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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.25288 |
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| _version_ | 1866914513460133888 |
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| author | Holland, Jonathan |
| author_facet | Holland, Jonathan |
| contents | 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 $(p,q)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_25288 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Reciprocity and the Maslov Phase Holland, Jonathan Number Theory Representation Theory 11A15 (primary), 53D12 (secondary) 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 $(p,q)$. |
| title | Reciprocity and the Maslov Phase |
| topic | Number Theory Representation Theory 11A15 (primary), 53D12 (secondary) |
| url | https://arxiv.org/abs/2604.25288 |