Counting rational points on cubic and quartic surfaces
نویسندگان
چکیده
منابع مشابه
Rational Points on Cubic Surfaces
Let k be an algebraic number eld and F (x0; x1; x2; x3) a non{singular cubic form with coeecients in k. Suppose that the pro-jective cubic k{surface X P 3 k given by F = 0 contains three coplanar lines deened over k, and let U (k) be the set of k{points on X which does not lie on any line on X. We show that the number of points in U (k), with height at most B, is OF;"(B 4=3+") for any " > 0.
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R0<b162R0 gcd(b1, N )1/2 R 0 (HP) . The second line is false and in fact one has M1 = 1 in Proposition 3. The author is very grateful to Professor Hongze Li for drawing his attention to this flaw. The error can be fixed by introducing an average over b1 into the statement of Proposition 3. This allows us to recover the main theorem in [1], and also [2, Lemma 11], via the following modification....
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Let V be a non-singular surface defined over Q which is embedded in projective space P by means of anticanonical divisors, and let U be the open subset of V obtained by deleting the lines on V . For any point P in U(Q) denote by h(P ) the height of P . In this paper h will usually be the standard height h1(P ) = max(|x0|, . . . , |xn|) where P = (x0, . . . , xn) for integers xi with highest com...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 2003
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa108-3-7