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

1999/03/18 by Franz Lemmermeyer, Lemmermeyer, Franz · 1 citation
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #math.NT

paper · pdf · doi:10.48550/arxiv.math/9903207

arxiv created 1999/03/18 · arxiv updated 2009/12/01

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 ax⁴ + by⁴ = 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 (including FLT for n = 7) from a modern point of view.

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