By I.G. Macdonald

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Bombieri and H. P. F. Swinnerton-Dyer, On the local zeta function of a cubic threefold, Ann. Scuola Norm. Sup. Pisa (3) 21 (1967) 1–29 [23] H. P. F. C. (1967), pp. 280–291 [24] H. P. F. Swinnerton-Dyer, A4 +B 4 = C 4 +D4 revisited, J. London Math. Soc. 43 (1968) 149–151 [25] P. Swinnerton-Dyer, The conjectures of Birch and Swinnerton-Dyer, and of Tate, in Proc. Conf. Local Fields (Driebergen 1966), Springer, Berlin (1967), pp. 132–157 [26] P. Swinnerton-Dyer, The use of computers in the theory of numbers, in Proc.

Soc. (2) 29 (1984) 509–520 [59] Arnaud Beauville, Jean-Louis Colliot-Th´el`ene, Jean-Jacques Sansuc et Peter Swinnerton-Dyer, Vari´et´es stablement rationnelles non rationnelles, Ann. of Math. (2) 121 (1985) 283–318 [60] H. P. F. Swinnerton-Dyer, The ﬁeld of deﬁnition of the N´eron-Severi group, in Studies in pure mathematics in memory of Paul Tur´an, Birkh¨auser, Basel (1983), pp. 719–731 [61] Jean-Louis Colliot-Th´el`ene, Jean-Jacques Sansuc and Peter SwinnertonDyer, Intersections of two quadrics and Chˆatelet surfaces.

Suppose now that r ≥ ρh (a). 1 yields |1 − α |v ≥ H (α )−rκ . 1 of [3] yields the cleaner bound stated here. e. h(α ) < 1/Dv∗ or equivalently h(Aξ) < r/Dv∗ . 10) for h(Aξ). 1. 11) h (ξi ), i=1 tDv∗ . 7). 11), r ≥ 4 and h(ξ) ≤ h(Aξ) + h (A), we have 4 4 1 1 r + h(ξ) ≤ 2 h (A) + r + h(Aξ). 12) 2ρ r 2ρ r Now we choose N to be N = 2(pfv − 1)Q 1 + max 8ρh (A), 16ρh(Aξ) . 10), holds with this choice of N . 10) we have h (a) ≤ h(Aξ) ≤ (Dv∗ )−1 2(pfv − 1)Q 1 + max 8ρh (A), 16ρh(Aξ) + 3. The ﬁrst alternative for the maximum easily yields h(Aξ) ≤ 16pfv ρh (A)Q, because ρh (A)Q is fairly large (use ρ ≥ 33Dv∗ to get ρh (A)Q ≥ (66t)t ), hence the small corrections in going from 1 + max to max and in removing the ceiling brackets and the constant 3 are easily absorbed in replacing pfv −1 by pfv .