The Hasse–Weil zeta function is a mathematical tool used in number theory and algebraic geometry, particularly in the study of algebraic varieties over finite fields and their properties. It generalizes the classical Riemann zeta function and serves as an important object in understanding the distribution of points on algebraic varieties defined over finite fields.
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The Hasse–Weil zeta function of an algebraic variety over a number field is assembled from local point-counting factors at finite places. At a good place with residue field of size , the factor is , where . Defining all bad-place factors requires a specified integral model or convention. The zeta function of an elliptic curve over a finite field is a local factor, rather than this global product.