Rational points on nonsingular cubic hypersurfaces

1994 ◽  
Vol 75 (2) ◽  
pp. 409-466 ◽  
Author(s):  
Christopher M. Skinner
Keyword(s):  
Rational Points ◽  
Cubic Hypersurfaces

The Lang–Weil Estimate for Cubic Hypersurfaces

2013 ◽  
Vol 56 (3) ◽  
pp. 500-502 ◽  
Author(s):  
T. D. Browning
Keyword(s):  
Rational Points ◽  
Cubic Hypersurfaces ◽  
Weil Estimate

AbstractAn improved estimate is provided for the number of 𝔽q-rational points on a geometrically irreducible, projective, cubic hypersurface that is not equal to a cone.

scholarly journals Rational points on cubic hypersurfaces that split off a form. With an appendix by J.-L. Colliot-Thélène

2010 ◽  
Vol 146 (4) ◽  
pp. 853-885 ◽  
Author(s):  
T. D. Browning
Keyword(s):  
Rational Points ◽  
Cubic Form ◽  
Cubic Hypersurfaces

AbstractLet X be a projective cubic hypersurface of dimension 11 or more, which is defined over ℚ. We show that X(ℚ) is non-empty provided that the cubic form defining X can be written as the sum of two forms that share no common variables.

scholarly journals How to calculate the proportion of everywhere locally soluble diagonal hypersurfaces

2021 ◽  
pp. 1-17
Author(s):  
Yoshinosuke Hirakawa ◽  
Yoshinori Kanamura
Keyword(s):  
Worked Examples ◽  
Rational Points ◽  
Product Formula ◽  
Fixed Degree ◽  
Cubic Hypersurfaces

In this paper, we establish a strategy for the calculation of the proportion of everywhere locally soluble diagonal hypersurfaces of [Formula: see text] of fixed degree. Our strategy is based on the product formula established by Bright, Browning and Loughran. Their formula reduces the problem into the calculation of the proportions of [Formula: see text]-soluble diagonal hypersurfaces for all places [Formula: see text]. As worked examples, we carry out our strategy in the cases of quadratic and cubic hypersurfaces. As a consequence, we prove that around [Formula: see text] of diagonal cubic fourfolds have [Formula: see text]-rational points under a hypothesis on the Brauer–Manin obstruction.

RATIONAL POINTS ON CUBIC HYPERSURFACES THAT SPLIT INTO FOUR FORMS

Mathematika ◽  
2016 ◽  
Vol 62 (2) ◽  
pp. 430-440 ◽  
Author(s):  
Boqing Xue ◽  
Lilu Zhao
Keyword(s):  
Rational Points ◽  
Cubic Hypersurfaces

scholarly journals Rational lines on cubic hypersurfaces

2020 ◽  
pp. 1-14
Author(s):  
JULIA BRANDES ◽  
RAINER DIETMANN
Keyword(s):  
Recent Result ◽  
Single Point ◽  
Rational Points ◽  
Cubic Hypersurfaces ◽  
Rational Line

Abstract We show that any smooth projective cubic hypersurface of dimension at least 29 over the rationals contains a rational line. A variation of our methods provides a similar result over p-adic fields. In both cases, we improve on previous results due to the second author and Wooley. We include an appendix in which we highlight some slight modifications to a recent result of Papanikolopoulos and Siksek. It follows that the set of rational points on smooth projective cubic hypersurfaces of dimension at least 29 is generated via secant and tangent constructions from just a single point.

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