- The paper constructs an arithmetic analogue of quantum local systems by applying p-adic étale cohomology to moduli spaces of curves.
- It introduces the arithmetic Heisenberg local system as a non-split extension that captures intricate Galois representations via a relative Puiseux section method.
- The study offers new insights into mapping class group symmetries and highlights potential applications in number theory and arithmetic geometry.
An Overview of Arithmetic Quantum Local Systems Over the Moduli of Curves
The paper "Arithmetic Quantum Local Systems Over the Moduli of Curves" presents the construction of an arithmetic analogue of quantum local systems within the context of the moduli of curves. These systems, traditionally derived from topological quantum field theories or Hitchin connections in a geometric setting, are extended to an arithmetic framework. The primary objective of this paper is to elucidate the methodology and implications of such constructions, with particular attention to the role of p-adic étale local systems.
Key Concepts and Construction
The study begins by recognizing the mapping class groups $\Mod_{g,n}$ as the topological fundamental groups of moduli stacks of compact Riemann surfaces. The classical quantum representations provide rich structures that do not necessarily factor through the Torelli groups. This paper advances by constructing an arithmetic analogue of the quantum representations utilizing p-adic étale local systems on Mg,n, the moduli space of curves of genus g with marked points over a number field K.
The core contribution is the development of the arithmetic Heisenberg local system $\rho_{Q,p}^{\Heis}$ on M=Mg,1,K. This object is defined as a non-split extension of Zp-linear duals of the cohomological data of universal curves, encoded via an extension: $0 \rightarrow Z_{p}(1) \rightarrow \rho_{Q,p}^{\Heis} \rightarrow H^{1}(C/M, Z_{p})^{\vee} \rightarrow 0.$
The construction leverages a relative version of the Puiseux section technique, offering a systematic approach for generating a class within the Selmer group of a curve's first geometric étale cohomology.
Implications and Theoretical Insights
The established arithmetic local systems yield Galois representations that bear significant arithmetic promises. For instance, they provide insight into the modular representations of mapping class groups and symmetry in number theory.
One of the most noteworthy elements is the delivery of an extension class $c_{Q,x}^{\Heis} \in H^{1}(K, H^{1}(C'_K, Z_p)(1))$, which emerges as potentially non-zero depending on the choice of rational points Q and x on the moduli space. These classes are speculated to encompass profound arithmetical insights and ramifications, similar to known phenomena in the context of conformal block local systems and the Knizhnik-Zamolodchikov equations.
Potential Applications and Future Horizon
The arithmetic properties of the quantum local systems, especially through their Galois cohomology classes, suggest numerous intriguing possibilities in number theory and arithmetic geometry. As demonstrated by historical precedents such as the Knizhnik-Zamolodchikov equations, these structures could pave the way for new progress in understanding the interaction between arithmetic and geometric facets of the moduli of curves.
Looking forward, further computation and verification of the cohomological classes associated with various rational points could elucidate additional properties and applications, potentially bridging connections to other domains like motivic theory or cryptographic protocols using modular curves.
This paper stands as a foundational effort to harness the profound potential of arithmetic analogues of well-explored geometric and topological phenomena, with the hope of illuminating unexplored arithmetic territories inherent in the moduli of curves. Future research could provide deeper insight into the crystalline properties of these local systems, especially under good reduction conditions.