arXiv subjects

Pierre L. L. Morain

Publications and source records attributed to Pierre L. L. Morain.

4 recordsLinked to original sources

arXiv2026

Computations of higher elliptic units in optimal settings

In this paper we present a simplified form of a conjecture on the construction of generalised elliptic units above number fields with exactly one complex place. They are conjectural algebraic numbers which are obtained as special values of higher elliptic Gamma functions. These functions form a collection of multivariate meromorphic functions which were studied in the late 1990s and early 2000s in mathematical physics. Our construction extends the scheme of a recent article by Bergeron, Charollois and García where they constructed conjectural elliptic units above complex cubic fields using the elliptic Gamma function. The higher elliptic units we construct are expected to generate specific abelian extensions of the base field where they are evaluated, thus giving a conjectural solution to Hilbert's 12th problem for the number fields with exactly one complex place. We provide several examples to support our conjecture in optimal settings for number fields of degree 3, 4, 5 and 6.

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arXiv2026

Geometric families of multiple elliptic Gamma functions and arithmetic applications, II

This is the second paper in a series where we study arithmetic applications of the multiple elliptic Gamma functions originated in mathematical physics. In the first article in this series we defined geometric families of these functions and proved that these families satisfied coboundary relations involving an attached collection of Bernoulli rational functions. The main purpose of the present paper is to show that smoothed versions of our geometric elliptic Gamma functions give rise to partial modular symbols for congruence subgroups of $\mathrm{SL}_{n}(\mathbb{Z})$ for $n \geq 2$ which restrict to $(n-2)$-cocycles on tori in $\mathrm{SL}_{n}(\mathbb{Z})$ coming from groups of totally positive units in number fields. To achieve this, we show that the associated smoothed Bernoulli rational functions reduce to smoothed higher Dedekind sums with uniformly bounded denominators.

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arXiv2026

Geometric families of multiple elliptic Gamma functions and arithmetic applications, I

This is the first paper in a series where we study arithmetic applications of the multiple elliptic Gamma functions originated from mathematical physics. The main purpose of this paper is the introduction of a framework for applications of these functions to Hilbert's 12th problem for general number fields with exactly one complex place following recent work by Bergeron, Charollois and García. Namely, we define geometric families of the multiple elliptic Gamma functions, upgrading the construction carried out by Felder, Henriques, Rossi and Zhu for rank $3$ lattices to lattices of higher ranks. These functions enjoy transformation properties under an action of the special linear group $\mathrm{SL}_n(\mathbb{Z})$ for $n \geq 2$ involving some Bernoulli rational functions as their so-called modularity defect. A second purpose of this paper is to use this collection of Bernoulli rational functions to construct $(n-1)$-cocycles for specific subgroups of $\mathrm{SL}_n(\mathbb{Z})$ associated to units groups in totally real number fields and use these cocycles to compute partial zeta values at $s=0$.

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arXiv2024

Elliptic Units Above Fields With Exactly One Complex Place

In this work we explore the construction of abelian extensions of number fields with exactly one complex place using multivariate analytic functions in the spirit of Hilbert's 12th problem. To this end we study the special values of the multiple elliptic Gamma functions introduced in the early 2000s by Nishizawa following the work of Felder and Varchenko on Ruijsenaars' elliptic Gamma function. We construct geometric variants of these functions enjoying transformation properties under an action of $\mathrm{SL}_{d}(\mathbb{Z})$ for $d \geq 2$. The evaluation of these functions at points of a degree $d$ field $\mathbb{K}$ with exactly one complex place following the scheme of a recent article by Bergeron, Charollois and García (arXiv:2311.04110) seems to produce algebraic numbers. More precisely, we conjecture that such infinite products yield algebraic units in abelian extensions of $\mathbb{K}$ related to conjectural Stark units and we provide numerical evidence to support this conjecture for cubic, quartic and quintic fields.

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