Publications de Pierre L. L. Morain
Hide abstracts.
Preprints
- Geometric families of multiple elliptic Gamma functions and arithmetic applications, I - 2025 (Submitted for publication). Link.
Abstact: 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 SLn(Z) for n ≥ 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 SLn(Z)
associated to groups of units in
totally real number fields and use these cocycles to compute partial zeta values at s = 0.
- Geometric families of multiple elliptic Gamma functions and arithmetic applications, II - 2026 (Accepted for publication in the Annales Mathématiques du Québec, special volume in honour of Henri Darmon). Link to preprint version.
Abstract: 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 SLn(Z) for n ≥ 2 which restrict to (n-2)-cocycles on tori in
SLn(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.
- Computations of higher elliptic units in optimal settings - 2026 (Submitted for publication). Link.
Abstract: 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.
- Geometric families of multiple elliptic Gamma functions and arithmetic applications, III - 2026. Link.
Abstract: This is the third paper in a series where we study arithmetic applications of the higher elliptic
Gamma functions.
In the first two articles in this series, we defined geometric families of these functions and proved that
they satisfy coboundary
and cocycle relations under the action of special linear groups and associated congruence subgroups.
The main purpose of the
present paper is to present a construction of conjectural higher elliptic units above number fields
with exactly one complex
place as special values of higher elliptic Gamma functions, upgrading the construction carried out
by Bergeron, Charollois and
García for complex cubic fields. These higher elliptic units are obtained by evaluating a
multiplicative (n-2)-cocycle built from
higher elliptic Gamma functions against a (n-2)-cycle associated to some group
of totally positive units of a given number field
of degree n with exactly one complex place. We conjecture that these
higher elliptic units are algebraic units which belong to
prescribed abelian extensions of the base field where they are
evaluated and that they satisfy a Kronecker limit formula which
relates the logarithm of their modulus to values of
derivatives of partial zeta functions at s = 0 in the base field. We showcase our
conjecture on various examples for number
fields of degree 3, 4 and 5.
Working papers / Documents de travail
- Elliptic units above number fields with exactly one complex place - 2024 (Superseeded by the series of papers "Geometric families..."). Link.
Mémoires
- Arithmétiques des corps cyclotomiques et théorie d'Iwasawa - Mémoire de M1 effectué sous la direction d'Olivier Fouquet, LMB.
- Multiple elliptic Gamma functions and Eisenstein series - Mémoire de M2 effectué sous la direction de Pierre Charollois, IMJ-PRG.
- Higher elliptic Gamma functions, cohomology of linear groups and higher elliptic units - Mémoire de thèse effectuée sous la direction de Pierre Charollois (IMJ-PRG) et Antonin Guilloux (IMJ-PRG, INRIA) : pdf.
Abstract: Multiple elliptic Gamma functions are functions originated from mathematical physics which
form a hierarchy of
multivariate meromorphic functions satisfying modular transformation properties for special
linear groups SLn(Z) involving
an attached hierarchy of Bernoulli rational functions. In this
work we study the connections between these functions and
the arithmetic of number fields and especially the
possibility of relating invariants in class field theory to special values of
multiple elliptic Gamma functions.
To achieve this, we define geometric families of the multiple elliptic Gamma functions
attached to families of
linear forms on integral lattices and recast the transformation properties satisfied by these functions
in terms
of cocycle relations for SLn(Z). This construction of higher elliptic Gamma functions upgrades
the construction
carried out by Felder, Henriques, Rossi and Zhu for the elliptic Gamma function to the whole
hierarchy of multiple elliptic
Gamma functions. We also define geometric families of Bernoulli rational functions
in a similar way and show that these
functions form a collection of (n-1)-cocycles for groups of totally
positive units in number fields and which may be used
to evaluated partial zeta functions at s=0 in totally
real number fields. Moreover, we investigate the cocycle properties of
both collections of functions for
congruence subgroups in SLn(Z) by performing a standard smoothing operation, and show
that
they simplify considerably.
The main goal of this work is the construction of conjectural higher elliptic units above number fields with
exactly one complex
place as special values of higher elliptic Gamma functions, upgrading the construction
carried out by Bergeron, Charollois and
García for complex cubic fields. These higher elliptic units are
obtained by evaluating the multiplicative (n-2)-cocycle for a
congruence subgroup in SLn(Z)
built from higher elliptic Gamma functions against a (n-2)-cycle on the group of totally positive
units of a
given number field of degree n with exactly one complex place. We conjecture that these higher elliptic units
are indeed
algebraic units which belong to prescribed abelian extensions of the base field where they are
evaluated and that they satisfy a
Kronecker limit formula which relates the logarithm of their modulus to values of
derivatives of partial zeta functions at s = 0 in
the base field. We present algorithms we used to produce
examples of higher elliptic units and examples of such computations for
number fields of degree 3, 4, 5 and 6.