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Corrigenda Arithmetic of Diagonal Quartic Surfaces, II
Author(s) -
SwinnertonDyer Peter
Publication year - 2002
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611502013990
Subject(s) - mathematics , quartic function , diagonal , arithmetic , quartic surface , algebra over a field , discrete mathematics , pure mathematics , geometry
Equation (14) should read d 0 i / d jk = a 0 a i /θ=θ/ a j a k (that is, the minus signs should be deleted). Equation (34) should read F 0 i F jk = L ( d ij d 0 k , c 0 i )= L ( a i a j θ,c 0 i )= ∏ v ∈ B 1( a i a j θ , c 0 i) v(that is, the subscript k in the two right‐hand expressions should be replaced by j ). There are systematic errors in the calculations of F ij F ik F il in §7, and therefore Lemma 12 (and the last paragraphs of its proof) are wrong. In fact F ij F ik F il = F 0 i F jk are the three elements (43) of the Brauer group are equal to the element (44) up to multiplication by a constant algebra; this has been shown directly by Colliot‐Thélène, and in the case when none of the ± a i a j is a square it also follows from the comprehensive tables contained in Martin Bright's Cambridge PhD thesis. Notwithstanding this, the surface (46) is indeed insoluble in Q. This follows from the fact that F 01 =−1 for it, as can easily be checked by direct computation. 2000 Mathematical Subject Classification : 10B10.