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Classicality for small slope overconvergent automorphic forms on some compact PEL Shimura varieties of type C

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We study the rigid cohomology of the ordinary locus in some compact PEL Shimura varieties of type C with values in automorphic local systems and use it to prove a small slope criterion for classicality of overconvergent Hecke eigenforms. This generalises work of Coleman, and is a first step in an ongoing project to extend the cohomological approach to classicality to higher-dimensional Shimura varieties.

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The author would like to thank his PhD supervisor Kevin Buzzard for suggesting this problem and for his constant help and encouragement during every aspect of this project. He would also like to thank his second supervisor Toby Gee for valuable advice during the write-up, as well as Wansu Kim, Christopher Lazda, James Newton, Shu Sasaki and Teruyoshi Yoshida for many helpful discussions relating to this work, and Francesco Baldassarri and Bernard Le Stum for answering questions about rigid and overconvergent de Rham cohomology. The author wishes to thank the Engineering and Physical Sciences Research Council for supporting him throughout his doctoral studies. It is also a pleasure to thank the Fields Institute, where part of the write-up of this paper was done, for their support and hospitality as well as for the excellent working conditions provided. Finally the author wishes to thank the anonymous referee for correcting some typos and for insightful comments. In the first version of this paper, \(p\) was assumed to be inert in \(F\). The author wishes to sincerely thank the referee for pointing out that the methods should extend to the case of \(p\) unramified, and for urging the author to investigate the general case.

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Correspondence to Christian Johansson.

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Johansson, C. Classicality for small slope overconvergent automorphic forms on some compact PEL Shimura varieties of type C. Math. Ann. 357, 51–88 (2013). https://doi.org/10.1007/s00208-013-0900-y

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Mathematics Subject Classification (2000)

  • 11F33
  • 11G18
  • 14F30