Arithmetic quantum local systems over the moduli of curves
Gyujin Oh
TL;DR
This work constructs arithmetic analogues of quantum local systems on the moduli of curves by producing a $p$-adic étale Heisenberg local system $ ho_{Q,p}^{Het}$ of rank $2g+1$ over the moduli $M_{g,1,K}$ from a $K$-rational point $Q$, organized as a non-split extension $0\rightarrow \mathbb{Z}_{p}(1)\rightarrow \rho_{Q,p}^{Het}\rightarrow H^{1}_{et}( ext{C}/\mathcal{M})^{\vee}\rightarrow 0$. Central to the construction is a relative Puiseux-section framework that extends Matsumoto’s tangential basepoint ideas to families, enabling a splitting of the étale homotopy exact sequence for configuration spaces in §2 and §3.1. By passing to the metabelianization, the inherently non-linear Heisenberg structure becomes a linear parabolic action, yielding a subrepresentation $\rho_{Q,\mathrm{sub},p}^{Het}$ (a cyclotomic character) and a rank-$2g$ quotient $\rho_{Q,\mathrm{quo},p}^{Het}$, with the quotient identified with $(R^{1}\phi_{*,et}\mathbb{Z}_{p})^{\vee}$. Restricting to $L$-rational points gives Heisenberg extension classes $c_{Q,x}^{Het}$ in $H^{1}(L, H^{1}_{et}(C'_{\bar{L}}, \mathbb{Z}_{p})(1))$, unramified outside a finite set, providing a systematic source of Selmer-type invariants and potential links to arithmetic analogues of Knizhnik–Zamolodchikov structures and related motives.
Abstract
We construct an arithmetic analogue of the quantum local systems on the moduli of curves, and study its basic structure. Such an arithmetic local system gives rise to a uniform way of assigning a Galois cohomology class of the first geometric étale cohomology of a smooth proper curve over a number field.