Symplectic Monodromy and Horizontal Sato--Tate for Reciprocal Qudit Wigner Functions

Abstract

For odd primes $p$, reciprocal-phase qudit states lead, away from one exceptional phase-space row, to the two-parameter exponential sums: $$T_p(A,B) = \sum_{t \neq \pm 1} e_p\left(\frac{A t}{t^2-1} + B t\right)$$ A companion paper established the exact Wigner dictionary, generic cohomological rank four, an exact fourth-moment factorization, and a conditional $\mathrm{USp}(4)$ interpretation. In this sequel we prove the missing monodromy and equidistribution theorems. The involution $t \mapsto -t$ gives an alternating self-duality, so the geometric monodromy lies in $\mathrm{Sp}_4$. The previously obtained fourth-moment factorization has exactly four top-dimensional geometric components, which yields the geometric moment $M_{2,2}=3$ over finite extensions. A Fourier–Deligne specialization at $B=0$ has rank drop $4 \to 3$ and forces a symplectic transvection, hence infinite monodromy. Larsen's alternative, in the form recorded by Katz–Tiep, then gives: $$G_{\mathrm{geom}} = \mathrm{Sp}_4$$ After a geometrically constant weight normalization, the arithmetic and geometric monodromy groups coincide. Uniform complexity bounds from quantitative sheaf theory imply horizontal equidistribution of Frobenius conjugacy classes in $\mathrm{USp}(4)$ as $p \to \infty$. Consequently, the normalized Wigner values converge to the standard $\mathrm{USp}(4)$ trace law. Since the normalized traces are uniformly bounded, the mana asymptotic becomes unconditional: $$\mathcal{M}(R_c) = \frac{1}{2} \log p + \log\left(\frac{4096}{525 \pi^2}\right) + o(1)$$ The monodromy proof uses the earlier fourth-moment factorization but is independent of the $K3$ and CM-modularity results of the companion paper.