Tokyo Journal of Mathematics

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

Yingpu DENG, Chang LV, and Junchao SHENTU

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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.

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

First available in Project Euclid: 20 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11D09: Quadratic and bilinear equations 11E12: Quadratic forms over global rings and fields 11D57: Multiplicative and norm form equations
Secondary: 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17] 11R37: Class field theory


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.

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