Reciprocity and the Maslov Phase

Abstract

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 $(L_\infty,L_a,L_0)$ is the one-dimensional Weil index $\gamma_v(a)$. The local Hilbert symbol appears as the defect of strict multiplicativity of these phases: \[ (a,b)_v = \frac{\gamma_v(a)\gamma_v(b)}{\gamma_v(1)\gamma_v(ab)}. \] The global step compares the local and adelic realizations of a single Bruhat word for the diagonal torus elements $m(a)\in \operatorname{SL}_2(\mathbb Q)$. 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 \[ \mu_v(a,b) = \frac{\gamma_v(a)\gamma_v(b)}{\gamma_v(1)\gamma_v(ab)}. \] 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 $(p,q)$.