The Hardy—Littlewood circle method

Abstract

One of the most significant all-purpose tools available in the study of rational points on higher-dimensional algebraic varieties is the Hardy—Littlewood circle method. In this chapter we will illustrate the power of this technique both as a theoretical tool and as a heuristic tool. In Section 8.2 we will establish Birch’s Theorem 1.1 in the case d=4 of quartic forms. Here, as in most applications of the circle method, the number of variables needed is rather large compared to the degree. Nonetheless, the circle method can still be used as a purely heuristic tool when the number of variables is smaller. Thus, in Section 8.3, we will provide some evidence for Manin’s Conjecture 2.3 in the setting of diagonal cubic surfaces.