Reciprocity and the Maslov Phase

We give a metaplectic proof of Hilbert reciprocity, and hence of quadratic reciprocity, in which the local phase is the Kashiwara–Maslov phase of a triple of Lagrangians. In rank two the phase of the ordered triple $(L_{\mathrm{\infty }},L_a,L_0)$ is the one-dimensional Weil index ${\gamma }_v(a)$. The local Hilbert symbol appears as the defect of strict multiplicativity of these phases: $$(a,b{)}_v=\frac{{\gamma }_v(a){\gamma }_v(b)}{{\gamma }_v(1){\gamma }_v(ab)}\mathrm{.}$$ The global step compares the local and adelic realizations of a single Bruhat word for the diagonal torus elements $m(a)\in {\mathrm{SL}}_2(\mathbb{Q})$. Locally the raw Bruhat-word lift carries the normalization factor determined by the chosen quadratic convention. These operators form a projective representation of the diagonal torus with defect $${\mu }_v(a,b)=\frac{{\gamma }_v(a){\gamma }_v(b)}{{\gamma }_v(1){\gamma }_v(ab)}\mathrm{.}$$ For rational adelic data, the normalized Bruhat word is multiplicative. The reciprocity law states that the total defect ${\prod }_v{\mu }_v(a,b)$ is $1$. Combined with the local bridge above, this yields Hilbert reciprocity, while quadratic reciprocity is then the specialization to the pair of odd primes $(p,q)$.