### A Note on P'epin's counter examples to the Hasse principle for curves of genus 1, by Franz Lemmermeyer

In a series of articles published in the C.R. Paris more
than a century ago, T. P\'epin 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 (including FLT for $n = 7$) from a
modern point of view.

Franz Lemmermeyer <lemmerm@mpim-bonn.mpg.de>