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A Note on Pépin's counter examples to the Hasse principle for curves of genus 1

Franz Lemmermeyer
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Abstract

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 $ax^4 + by^4 = z^2$. 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.

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Franz Lemmermeyer. 1999-03-18. A Note on Pépin's counter examples to the Hasse principle for curves of genus 1. https://arxiv.org/abs/math/9903207

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