Diophantine equations defined by binary quadratic forms over rational function fields

We study the ``imaginary" binary quadratic form equations ax^2+bxy+cy^2+g=0 over k[t] in rational function fields, showing that a condition with respect to the Artin reciprocity map, is the only obstruction to the local-global principle for integral solutions of the equation.Comment: 14 pages, git commit 20170117/57d58b6, to appear in Acta Arithmetic

Representing Primes as the Form x2+ny2x^2+ny^2 in Some Imaginary Quadratic Fields

We give criteria of the solvability of the diophantine equation $p=x^2+ny^2$ over some imaginary quadratic fields where $p$ is a prime element. The criteria becomes quite simple in special cases.Comment: 8 pages, This paper has been withdrawn by the author since it was merged into the article arXiv:1405.5776 on August 8, 201