对埃西尔斯坦证明二次互反律的involution视角
An involutive perspective on Eisenstein's proof of quadratic reciprocity
AI总结：
基于埃西尔斯坦的几何证明，明确其内在involution对称性，通过Legendre符号作为有限矩形中的格点计数，构造简单无固定点的involution，揭示二次互反律从基本组合配对原理中浮现。
摘要：
我们重新审视埃西尔斯坦对二次互反律的几何证明，并明确其内在的involution对称性。基于高斯引理，我们将Legendre符号解释为有限矩形中的格点计数，并构造一个简单的无固定点的involution，对应矩形的中心对称性，交换点上方和下方的点。这种改写强调了involution对称性，并将经典证明置于Zagier型involution论证的范畴内。该方法展示了二次互反律如何从一个基本的组合配对原理中浮现。
英文摘要：
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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