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

Franz Lemmermeyer|ArXiv.org|Mar 18, 1999

Algebraic Geometry and Number Theory8 references4 citations

TL;DR

This paper proves T. Pépin's 19th-century claims about counterexamples to the Hasse principle for genus 1 curves using the structure of 2-class groups in imaginary quadratic fields. It provides a modern algebraic number theory framework to validate Pépin's results and connects them to related problems like Fermat's Last Theorem for n=7.

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.

Motivation & Objective

• To validate T. Pépin's unproven claims from the 1890s about diophantine equations of the form $ax^4 + by^4 = z^2$.
• To explain the solvability conditions of these equations using the 2-class group structure of imaginary quadratic number fields.
• To reinterpret classical results on the Hasse principle for genus 1 curves in a modern algebraic number theory context.
• To connect Pépin's results to related number-theoretic problems, including Fermat's Last Theorem for $n=7$.

Proposed method

• Analyzes the 2-class group structure of imaginary quadratic fields to determine the solvability of $ax^4 + by^4 = z^2$.
• Applies class field theory techniques to relate the existence of rational points on genus 1 curves to ideal class group properties.
• Uses descent methods and 2-descent techniques to study rational solutions of quartic Diophantine equations.
• Reconstructs Pépin's original arguments using modern terminology and algebraic tools.
• Establishes a correspondence between the triviality of certain 2-torsion elements in class groups and the existence of rational points.
• Demonstrates that Pépin's counterexamples to the Hasse principle arise from nontrivial 2-class groups in specific imaginary quadratic fields.

Experimental results

Research questions

• RQ1 How can Pépin's unproven claims about the solvability of $ax^4 + by^4 = z^2$ be rigorously established?
• RQ2 What role does the 2-class group of an imaginary quadratic field play in determining the Hasse principle violation for genus 1 curves?
• RQ3 In what way do the structural properties of class groups explain the existence of local solutions without global ones?
• RQ4 How do Pépin's results connect to the broader framework of Fermat's Last Theorem for $n=7$?
• RQ5 Can classical counterexamples to the Hasse principle be systematically reconstructed using modern class field theory?

Key findings

• Pépin's claimed counterexamples to the Hasse principle for genus 1 curves are rigorously validated using the 2-class group structure of imaginary quadratic fields.
• The solvability of $ax^4 + by^4 = z^2$ is determined by the triviality of specific 2-torsion elements in the ideal class group of an associated imaginary quadratic field.
• The paper provides a modern proof of Pépin's results, showing that the absence of rational points despite local solvability stems from nontrivial 2-class groups.
• The method successfully explains the failure of the Hasse principle in these cases through algebraic number theory tools.
• The approach offers a conceptual framework that extends to other Diophantine equations, including those related to Fermat's Last Theorem for $n=7$.
• The paper demonstrates that Pépin's results, though announced over a century ago, are consistent with and can be derived from contemporary class field theory.