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- W2000343739 abstract "LetK be a number field. Denote byV 3 a split Del Pezzo surface of degree six overK and by ω its canonical divisor. Denote byW 3 the open complement of the exceptional lines inV 3. LetN W s(−ω, X) be the number ofK-rational points onW 3 whose anticanonical heightH −ω is bounded byX. Manin has conjectured that asymptoticallyN W 3(−ω, X) tends tocX(logX)3, wherec is a constant depending only on the number field and on the normalization of the height. Our goal is to prove the following theorem: For each number fieldK there exists a constantc K such thatN W 3(−ω, X)≤cKX(logX)3+2r , wherer is the rank of the group of units ofO K. The constantc K is far from being optimal. However, ifK is a purely imaginary quadratic field, this proves an upper bound with a correct power of logX. The proof of Manin's conjecture for arbitrary number fields and a precise treatment of the constants would require a more sophisticated setting, like the one used by [Peyre] to prove Manin's conjecture and to compute the correct asymptotic constant (in some normalization) in the caseK=ℚ. Up to now the best result for arbitraryK goes back, as far as we know, to [Manin-Tschinkel], who gives an upper boundN W 3(−ω,X)≤cXl+ε. The author would like to express his gratitude to Daniel Coray and Per Salberger for their generous and indispensable support." @default.
- W2000343739 created "2016-06-24" @default.
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- W2000343739 date "1995-12-01" @default.
- W2000343739 modified "2023-09-26" @default.
- W2000343739 title "Rational points of bounded height on Del Pezzo surfaces of degree six" @default.
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- W2000343739 doi "https://doi.org/10.1007/bf02566015" @default.
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