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p-Adic q-Expansion Principles on Unitary Shimura Varieties

  • Ana Caraiani
  • Ellen Eischen
  • Jessica Fintzen
  • Elena Mantovan
  • Ila Varma
Conference paper
Part of the Association for Women in Mathematics Series book series (AWMS, volume 3)

Abstract

We formulate and prove certain vanishing theorems for p-adic automorphic forms on unitary groups of arbitrary signature. The p-adic q-expansion principle for p-adic modular forms on the Igusa tower says that if the coefficients of (sufficiently many of) the q-expansions of a p-adic modular form f are zero, then f vanishes everywhere on the Igusa tower. There is no p-adic q-expansion principle for unitary groups of arbitrary signature in the literature. By replacing q-expansions with Serre–Tate expansions (expansions in terms of Serre–Tate deformation coordinates) and replacing modular forms with automorphic forms on unitary groups of arbitrary signature, we prove an analogue of the p-adic q-expansion principle. More precisely, we show that if the coefficients of (sufficiently many of) the Serre–Tate expansions of a p-adic automorphic form f on the Igusa tower (over a unitary Shimura variety) are zero, then f vanishes identically on the Igusa tower.This paper also contains a substantial expository component. In particular, the expository component serves as a complement to Hida’s extensive work on p-adic automorphic forms.

Keywords

Modular Form Unitary Group Abelian Variety Automorphic Form Closed Immersion 
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.

Notes

Acknowledgements

We are grateful to L. Long, R. Pries, and K. Stange for organizing the Women in Numbers 3 workshop and facilitating this collaboration. We would like to thank the referee for carefully reading the paper and providing many helpful comments, including suggestions for how to improve the introduction. We would also like to thank M. Harris, H. Hida, and K.-W. Lan for answering questions about q-expansion principles. We are grateful to the Banff International Research Station for creating an ideal working environment.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Ana Caraiani
    • 1
  • Ellen Eischen
    • 2
  • Jessica Fintzen
    • 3
  • Elena Mantovan
    • 4
  • Ila Varma
    • 1
  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Department of MathematicsUniversity of OregonEugeneUSA
  3. 3.Department of MathematicsHarvard UniversityCambridgeUSA
  4. 4.Department of MathematicsCalTech, PasadenaUSA

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