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 $γ_v(a)$. The local Hilbert symbol appears as the defect of strict multiplicativity of these phases: [ (a,b)_v = \frac{γ_v(a)γ_v(b)}{γ_v(1)γ_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 [ μ_v(a,b) = \frac{γ_v(a)γ_v(b)}{γ_v(1)γ_v(ab)}. ] For rational adelic data, the normalized Bruhat word is multiplicative. The reciprocity law states that the total defect $\prod_vμ_v(a,b)$ 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 $(p,q)$.

• The paper gives a metaplectic proof of Hilbert reciprocity by identifying the Kashiwara–Maslov phase of Lagrangian triples with the local Weil index, linking symplectic geometry to arithmetic.
• The local multiplicativity defect of Weil indices is exactly the Hilbert symbol, so normalized Bruhat-word operators form a projective torus representation with cocycle $(a,b)_v$.
• The global comparison of local and adelic Bruhat words forces the product of local defects to equal one, with the 2-adic factor producing the supplementary sign in quadratic reciprocity.

This paper by Jonathan Holland gives a metaplectic proof of Hilbert reciprocity, and hence quadratic reciprocity, in which the local reciprocity data arise as Kashiwara–Maslov phases of triples of Lagrangians in a symplectic plane (2604.25288). The central identification is that for the ordered triple $(L_\infty, L_a, L_0)$ of pairwise transverse Lagrangian lines, the local Maslov phase equals the one-dimensional Weil index $\gamma_v(a)$, and the Hilbert symbol $(a,b)_v$ appears as the defect of strict multiplicativity of these phases. The global argument compares local and adelic realizations of a single Bruhat word for diagonal torus elements $m(a) \in \mathrm{SL}_2(\mathbb{Q})$, showing that the total defect $\prod_v \mu_v(a,b)$ is trivial.

The rank-two Maslov cocycle as Weil index

The paper works over a field $K$ of characteristic different from $2$, with $V = K^2$ and Lagrangian lines $L_a := K(1,a)$ together with $L_\infty := K(0,1)$. For a triple of Lagrangians, the Kashiwara space is

$\gamma_v(a)$0

equipped with the quadratic form $\gamma_v(a)$1, which is well-defined on this subspace since the three pairwise pairings agree there. Two explicit computations anchor the theory: for distinct finite slopes $\gamma_v(a)$2,

$\gamma_v(a)$3

while for the triple involving infinity,

$\gamma_v(a)$4

The order of the triple matters: reversing to $\gamma_v(a)$5 yields $\gamma_v(a)$6. Combined with the rank-two metaplectic cocycle formula — proved in the appendix via an explicit kernel computation on the Heisenberg group, where disintegration along the addition map restricts the oscillatory phase to the closed-triangle space — one obtains the key corollary:

$\gamma_v(a)$7

i.e., the one-dimensional Weil index is literally a Kashiwara–Maslov phase. This is the geometric source of all local arithmetic in the paper.

Local multiplicativity defect is the Hilbert symbol

Fixing standard additive characters ($\gamma_v(a)$8 at $\gamma_v(a)$9, $(a,b)_v$0 at $(a,b)_v$1) and self-dual measures, the Weil index $(a,b)_v$2 is defined through the Fourier transform identity for quadratic Gaussians; it depends only on the square class of $(a,b)_v$3 and has modulus $(a,b)_v$4. The local bridge theorem states

$(a,b)_v$5

proved from the Hasse-invariant formula $(a,b)_v$6 applied to the binary form $(a,b)_v$7. Consequently the Hilbert symbol measures exactly the failure of the Maslov phases to multiply strictly. The appendix also computes the scalar of the local Bruhat word $(a,b)_v$8 as $(a,b)_v$9 times the standard scaling operator, so the normalized operators $m(a) \in \mathrm{SL}_2(\mathbb{Q})$0 satisfy

$m(a) \in \mathrm{SL}_2(\mathbb{Q})$1

with the factors of $m(a) \in \mathrm{SL}_2(\mathbb{Q})$2 canceling by bilinearity and symmetry of the Hilbert symbol. Thus the normalized Bruhat-word lifts form a projective representation of the diagonal torus whose cocycle is the Hilbert symbol itself.

