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After a brief introduction to the classical theory of binary quadratic formswe use these results for proving (most of) the claims made by P\'epin in aseries of articles on unsolvable quartic diophantine equations, and forconstructing families of counterexamples to the Hasse Principle for curves ofgenus 1 defined by equations of the form $ax^4 + by^4 = z^2$.

Binary Quadratic Forms and Counterexamples to Hasse's Local-Global Principle