CM periods, CM regulators and hypergeometric functions, I

We study the $H^2$ of certain surfaces with complex multiplication by a cyclotomic field. The periods are written in terms of values of the gamma function and the conjecture of Gross-Deligne is verified. The regulators of certain $K_1$-elements are written in terms of values of hypergeometric functions ${}_3F_2$, and we prove their non-vanishing.