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arXiv · 2603.16611

An involutive perspective on Eisenstein's proof of quadratic reciprocity

Jean-Christophe Pain
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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.

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Jean-Christophe Pain. 2026-03-17. An involutive perspective on Eisenstein's proof of quadratic reciprocity. https://arxiv.org/abs/2603.16611

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