## Involve: A Journal of Mathematics

• Involve
• Volume 12, Number 2 (2019), 321-327.

### Failure of strong approximation on an affine cone

#### Abstract

We use the Brauer–Manin obstruction to strong approximation on a punctured affine cone to explain why some mod $p$ solutions to a homogeneous Diophantine equation of degree $2$ cannot be lifted to coprime integer solutions.

#### Article information

Source
Involve, Volume 12, Number 2 (2019), 321-327.

Dates
Revised: 29 January 2018
Accepted: 13 March 2018
First available in Project Euclid: 25 October 2018

https://projecteuclid.org/euclid.involve/1540432920

Digital Object Identifier
doi:10.2140/involve.2019.12.321

Mathematical Reviews number (MathSciNet)
MR3864220

Zentralblatt MATH identifier
06980504

#### Citation

Bright, Martin; Kok, Ivo. Failure of strong approximation on an affine cone. Involve 12 (2019), no. 2, 321--327. doi:10.2140/involve.2019.12.321. https://projecteuclid.org/euclid.involve/1540432920

#### References

• M. J. Bright, Computations on diagonal quartic surfaces, Ph.D. thesis, University of Cambridge, 2002, http://www.boojum.org.uk/maths/quartic-surfaces/thesis.pdf.
• M. Bright, “The Brauer–Manin obstruction on a general diagonal quartic surface”, Acta Arith. 147:3 (2011), 291–302.
• J.-L. Colliot-Thélène and F. Xu, “Strong approximation for the total space of certain quadratic fibrations”, Acta Arith. 157:2 (2013), 169–199.
• D. Eisenbud and J. Harris, The geometry of schemes, Graduate Texts in Mathematics 197, Springer, 2000.
• T. J. Ford, “The Brauer group of an affine cone”, J. Pure Appl. Algebra 155:1 (2001), 29–40.
• R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, Springer, 1977.
• S. Lindqvist, “Weak approximation results for quadratic forms in four variables”, preprint, 2017.
• J. S. Milne, Étale cohomology, Princeton Mathematical Series 33, Princeton University Press, 1980.