Generalizations of rigid analytic Picard theorems

Lin, Chien-Wei; Wang, Julie

doi:10.1090/S0002-9939-09-10038-2

Generalizations of rigid analytic Picard theorems
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by Chien-Wei Lin and Julie Tzu-Yueh Wang PDF

Proc. Amer. Math. Soc. 138 (2010), 133-139 Request permission

Abstract:

Berkovich’s Picard theorem states that there are no non-constant analytic maps from the affine line to the complement of two points on a nonsingular projective curve. The purpose of this article is to find generalizations of this result in higher dimensional varieties.

References

Ta Thi Hoai An, A defect relation for non-Archimedean analytic curves in arbitrary projective varieties, Proc. Amer. Math. Soc. 135 (2007), no. 5, 1255–1261. MR 2276632, DOI 10.1090/S0002-9939-06-08591-1
An, T. T. H.; Cherry, W.; Wang, J. T.-Y., Algebraic degeneracy of non-Archimedean analytic maps, Indagationes Mathematicae 19 (2008), no. 3, 481-492.
Ta Thi Hoai An, Julie Tzu-Yueh Wang, and Pit-Mann Wong, Non-Archimedean analytic curves in the complements of hypersurface divisors, J. Number Theory 128 (2008), no. 8, 2275–2281. MR 2394821, DOI 10.1016/j.jnt.2007.10.011
Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, 1990. MR 1070709, DOI 10.1090/surv/033
Pietro Corvaja and Umberto Zannier, A subspace theorem approach to integral points on curves, C. R. Math. Acad. Sci. Paris 334 (2002), no. 4, 267–271 (English, with English and French summaries). MR 1891001, DOI 10.1016/S1631-073X(02)02240-9
P. Corvaja and U. Zannier, On integral points on surfaces, Ann. of Math. (2) 160 (2004), no. 2, 705–726. MR 2123936, DOI 10.4007/annals.2004.160.705
János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959, DOI 10.1017/CBO9780511662560
Levin, A., Generalizations of Siegel’s and Picard’s theorems, Ann. of Math., to appear.
Yuancheng Liu and Min Ru, Degeneracy of holomorphic curves in surfaces, Sci. China Ser. A 48 (2005), no. suppl., 156–167. MR 2156497, DOI 10.1007/BF02884702
Alain M. Robert, A course in $p$-adic analysis, Graduate Texts in Mathematics, vol. 198, Springer-Verlag, New York, 2000. MR 1760253, DOI 10.1007/978-1-4757-3254-2
Min Ru, A note on $p$-adic Nevanlinna theory, Proc. Amer. Math. Soc. 129 (2001), no. 5, 1263–1269. MR 1712881, DOI 10.1090/S0002-9939-00-05680-X