[For the entire collection see Zbl 0542.00006.]
Let A be an abelian scheme over a discrete valuation ring R with K \(= field\) of fractions of R and with perfect residue field k of characteristic p, and S be the completion of \({\mathfrak O}_{A,0}\). For a divisor X on A not passing through the origin 0, by using the relation given by the theorem of cube, we can define a theta-series \(\theta_{X_ K}\) as an element of \(S_ K\). The author shows that if \(R=W(k)\), \(p\neq 0,2\); if the special fibre of A is ordinary and if X is totally symmetric, then \(\theta_ X=\theta_{X_ K}\) can be given as an element of S. Moreover, by using this result, he shows that the canonical decomposition of the first De Rham cohomology of A is related to the theta series.