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Mathematics of Computation

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The density of zeros of forms for which weak approximation fails

Author: D. R. Heath-Brown
Journal: Math. Comp. 59 (1992), 613-623
MSC: Primary 11G35; Secondary 11D25, 11P55
MathSciNet review: 1146835
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Abstract: The weak approximation principle fails for the forms $ {x^3} + {y^3} + {z^3} = k{w^3}$ , when $ k = 2$ or 3. The question therefore arises as to what asymptotic density one should predict for the rational zeros of these forms. Evidence, both numerical and theoretical, is presented, which suggests that, for forms of the above type, the product of the local densities still gives the correct global density.

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Keywords: Cubic surfaces, weak approximation, Brauer-Manin obstruction, Hardy-Littlewood formula, asymptotic estimates
Article copyright: © Copyright 1992 American Mathematical Society