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p-adic q-expansion principles on unitary Shimura varieties

Caraiani, Ana and Eischen, Ellen and Fintzen, Jessica and Mantovan, Elena and Varma, Ila (2016) p-adic q-expansion principles on unitary Shimura varieties. In: Directions in Number Theory. Association for Women in Mathematics. No.3. Springer , Cham, pp. 197-243. ISBN 978-3-319-30974-3. http://resolver.caltech.edu/CaltechAUTHORS:20150424-124825553

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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.


Item Type:Book Section
Related URLs:
URLURL TypeDescription
https://dx.doi.org/10.1007/978-3-319-30976-7_7DOIArticle
https://link.springer.com/chapter/10.1007%2F978-3-319-30976-7_7PublisherArticle
http://rdcu.be/twg0PublisherFree ReadCube access
http://arxiv.org/abs/1411.4350arXivDiscussion Paper
Additional Information:© 2016 Springer International Publishing Switzerland. First Online: 27 September 2016. Ana Caraiani’s research is partially supported by NSF Postdoctoral Fellowship DMS-1204465 and NSF Grant DMS-1501064. Ellen Eischen’s research is partially supported by NSF Grants NSF DMS-1249384 and DMS-1559609. Jessica Fintzen’s research is partially supported by the Studienstiftung des deutschen Volkes. Elena Mantovan’s research is partially supported by NSF Grant DMS-1001077. Ila Varma’s research is partially supported by a National Defense Science and Engineering Fellowship. 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.
Funders:
Funding AgencyGrant Number
NSF Postdoctoral FellowshipDMS-1204465
NSFDMS-1501064
NSFDMS-1249384
NSFDMS-1559609
Studienstiftung des deutschen VolkesUNSPECIFIED
NSFDMS-1001077
National Defense Science and Engineering Graduate (NDSEG) FellowshipUNSPECIFIED
Record Number:CaltechAUTHORS:20150424-124825553
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20150424-124825553
Official Citation:Caraiani A., Eischen E., Fintzen J., Mantovan E., Varma I. (2016) p-Adic q-Expansion Principles on Unitary Shimura Varieties. In: Eischen E., Long L., Pries R., Stange K. (eds) Directions in Number Theory. Association for Women in Mathematics Series, vol 3. Springer, Cham
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:56958
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:24 Apr 2015 21:01
Last Modified:16 Jun 2017 20:27

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