Global cancellation via theta invariance

The global step rests on three classical inputs: triviality of the adelic character on the diagonal copy of $m(a) \in \mathrm{SL}_2(\mathbb{Q})$3, the product formula $m(a) \in \mathrm{SL}_2(\mathbb{Q})$4, and Poisson summation. These give invariance of the theta distribution $m(a) \in \mathrm{SL}_2(\mathbb{Q})$5 under each generator $m(a) \in \mathrm{SL}_2(\mathbb{Q})$6, $m(a) \in \mathrm{SL}_2(\mathbb{Q})$7, and $m(a) \in \mathrm{SL}_2(\mathbb{Q})$8. Since the Bruhat matrix identity $m(a) \in \mathrm{SL}_2(\mathbb{Q})$9 holds in $\prod_v \mu_v(a,b)$0, Schur's lemma gives $\prod_v \mu_v(a,b)$1 on the irreducible adelic Schrödinger representation, and testing against a product function with $\prod_v \mu_v(a,b)$2 forces $\prod_v \mu_v(a,b)$3, so $\prod_v \mu_v(a,b)$4 exactly.

A restricted tensor comparison shows $\prod_v \mu_v(a,b)$5, with the tensor products well-defined because at almost all finite places $\prod_v \mu_v(a,b)$6 acts as the scaling operator fixing the standard vector $\prod_v \mu_v(a,b)$7. Comparing the strictly multiplicative global operators $\prod_v \mu_v(a,b)$8 with the projective local multiplication law then forces

$\prod_v \mu_v(a,b)$9

and multiplying the local bridge over all places yields Hilbert reciprocity $K$0 immediately. The proof structure is clean: no case analysis enters the global step, only the exact equality of two realizations of one group-theoretic word.

Quadratic reciprocity and the role of the place 2

Specializing to distinct odd primes $K$1, the local evaluations are standard: the symbol is trivial at all primes $K$2 and at the real place; it equals $K$3 at $K$4 and $K$5 at $K$6; and at $K$7 it is $K$8, obtained from the explicit formula $K$9 on $2$0. Substituting into the product formula gives

$2$1

The paper's interpretive claim is precise: the classical supplementary sign is the $2$2-adic component of the global cancellation law for Kashiwara–Maslov phases, not an artifact of a particular computation.

The real warm-up and Gauss sums

Before the adelic machinery, the paper develops a classical model on $2$3 coupled to the lattice state $2$4, which satisfies $2$5 by Poisson summation. For odd $2$6, the residue-class combs span a finite sector preserved by the phase operator $2$7, and the transport coefficient $2$8 — the unshifted coefficient of the finite operator $2$9 — is exactly the quadratic Gauss sum $V = K^2$0. Two identities combine: numerator variation gives $V = K^2$1, while theta transformation gives $V = K^2$2 with $V = K^2$3 or $V = K^2$4 according as $V = K^2$5 or $V = K^2$6. Via the Chinese remainder decomposition $V = K^2$7, the reciprocity sign emerges as the quotient of metaplectic square-root phases attached to $V = K^2$8, $V = K^2$9, and $L_a := K(1,a)$0. Notably, the purely real sign $L_a := K(1,a)$1 recording cyclic ordering carries no arithmetic data; the arithmetic enters only through metaplectic transport tested against the integer lattice.

Limitations and scope

The paper is candid about its inputs. The rank-two identification of the Maslov cocycle with the Weil index is a standard metaplectic fact, cited from Rao and Lion–Vergne rather than reproved beyond the normalization-fixing kernel computation. The local multiplicativity law is taken as the binary specialization of the Hasse-invariant formula, so the "new" content lies in the global cancellation argument, not the local evaluations. The proof is specific to $L_a := K(1,a)$2: the global step uses the product formula, Poisson summation for $L_a := K(1,a)$3, and the explicit Hilbert-symbol evaluations at $L_a := K(1,a)$4 and $L_a := K(1,a)$5. Whether the same Bruhat-word comparison extends to general number fields, or whether higher-rank analogues of the defect cancellation yield higher reciprocity laws, are questions the paper does not address. The dependence on the quadratic convention $L_a := K(1,a)$6 is handled explicitly but means the scalars (e.g., $L_a := K(1,a)$7 versus $L_a := K(1,a)$8) shift under other normalizations, even though the final reciprocity statement does not.

Conclusion

The paper recasts quadratic reciprocity as a statement about the geometry of Lagrangian triples: locally, the Hilbert symbol is the multiplicativity defect of Kashiwara–Maslov phases $L_a := K(1,a)$9; globally, the defects of a single Bruhat word cancel adelically because the word realizes a genuine torus element on the theta distribution. The supplementary sign $L_\infty := K(0,1)$0 is thereby identified as the $L_\infty := K(0,1)$1-adic contribution to this cancellation. The result is a conceptually economical proof whose technical weight is concentrated in standard local inputs, leaving the extension beyond $L_\infty := K(0,1)$2 as the natural open direction.