Level structures on abelian varieties, Kodaira dimensions, and Lang’s conjecture. (English) Zbl 1405.14115
Let \(K\) be a number field. Lang’s conjecture says that if \(X\) is a positive-dimensional algebraic variety of general type over \(K\), then \(X(K)\) is not Zariski-dense in \(\mathbb{C}\). Let \(p\) be a prime number, and \(g\) a positive integer. Assuming Lang’s conjecture, the authors show that there exists a positive integer \(r\) such that no principally polarized \(g\)-dimensional abelian variety over \(K\) has full level-\(p^r\) structure. The authors rely on a result of K. Zuo [Asian J. Math. 4, No. 1, 279–301 (2000; Zbl 0983.32020)] to show that certain subvarieties of the moduli space of principally polarized abelian varieties of dimension \(g\) with level-\(m\) structure are of general type for sufficiently large \(m\), and thus satisfy the hypotheses of Lang’s conjecture. They point out that a recent result of Y. Brunebarbe [“A strong hyperbolicity property of locally symmetric varieties”, Preprint, arXiv:1606.03972] leads to stronger results. They also point out how the restriction to principally polarized abelian varieties may be removed, leaving the details to the reader.
Reviewer’s note: The statement of the main theorem in the published abstract omits the essential hypothesis that the abelian varieties be \(g\)-dimensional.
Reviewer’s note: The statement of the main theorem in the published abstract omits the essential hypothesis that the abelian varieties be \(g\)-dimensional.
Reviewer: Salman Abdulali (Greenville)
MSC:
| 14K10 | Algebraic moduli of abelian varieties, classification |
| 14K15 | Arithmetic ground fields for abelian varieties |
| 11G18 | Arithmetic aspects of modular and Shimura varieties |
| 11G10 | Abelian varieties of dimension \(> 1\) |
Citations:
Zbl 0983.32020Software:
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