## p-Adic q-Expansion Principles on Unitary Shimura Varieties

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 Serreâ€“Tate expansions (expansions in terms of Serreâ€“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 Serreâ€“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 Hidaâ€™s extensive work on p-adic automorphic forms.

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*Association for Women in Mathematics Series*(Vol. 3, pp. 197–243). https://doi.org/10.1007/978-3-319-30976-7_7

*Association for Women in Mathematics Series*, 3:197–243, 2016. https://doi.org/10.1007/978-3-319-30976-7_7.

*Association for Women in Mathematics Series*, vol. 3, 2016, pp. 197–243.

*Scopus*, doi:10.1007/978-3-319-30976-7_7.