arXiv subjects

L. Demangos

Publications and source records attributed to L. Demangos.

6 recordsLinked to original sources

arXiv2024

Explicit Class Field Theory for Orders in Global Function Fields

This paper develops explicit class field theory for orders: of rank 1 in any global function field -- Hayes theory -- and of rank 2 in real quadratic function fields -- Real Multiplication. The essential ingredient in the development of the Hayes Theory is an orders version of Shimura's Main Theorem on Complex Multiplication. The section on Real Multiplication for orders uses values of the quantum modular invariant to generate the Hilbert class field of a rank 2 order contained in the integral closure of $\mathbb{F}_{q}[T]$.

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arXiv2020

Modular Invariant of Rank 1 Drinfeld Modules and Class Field Generation

The modular invariant of rank 1 Drinfeld modules is introduced and used to formulate and prove an exact analog of the Weber-Fueter theorem for global function fields. The main ingredient in the proof is a version of Shimura's Main Theorem of Complex Multiplication for global function fields, which is also proved here.

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arXiv2019

Quantum Drinfeld Modules I: Quantum Modular Invariant and Hilbert Class Fields

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 \[ j^{\rm qt}: k_{\infty}\multimap k_{\infty}\] 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 $j^{\rm qt}(f)$.

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arXiv2018

Quantum j-invariant in positive characteristic I: Definition and Convergence

We introduce the quantum $j$-invariant in positive characteristic as a multi-valued, modular-invariant function of a local function field. In this paper, we concentrate on basic definitions and questions of convergence. Note: This version contains a correction to the published version of Theorem 3. The error that was found is not in any way serious, but its correction does change slightly the statement of Theorem 3 appearing in the published version. Otherwise, it has no impact on this article nor any of its sequels.

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arXiv2017

Quantum Drinfeld Modules and Ray Class Fields of Real Quadratic Global Function Fields

This is the second in a series of two papers presenting a solution to Hilbert's 12th problem for real quadratic function fields in positive characteristic, in the sense of proving an analog of the Theorem of Weber-Fueter. We also offer a conjectural treatment of the number field case using quasicrystal counterparts of the constructions used in function fields.

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arXiv2016

Quantum $j$-Invariant in Positive Characteristic II: Formulas and Values at the Quadratics

The multi-valued quantum $j$-invariant in positive characteristic is studied at quadratic elements. For every quadratic $f$, an explicit expression for each of the values of $j^{\rm qt}(f)$ is given as a limit of rational functions of $f$. It is proved that the number of values of $j^{\rm qt}(f)$ is finite and that none of these values is $\infty$.

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