A note on pépin’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. PéPIN announced a list of “theorems” concerning the solvability of diophantine equations of the type ax⁴ +by ⁴ = z². 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.