In the last few years, it has become evident that the study of the zeta-function of an algebraic variety can yield valuable information about that variety. These facts grew out of the attempt to apply Siegel's works on quadratic forms to elliptic curves. Many extensive computations concerning rational points on elliptic curves were done by Birch and the author and through these works they published many important conjectures. Then J. Tate has put forward and produced evidence for these conjectures. The present article is an exposition of those conjectures on zeta-functions and $L$-series of (1) elliptic curves over $Q$, (2) elliptic curves with complex multiplication, (3) elliptic curves parametrized by modular functions, (4) abelian varieties over algebraic number fields, (5) varieties over finite fields, and (6) varieties over algebraic number fields.

Reviewed by Y. Nakai