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Association for Women in Mathematics Series

p-Adic q-Expansion Principles on Unitary Shimura Varieties

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Caraiani, A; Eischen, E; Fintzen, J; Mantovan, E; Varma, I
January 1, 2016

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.

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Publication Date

January 1, 2016

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3

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197 / 243
 

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Caraiani, A., Eischen, E., Fintzen, J., Mantovan, E., & Varma, I. (2016). p-Adic q-Expansion Principles on Unitary Shimura Varieties. In Association for Women in Mathematics Series (Vol. 3, pp. 197–243). https://doi.org/10.1007/978-3-319-30976-7_7
Caraiani, A., E. Eischen, J. Fintzen, E. Mantovan, and I. Varma. “p-Adic q-Expansion Principles on Unitary Shimura Varieties.” In Association for Women in Mathematics Series, 3:197–243, 2016. https://doi.org/10.1007/978-3-319-30976-7_7.
Caraiani A, Eischen E, Fintzen J, Mantovan E, Varma I. p-Adic q-Expansion Principles on Unitary Shimura Varieties. In: Association for Women in Mathematics Series. 2016. p. 197–243.
Caraiani, A., et al. “p-Adic q-Expansion Principles on Unitary Shimura Varieties.” Association for Women in Mathematics Series, vol. 3, 2016, pp. 197–243. Scopus, doi:10.1007/978-3-319-30976-7_7.
Caraiani A, Eischen E, Fintzen J, Mantovan E, Varma I. p-Adic q-Expansion Principles on Unitary Shimura Varieties. Association for Women in Mathematics Series. 2016. p. 197–243.

DOI

Publication Date

January 1, 2016

Volume

3

Start / End Page

197 / 243