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An involutive perspective on Eisenstein's proof of quadratic reciprocity

Jean-Christophe Pain

Abstract

We revisit Eisenstein's geometric proof of quadratic reciprocity and make explicit the involutive symmetry underlying Eisenstein's lattice-point argument. Building on Gauss's lemma, we interpret the Legendre symbols as counts of lattice points in a finite rectangle and construct a simple fixed-point-free involution corresponding to the central symmetry of the rectangle, which exchanges points above and below the line $qx=py$. This reformulation highlights the involutive symmetry and places the classical proof in the spirit of Zagier-type involutive arguments. The approach shows how the reciprocity law emerges from an elementary combinatorial pairing principle.

An involutive perspective on Eisenstein's proof of quadratic reciprocity

Abstract

We revisit Eisenstein's geometric proof of quadratic reciprocity and make explicit the involutive symmetry underlying Eisenstein's lattice-point argument. Building on Gauss's lemma, we interpret the Legendre symbols as counts of lattice points in a finite rectangle and construct a simple fixed-point-free involution corresponding to the central symmetry of the rectangle, which exchanges points above and below the line . This reformulation highlights the involutive symmetry and places the classical proof in the spirit of Zagier-type involutive arguments. The approach shows how the reciprocity law emerges from an elementary combinatorial pairing principle.
Paper Structure (5 sections, 3 theorems, 45 equations)

This paper contains 5 sections, 3 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. The classical parity proof of quadratic reciprocity via Gauss's lemma
  3. Proof based on involution arguments
  4. Klein four group acting on the lattice rectangle
  5. Conclusion

Key Result

Theorem 2.1

Let $p$ and $q$ be distinct odd primes. Then

Theorems & Definitions (6)

  • Theorem 2.1: Quadratic reciprocity
  • Lemma 2.2: Gauss's Lemma
  • proof : Proof of quadratic reciprocity
  • Proposition 3.1: Symmetry of the lattice model
  • Remark 3.2
  • Remark 3.3