Duke Mathematical Journal

On the Hasse principle for quartic hypersurfaces

O. Marmon and P. Vishe

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Abstract

We establish the Hasse principle for smooth projective quartic hypersurfaces of dimension greater than or equal to 28 defined over Q.

Article information

Source
Duke Math. J., Volume 168, Number 14 (2019), 2727-2799.

Dates
Received: 20 December 2017
Revised: 19 December 2018
First available in Project Euclid: 18 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1568794037

Digital Object Identifier
doi:10.1215/00127094-2019-0025

Subjects
Primary: 11D72: Equations in many variables [See also 11P55]
Secondary: 11P55: Applications of the Hardy-Littlewood method [See also 11D85] 14G05: Rational points

Keywords
circle method Hasse principle quartic hypersurfaces rational points analytic number theory forms in many variables

Citation

Marmon, O.; Vishe, P. On the Hasse principle for quartic hypersurfaces. Duke Math. J. 168 (2019), no. 14, 2727--2799. doi:10.1215/00127094-2019-0025. https://projecteuclid.org/euclid.dmj/1568794037


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References

  • [1] B. J. Birch, Forms in many variables, Proc. Roy. Soc. Ser. A 265 (1961/1962), no. 1321, 245–263.
  • [2] T. D. Browning and D. R. Heath-Brown, Rational points on quartic hypersurfaces, J. Reine Angew. Math. 629 (2009), 37–88.
  • [3] T. D. Browning and S. M. Prendiville, Improvements in Birch’s theorem on forms in many variables, J. Reine Angew. Math. 731 (2017), 203–234.
  • [4] T. D. Browning and P. Vishe, Cubic hypersurfaces and a version of the circle method for number fields, Duke Math. J. 163 (2014), no. 10, 1825–1883.
  • [5] T. D. Browning and P. Vishe, Rational curves on smooth hypersurfaces of low degree, Algebra Number Theory 11 (2017), no. 7, 1657–1675.
  • [6] H. Davenport, On a principle of Lipschitz, J. London Math. Soc. 26 (1951), no. 3, 179–183.
  • [7] W. Duke, J. Friedlander, and H. Iwaniec, Bounds for automorphic $L$-functions, Invent. Math. 112 (1993), no. 1, 1–8.
  • [8] S. R. Ghorpade and G. Lachaud, Étale cohomology, Lefschetz theorems and number of points of singular varieties over finite fields, Mosc. Math. J. 2 (2002), no. 3, 589–631.
  • [9] M. A. Hanselmann, Rational points on quartic hypersurfaces, Ph.D. dissertation, Ludwig-Maximilians-Universität München, Munich, 2012.
  • [10] D. R. Heath-Brown, Cubic forms in ten variables, Proc. Lond. Math. Soc. (3) 47 (1983), no. 2, 225–257.
  • [11] D. R. Heath-Brown, A new form of the circle method, and its application to quadratic forms, J. Reine Angew. Math. 481 (1996), 149–206.
  • [12] D. R. Heath-Brown, Cubic forms in $14$ variables, Invent. Math. 170 (2007), no. 1, 199–230.
  • [13] C. Hooley, On the number of points on a complete intersection over a finite field, with an appendix by N. M. Katz, J. Number Theory 38 (1991), no. 3, 338–358.
  • [14] C. Hooley, On octonary cubic forms, Proc. Lond. Math. Soc. (3) 109 (2014), no. 1, 241–281.
  • [15] N. M. Katz, Estimates for “singular” exponential sums, Int. Math. Res. Not. IMRN 1999, no. 16, 875–899.
  • [16] H. D. Kloosterman, On the representation of numbers in the form $ax^{2}+by^{2}+cz^{2}+dt^{2}$, Acta Math. 49 (1927), no. 3, 407–464.
  • [17] O. Marmon, The density of integral points on complete intersections, with an appendix by P. Salberger, Q. J. Math. 59 (2008), no. 1, 29–53.
  • [18] O. Marmon, The density of integral points on hypersurfaces of degree at least four, Acta Arith. 141 (2010), no. 3, 211–240.
  • [19] R. Munshi, Pairs of quadrics in $11$ variables, Compos. Math. 151 (2015), no. 7, 1189–1214.
  • [20] B. Poonen and J. F. Voloch, “Random Diophantine equations,” with appendices by J.-L. Colliot-Thélène and N. M. Katz, in Arithmetic of Higher-Dimensional Algebraic Varieties (Palo Alto, CA, 2002), Progr. Math. 226, Birkhäuser Boston, Boston, 2004, 175–184.
  • [21] P. Salberger, Integral points on hypersurfaces of degree at least three, unpublished, 2006.
  • [22] J.-P. Serre, Lectures on the Mordell-Weil theorem, 3rd ed., Vieweg, Braunschweig, 1997.
  • [23] P. Swinnerton-Dyer, Arithmetic of diagonal quartic surfaces, II, Proc. Lond. Math. Soc. (3) 80 (2000), no. 3, 513–544.
  • [24] R. C. Vaughan, A new iterative method in Waring’s problem, Acta Math. 162 (1989), no. 1–2, 1–71.