## Tokyo Journal of Mathematics

### On the Integral Representation of Binary Quadratic Forms and the Artin Condition

#### Abstract

For diophantine equations of the form $ax^2+bxy+cy^2+g=0$ over $\mathbb{Z}$ whose coefficients satisfy some assumptions, we show that a condition with respect to the Artin reciprocity map, which we call the Artin condition, is the only obstruction to the local-global principle for integral solutions of the equation. Some concrete examples are presented.

#### Article information

Source
Tokyo J. Math., Volume 41, Number 2 (2018), 371-384.

Dates
First available in Project Euclid: 20 November 2017

https://projecteuclid.org/euclid.tjm/1511221568

Mathematical Reviews number (MathSciNet)
MR3908800

Zentralblatt MATH identifier
07053482

#### Citation

LV, Chang; SHENTU, Junchao; DENG, Yingpu. On the Integral Representation of Binary Quadratic Forms and the Artin Condition. Tokyo J. Math. 41 (2018), no. 2, 371--384. https://projecteuclid.org/euclid.tjm/1511221568

#### References

• D. A. Cox, Primes of the form $x^2+ ny^2$ : Fermat, class field theory, and complex multiplication, John Wiley & Sons, 1989.
• D. Harari, Le défaut d'approximation forte pour les groupes algébriques commutatifs, Algebra & Number Theory 2, no. 5 (2008), 595–611.
• C. Lv and Y. Deng, On orders in number fields: Picard groups, ring class fields and applications, Science China Mathematics 58, no. 8 (2015), 1627–1638.
• C. Lv and J. Shentu, On the integral representation of $ax^2 + by^2$ and the Artin condition, arXiv preprint arXiv: 1502.07457 (2015), 1–9.
• M. Newman, A note on an equation related to the Pell equation, American Mathematical Monthly 84, no. 5 (1977), 365–366.
• D. Wei, On the sum of two integral squares in quadratic fields $\mathbb Q(\sqrt{\pm p})$, Acta Arith. 147, no. 3 (2011), 253–260.
• D. Wei, On the diophantine equation $x^2-Dy^2=n$, Science China Mathematics 56, no. 2 (2013), 227–238.
• D. Wei, On the sum of two integral squares in the imaginary quadratic field $\mathbb Q(\sqrt{-2p})$, Science China Mathematics 57, no. 1 (2014), 49–60.
• D. Wei and F. Xu, Integral points for multi-norm tori, Proceedings of the London Mathematical Society 104, no. 5 (2012), 1019–1044.
• D. Wei and F. Xu, Integral points for groups of multiplicative type, Advances in Mathematics 232, no. 1 (2013), 36–56.