In our previous paper [cf. T. Terasoma, J. Math. Angew. Math. 433, 143-159 (1992; Zbl 0753.14022)], we showed that the determinant formula for the cohomology of a local system on a curve which corresponds to a finite image representation of fundamental group comes from an algebraic correspondence. To construct this correspondence, we used a so called symmetric construction for the locally constant sheaf on the symmetric product of curves. By using similar methods, we can interpret the Gauss multiplication formula in terms of algebraic cycles. We will discuss this correspondence in §1. In this paper, we will give an example of multiplication formula for hypergeometric functions and interpret this formula in geometric language. To find such relations, we will do some experimental computations and as a consequence, we get a nontrivial relation between periods in §2. In this section, we concentrate on the Gamma series expression given by I. M. Gel’fand, A. V. Zelevinski and H. M. Kapranov [Sov. Math., Dokl. 37, No. 3, 678-682 (1988); translation from Dokl. Akad. Nauk SSSR 300, No. 3, 529-534 (1988; Zbl 0667.33010)]. We will interpret the proof of §1 in terms of étale cohomology in §3. There we get a relation between character sums called multiplication formula for Gauss sums. Next section, we treat the similar relation between character sums analogous to the relation between periods established in §2. We will give only an outline of the proof of this relation (only the typical case we proved here). A detailed proof will be given in a forthcoming paper in a more general setting. In \(\S 5\), we will give a geometric interpretation of these equalities in cohomological language. In this section, we use the result of \(\S 4\).
For the entire collection see [Zbl 0838.00011].