[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].