Abstract
We prove new theorems that are higher-dimensional generalizations of the classical theorems of Siegel on integral points on affine curves and of Picard on holomorphic maps from $\mathbb{C}$ to affine curves. These include results on integral points over varying number fields of bounded degree and results on Kobayashi hyperbolicity. We give a number of new conjectures describing, from our point of view, how we expect Siegel’s and Picard’s theorems to optimally generalize to higher dimensions.
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@incollection{Vo5, MRKEY = {1748613},
AUTHOR = {Vojta, Paul},
EDITOR = {Schneider, Michael and Siu, Yum-Tong},
TITLE = {Nevanlinna theory and {D}iophantine approximation},
BOOKTITLE = {Several Complex Variables},
SERIES = {Math. Sci. Res. Inst. Publ.},
NUMBER = {37},
PAGES = {535--564},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1999},
MRCLASS = {11J97 (30D35 32H30)},
MRNUMBER = {2001j:11072},
ZBLNUMBER = {0960.32013},
MRREVIEWER = {William A. Cherry},
} -
[Co4]
U. Zannier, "On the integral points on the complement of ramification-divisors," J. Inst. Math. Jussieu, vol. 4, iss. 2, pp. 317-330, 2005.
@article{Co4, MRKEY = {2135140},
AUTHOR = {Zannier, Umberto},
TITLE = {On the integral points on the complement of ramification-divisors},
JOURNAL = {J. Inst. Math. Jussieu},
FJOURNAL = {Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l'Institute de Mathématiques de Jussieu},
VOLUME = {4},
YEAR = {2005},
NUMBER = {2},
PAGES = {317--330},
ISSN = {1474-7480},
MRCLASS = {11D72 (11G35 14G99)},
MRNUMBER = {2005m:11055},
DOI = {10.1017/S1474748005000071},
ZBLNUMBER = {1089.11021},
MRREVIEWER = {Clemens Fuchs},
} -
[Za]
O. Zariski, "The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface," Ann. of Math., vol. 76, pp. 560-615, 1962.
@article{Za, MRKEY = {0141668},
AUTHOR = {Zariski, Oscar},
TITLE = {The theorem of {R}iemann-{R}och for high multiples of an effective divisor on an algebraic surface},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {76},
YEAR = {1962},
PAGES = {560--615},
ISSN = {0003-486X},
MRCLASS = {14.20 (14.15)},
MRNUMBER = {25 \#5065},
DOI = {10.2307/1970376},
ZBLNUMBER = {0124.37001},
MRREVIEWER = {Y. Nakai},
}