Propriétés arithmétiques de fonctions thêta à plusieurs variables. (French) Zbl 0546.14029
Number theory, Proc. Journ. arith., Noordwijkerhout/Neth. 1983, Lect. Notes Math. 1068, 17-22 (1984).
[For the entire collection see Zbl 0535.00008.]
Following Barsotti’s construction, the author defines a canonical theta function associated to certain divisors on abelian varieties admitting multiplications by a quadratic imaginary number field. Transcendence properties of these theta functions are established, and are applied to the study of p-adic heights on elliptic curves with complex multiplications. In particular, their isotropic vectors and more general orthogonality relations are completely determined [see the author, Prog. Math. 22, 1-11 (1982; Zbl 0488.10031), for previous results in this direction].
Following Barsotti’s construction, the author defines a canonical theta function associated to certain divisors on abelian varieties admitting multiplications by a quadratic imaginary number field. Transcendence properties of these theta functions are established, and are applied to the study of p-adic heights on elliptic curves with complex multiplications. In particular, their isotropic vectors and more general orthogonality relations are completely determined [see the author, Prog. Math. 22, 1-11 (1982; Zbl 0488.10031), for previous results in this direction].
MSC:
14K25 | Theta functions and abelian varieties |
11J81 | Transcendence (general theory) |
14K22 | Complex multiplication and abelian varieties |
14G25 | Global ground fields in algebraic geometry |