[Paper Review] A Solution to the Real Multiplication Program in Positive Characteristic I: Quantum Modular Invariant and Hilbert Class Fields

L. Demangos, T. M. Gendron|arXiv (Cornell University)|Jul 8, 2016

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper solves a key aspect of Manin's Real Multiplication program in positive characteristic by introducing a quantum modular invariant $ j^{mqt} $, a multivalued modular function over $ k_{ otinfty} = \mathbb{F}_q((T^{-1})) $. It proves that the Hilbert class field $ H_{\mathcal{O}_K} $ of a real quadratic extension $ K = k(f) $ is generated over $ K $ by the product of the multivalues of $ j^{mqt}(f) $, providing an explicit class field theory construction in positive characteristic.

ABSTRACT

This is the first of a series of two papers in which we present a solution to Manin's Real Multiplication program -- an approach to Hilbert's 12th problem for real quadratic extensions of $\mathbb{Q}$ -- in positive characteristic, using quantum analogs of the exponential function and the modular invariant. In this first paper, we treat the problem of Hilbert class field generation. If $k=\mathbb{F}_{q}(T)$ and $k_{\infty}$ is the analytic completion of $k$, we introduce the quantum modular invariant \[ j^{ m qt}: k_{\infty}\multimap k_{\infty}\] as a multivalued, modular invariant function. Then if $K=k(f)\subset k_{\infty}$ is a real quadratic extension of $k$ where $f$ is a quadratic unit, we show that the Hilbert class field $H_{\mathcal{O}_{K}}$ (associated to $\mathcal{O}_{K}=$ integral closure of $\mathbb{F}_{q}[T]$ in $K$) is generated over $K$ by the product of the multivalues of $j^{ m qt}(f)$.

Motivation & Objective

• To address Hilbert's 12th problem for real quadratic extensions of $ \mathbb{F}_q(T) $ in positive characteristic.
• To develop a quantum modular invariant $ j^{mqt} $ as a multivalued, modular function over the completion $ k_{\infty} $.
• To construct the Hilbert class field $ H_{\mathcal{O}_K} $ of the ring of integers $ \mathcal{O}_K $ in a real quadratic extension $ K $.
• To establish that $ H_{\mathcal{O}_K} $ is generated over $ K $ by the product of the multivalues of $ j^{mqt}(f) $, where $ f $ is a quadratic unit.

Proposed method

• Introduce the quantum modular invariant $ j^{mqt}: k_{\infty} \multimap k_{\infty} $ as a multivalued, modular-invariant function over the completion $ k_{\infty} = \mathbb{F}_q((T^{-1})) $.
• Define $ k = \mathbb{F}_q(T) $, and consider a real quadratic extension $ K = k(f) $, where $ f $ is a quadratic unit in $ k_{\infty} $.
• Use the multivalued nature of $ j^{mqt}(f) $ to extract algebraic generators for the Hilbert class field $ H_{\mathcal{O}_K} $.
• Show that the product of the multivalues of $ j^{mqt}(f) $ generates $ H_{\mathcal{O}_K} $ over $ K $, using properties of quantum modular functions and class field theory in positive characteristic.
• Leverage the modular invariance of $ j^{mqt} $ to ensure compatibility with the Galois action on class fields.
• Establish that the construction is intrinsic to the arithmetic of $ \mathcal{O}_K $, relying on the structure of the unit group and the class group.

Experimental results

Research questions

• RQ1 Can a quantum modular invariant be constructed in positive characteristic to generate Hilbert class fields for real quadratic extensions of $ \mathbb{F}_q(T) $?
• RQ2 How does the multivalued nature of $ j^{mqt}(f) $ relate to the Galois group of the Hilbert class field extension?
• RQ3 What is the precise algebraic generator of $ H_{\mathcal{O}_K} $ over $ K $, and how is it encoded in the values of $ j^{mqt}(f) $?
• RQ4 Does the quantum modular invariant $ j^{mqt} $ satisfy the required modular transformation properties in positive characteristic?
• RQ5 Can the Hilbert class field of $ \mathcal{O}_K $ be explicitly generated using only the values of $ j^{mqt}(f) $, without relying on classical modular forms?

Key findings

• The quantum modular invariant $ j^{mqt} $ is defined as a multivalued, modular-invariant function on $ k_{\infty} = \mathbb{F}_q((T^{-1})) $, generalizing the classical modular $ j $-invariant to the quantum setting.
• For a real quadratic extension $ K = k(f) $ with $ f $ a quadratic unit, the Hilbert class field $ H_{\mathcal{O}_K} $ is generated over $ K $ by the product of the multivalues of $ j^{mqt}(f) $.
• The construction provides an explicit class field theory generator for $ H_{\mathcal{O}_K} $, solving a key case of Manin's Real Multiplication program in positive characteristic.
• The multivalued nature of $ j^{mqt}(f) $ encodes the full Galois action on the class field, ensuring that the product of its values generates the entire extension.
• The method relies on quantum analogs of the exponential function and modular invariants, offering a new approach to explicit class field theory in positive characteristic.
• The result establishes a bridge between quantum modular functions and arithmetic in global function fields, particularly for real quadratic extensions.