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- W1553033667 abstract "NF (P ) = N(P ) = #{x ∈ Z4 : F (x) = 0, |x| ≤ P}, where |x| is the Euclidean length of x. This paper is concerned with the behaviour of N(P ) as P tends to infinity. It is clear that if the surface F = 0 contains a rational line, then points on that line will contribute cP 2+O(P ) to N(P ) for an appropriate constant c > 0. It appears that these contributions play the dominant role in determining the behaviour of N(P ), and we therefore define N (0)(P ) to be the number of points x counted by N(P ), for which there is no rational line in the surface F = 0 which contains x. Of course it is quite possible that the surface F = 0 contains no rational lines, in which case N (0)(P ) = N(P ). Alternatively it may happen that all points lie on rational lines. This will certainly be the case whenever F is degenerate, but also occurs in examples such as F = WX2 − Y Z2. Here a point (a, b, c, d) with (b, d) 6= (0, 0) lies on the line b2W = d2Y , dX = bZ, while if b = d = 0, say, then the point is on the line X = Z = 0. In general, however, we may regard N (0)(P ) as counting “non-trivial” points on F = 0. Thus, taking" @default.
- W1553033667 created "2016-06-24" @default.
- W1553033667 creator A5086566271 @default.
- W1553033667 date "1997-01-01" @default.
- W1553033667 modified "2023-10-17" @default.
- W1553033667 title "The density of rational points on cubic surfaces" @default.
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- W1553033667 doi "https://doi.org/10.4064/aa-79-1-17-30" @default.
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