Abstract
We prove the Ax-Lindemann theorem for the coarse moduli space $\mathcal{A}_{g}$ of principally polarized abelian varieties of dimension $g\ge 1$. We affirm the André-Oort conjecture unconditionally for $\mathcal{A}_g$ for $g\le 6$, and under GRH for all $g$.
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TITLE = {{$G$}-Functions and Geometry},
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@book {PR, MRKEY = {1278263},
AUTHOR = {Platonov, Vladimir and Rapinchuk, Andrei},
TITLE = {Algebraic Groups and Number Theory},
SERIES = {Pure Appl. Math.},
VOLUME = {139},
PUBLISHER = {Academic Press},
ADDRESS = {Boston, MA},
YEAR = {1994},
PAGES = {xii+614},
ISBN = {0-12-558180-7},
MRCLASS = {11E57 (11-02 20Gxx)},
MRNUMBER = {1278263},
ZBLNUMBER = {0841.20046},
} -
[SANSUC]
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@article {SANSUC, MRKEY = {0631309},
AUTHOR = {Sansuc, J.-J.},
TITLE = {Groupe de {B}rauer et arithmétique des groupes algébriques linéaires sur un corps de nombres},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
VOLUME = {327},
YEAR = {1981},
PAGES = {12--80},
ISSN = {0075-4102},
CODEN = {JRMAA8},
MRCLASS = {12G05 (10C30 14G25 20G30 20G35)},
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@book {SAT, MRKEY = {0591460},
AUTHOR = {Satake, Ichir{ô}},
TITLE = {Algebraic Structures of Symmetric Domains},
SERIES = {Kanô Mem. Lect.},
VOLUME = {4},
PUBLISHER = {Iwanami Shoten},
ADDRESS = {Tokyo},
YEAR = {1980},
PAGES = {xvi+321},
MRCLASS = {32-02 (17C35 32Mxx 53C35)},
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@book {SIEGEL, MRKEY = {0164063},
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MRNUMBER = {0164063},
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VOLUME = {25},
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author={Ullmo, Emmanuel},
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NOTE={\emph{Compositio},
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author={Ullmo, Emmanuel and Yafaev, Andrei},
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YEAR = {2011},
NUMBER = {2},
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NOTE={\emph{Bulletin de la SMF},
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} -
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} -
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AUTHOR = {Wilkie, A. J.},
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VOLUME = {9},
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MRNUMBER = {1398816},
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ZBLNUMBER = {0892.03013},
DOI = {10.1090/S0894-0347-96-00216-0},
}