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Arithmetic of diagonal quartic surfaces, II
Author(s) -
SwinnertonDyer Peter
Publication year - 2000
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/s0024611500012302
Subject(s) - mathematics , quartic function , solubility , rational number , diagonal , order (exchange) , pure mathematics , square (algebra) , mathematics subject classification , hasse principle , combinatorics , algebraic number field , geometry , organic chemistry , chemistry , finance , economics
The author investigates the solubility in rationals of equations of the forma 0 X 0 4 + a 1 X 1 4 + a 2 X 2 4 + a 3 X 3 4 = 0 where a 0 a 1 a 2 a 3 is a square, building on the ideas which Colliot‐Thène, Skorobogatov and he have developed; see Invent. Math . 134 (1998) 579–650. He obtains sufficient conditions for solubility, which appear to be related to the absence of a Brauer–Manin obstruction. This represents the first large family of K3 surfaces which almost satisfy the Hasse principle, in the sense that the auxiliary condition which ensures that local solubility everywhere implies global solubility is nearly always satisfied. 1991 Mathematics Subject Classification : 10B10.