: Automorphic Forms, Representations and \(L\)-Functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977). Part 2, Proc. Sympos. Pure Math., XXXIII, pp. 247–289. Amer. Math. Soc., Providence, RI (1979)Google Scholar
  3. 3.
    Eischen, E.E.: \(p\)-adic differential operators on automorphic forms on unitary groups. Ann. Inst. Fourier (Grenoble) 62(1), 177–243 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Eischen, E.: A p-adic Eisenstein measure for vector-weight automorphic forms. Algebra Number Theory 8(10), 2433–2469 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Eischen, E.E.: A p-adic Eisenstein measure for unitary groups. J. Reine Angew. Math. 699, 111–142 (2015)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Eischen, E., Wan, X.: \(p\)-functions of certain cusp forms on definite unitary groups. J. Inst. Math. Jussieu 11, FirstView:1–40 (2014)Google Scholar
  7. 7.
    Harris, M.: \(L\)-functions and periods of polarized regular motives. J. Reine Angew. Math. 483, 75–161 (1997)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Hida, H.: Geometric Modular Forms and Elliptic Curves. World Scientific Publishing Co., Inc., River Edge (2000)CrossRefzbMATHGoogle Scholar
  9. 9.
    Hida, H.: \(p\)-adic Automorphic Forms on Shimura Varieties. Springer Monographs in Mathematics. Springer-Verlag, New York (2004)CrossRefzbMATHGoogle Scholar
  10. 10.
    Hida, H.: \(p\)-adic automorphic forms on reductive groups. Astérisque 298, 147–254 (2005) Automorphic forms. IGoogle Scholar
  11. 11.
    Harris, M., Li, J.-S., Skinner, C.M.: \(p\)-functions for unitary Shimura varieties. I. Construction of the Eisenstein measure. Doc. Math. (Extra Vol.), 393–464 (electronic) (2006)Google Scholar
  12. 12.
    Hua, L.K.: Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Translated from the Russian by Leo Ebner and Adam Korányi. American Mathematical Society, Providence, RI (1963)Google Scholar
  13. 13.
    Jantzen, J.C.: Representations of Algebraic Groups. Pure and Applied Mathematics, vol. 131. Academic Press Inc., Boston (1987)Google Scholar
  14. 14.
    Johnson, K.D.: On a ring of invariant polynomials on a Hermitian symmetric space. J. Algebra 67(1), 72–81 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Katz, N.: Travaux de Dwork. In: Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 409, pp. 167–200. Lecture Notes in Math., vol. 317. Springer, Berlin (1973)Google Scholar
  16. 16.
    Katz, N.M.: \(p\)-adic \(L\)-functions for CM fields. Invent. Math. 49(3), 199–297 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Langlands, R.P.: On the Functional Equations Satisfied by Eisenstein Series. Lecture Notes in Mathematics, vol. 544. Springer-Verlag, Berlin (1976)Google Scholar
  18. 18.
    Lan, K.-W.: Arithmetic Compactifications of PEL-Type Shimura Varieties, volume 36 of London Mathematical Society Monographs. Princeton University Press (2013)Google Scholar
  19. 19.
    Milne, J.: Introduction to Shimura Varieties. http://www.jmilne.org/math/ (2004)
  20. 20.
    Shimura, G.: Differential operators and the singular values of Eisenstein series. Duke Math. J. 51(2), 261–329 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Shimura, G.: On differential operators attached to certain representations of classical groups. Invent. Math. 77(3), 463–488 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Shimura, G.: Euler Products and Eisenstein Series, volume 93 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC (1997)Google Scholar
  23. 23.
    Shimura, G.: Zeta functions and Eisenstein series on classical groups. Proc. Natl. Acad. Sci. U.S.A. 94(21):11133–11137 (1997). Elliptic curves and modular forms (Washington, DC, 1996)Google Scholar
  24. 24.
    Shimura, G.: Arithmeticity in the Theory of Automorphic Forms. Mathematical Surveys and Monographs, vol. 82. American Mathematical Society, Providence, RI (2000)Google Scholar
  25. 25.
    Washington, L.C.: Introduction to Cyclotomic Fields, volume 83 of Graduate Texts in Mathematics, 2nd edn. Springer-Verlag, New York (1997)Google Scholar

Copyright information

© Fondation Carl-Herz and Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OregonEugeneUSA

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