# Generalizing Quadratic Reciprocity Law with Dilates

Eisenstein's proof of the Quadratic Reciprocity (QR) (and its Jacobi symbol generalization) both rely on counting lattice points in two congruent triangles. If we take $t$-dilates of these triangles, does an extension of Gauss' Lemma still hold with some dependence on $t$? That is, can we assign a Legendre symbol $(p|q)$, or some simple extension thereof depending on $t$, to $(-1)^{M(t)}$, where $M(t)$ is the number of lattice points in the $t$-dilate of the same triangle used in the standard proof and derive from it a "$t$-dilated" version of QR?

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what is a t-dilate? – quanta Mar 11 '11 at 17:20
Suppose $P$ is an $n$-polytope of the form $\text{conv}\{\mathbf{0}, a_1 \mathbf{e}_1, \dots, a_n \mathbf{e}_n \}$. Then a non-trivial $t$-dilate $t P$ is the convex hull $\text{conv}\{\mathbf{0}, a_1 t \mathbf{e}_1, \dots, a_n t \mathbf{e}_n \}$, where $t > 1$. The triangles above correspond to taking $n = 2$. – user02138 Mar 11 '11 at 17:33