An involutive perspective on Eisenstein's proof of quadratic reciprocity

TL;DR

This paper provides a new perspective on Eisenstein's geometric proof of quadratic reciprocity by explicitly constructing an involution that reveals the symmetry underlying the lattice-point counting argument.

Contribution

It introduces an explicit involutive symmetry in Eisenstein's proof, connecting lattice-point counts to classical reciprocity through combinatorial pairing.

Findings

Reveals an involutive symmetry in Eisenstein's proof

Connects lattice-point counting with quadratic reciprocity

Highlights the elementary combinatorial nature of the law

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.