Curves, function fields and the Riemann hypothesis (Q2744683)

According to the author, this booklet is a progress report purporting ``to establish the correspondence between curves and function fields in one variable'' and then to give a proof of the celebrated theorem of A. Weil, the ``Riemann Hypothesis'' for a smooth projective curve defined over a finite field. Apart from some elementary prerequisites from commutative algebra, the Riemann-Roch theorem for curves, and the theorem on the existence of a smooth model of a curve, which are stated without proof, the author's exposition is self-contained. Weil's ``Riemann Hypothesis'' is proved by a version of Stepanov's method going back to \textit{E. Bombieri} [Proc. Symp. Pure Math. 28, De Kalb 1974, 269-274 (1976; Zbl 0351.14009)]. The reader interested in this history of and the literature on this subject may consult, for instance, the monograph by \textit{S. A. Stepanov} [Arithmetic of algebraic curves, Monographs in Contemporary Mathematics (1994; Zbl 0862.11036)].