A note on Pepin's counter examples to the Hasse principle for curves of genus 1

In a series of articles published in the C.R. Paris more than a century ago, T. PEPIN announced a list of "theorems" concerning the solvability of diophantine equations of the type ax(4) + by(4) = z(2). In this article, we show how to prove these claims using the structure of 2-class groups of imaginary quadratic number fields. We will also look at a few related results from a modern point of view.