Discrete tomography and the Hodge conjecture for certain abelian varieties of CM-type. (English) Zbl 1106.14004

Summary: We study a problem in discrete tomography on \(\mathbb Z^n\), and show that there is an intimate connection between the problem and the study of the Hodge cycles on abelian varieties of CM-type. This connection enables us to apply our results in tomography to obtain several infinite families of abelian varieties for which the Hodge conjecture holds.


14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14K05 Algebraic theory of abelian varieties
39A10 Additive difference equations
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[1] R. E. Edwards, Fourier series, a modern introduction, vol. 2, 2nd. ed., Graduate Texts in Mathematics 85, Springer, New York, 1982. · Zbl 0599.42001
[2] F. Hazama, Algebraic cycles on certain abelian varieties and powers of special surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1985), no. 3, 487-520. · Zbl 0591.14006
[3] F. Hazama, Hodge cycles on abelian varieties with complex multiplication by cyclic CM-fields, J. Math. Sci. Univ. Tokyo 10 (2003), no. 4, 581-598. · Zbl 1061.14008
[4] K. A. Ribet, Division fields of abelian varieties with complex multiplication, Mém. Soc. Math. France (N.S.) 2 (1980), no. 2, 75-94. · Zbl 0452.14009
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