A Survey of Applications of the Circle Method to Rational Points
نویسنده
چکیده
of adèlic points under the product topology? Can one count the points in X(k) of bounded height? In favourable circumstances, the Hardy–Littlewood circle method can systematically provide answers to all of these questions. The focus of this survey will be upon the first and most basic of these questions: when is X(k) 6= ∅ for a given variety X defined over k? In considering what the circle method has to say about this we shall usually restrict our attention to varieties X defined over k that are geometrically integral, non-singular and projective. Recall that such a family is said to satisfy the Hasse principle if any variety in the family has a k-point as soon as it has a point in every completion of k. Sometimes we will drop the assumption on non-singularity, saying that the smooth Hasse principle holds for a family of such varieties when the Hasse principle holds for the smooth locus of X. Quadrics are among the first examples of families satisfying the Hasse principle. Beyond this simple setting, it is quite rare to find varieties which satisfy the Hasse principle. In dimension 2 we have the following counter-example.
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