On octonary cubic forms
type: Item
instance of: scholarly article
title: On octonary cubic forms
zbMATH Open document ID: 1343.11044
author: Christopher Hooley
publication date: 2014-08-08T00:00:00Z
Mathematics Subject Classification ID: 11D72
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 11P55
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 11E76
zbMATH DE Number: 6327948
zbMATH Keywords: cubic form
zbMATH Keywords: rational point
zbMATH Keywords: Hasse principle
zbMATH Keywords: zero
zbMATH Keywords: nonsingular
zbMATH Keywords: eight variables
zbMATH Keywords: conditional
zbMATH Keywords: Hasse-Weil \(L\)-function
zbMATH Keywords: Riemann hypothesis
MaRDI profile type: Publication
full work available at URL: pdt066
OpenAlex ID: W1967933835
cites work: The Distribution of Galois Groups and Hilbert's Irreducibility Theorem
cites work: Cubic forms in sixteen variables
cites work: La conjecture de Weil. I
cites work: Cubic Forms in Ten Variables
cites work: A new form of the circle method, and its application to quadratic forms.
cites work: The circle method and diagonal cubic forms
cites work: Cubic forms in 14 variables
cites work: On Waring's problem
cites work: On nonary cubic forms.
cites work: On nonary cubic forms: II.
cites work: On the number of points on a complete intersection over a finite field. (Appendix: Number of points on singular complete intersections by Nicholas M. Katz)
cites work: On nonary cubic forms: III.
DOI: 10.1112/PLMS/PDT066
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published in: Proceedings of the London Mathematical Society. Third Series
review text: This paper gives a conditional proof of the Hasse principle for nonsingular cubic forms defined over the rationals, with 8 (or more) variables. It represents significant progress in our handling of cubic forms via analytic methods.NEWLINENEWLINEGenerally, if \(C(x_1,\ldots,x_n)\) is a nonsingular cubic form with rational integer coefficients, then it was shown by the reviewer [Proc. Lond. Math. Soc. (3) 47, 225--257 (1983; Zbl 0494.10012)] that \(C=0\) always has a non-trivial integer solution as soon as \(n\geq 10\). This was improved by the author of the present paper [J. Reine Angew. Math. 386, 32--98 (1988; Zbl 0641.10019)], who was able to handle the corresponding situation for nonsingular cubics in \(n=9\) variables, for which the Hasse principle was established. Indeed in subsequent papers the smoothness condition was relaxed somewhat.NEWLINENEWLINEThe papers cited above used the circle method with Kloosterman's refinement. Thus one counts solutions of \(C=0\) using a generating function evaluated at rational points \(a/q\), and obtains a saving by averaging non-trivially over the numerators \(a\). In principle one can hope to perform a ``second Kloosterman refinement'', in which one extracts a further saving by averaging non-trivially over the denominators \(q\). The author first demonstrated the feasibility of this with the special cubic form in \(n=6\) variables, formed as a difference of sums of 3 cubes [Acta Math. 157, 49--97 (1986; Zbl 0614.10038)]. The necessarily cancellation as one sums over \(q\) is not currently available unconditionally, but requires some plausible assumptions about the Hasse-Weil \(L\)-functions attached to hyperplane slices of the variety \(C=0\).NEWLINENEWLINEThe present paper extends the above techniques to general nonsingular cubic hypersurfaces \(C=0\). If the method could be pushed through completely it would handle forms in \(n=7\) variables, but the current work contents itself with the case \(n=8\). There are many techni…
reviewed by: D. R. Heath-Brown
description: scientific article; zbMATH DE number 6327948