**Pierre CHAROLLOIS**

born April 21st @ Roanne (LO), France

A detailed CV can be found here : (CV.pdf),

Equipe de théorie des nombres

Case 247 - 4, place Jussieu -

75252 Paris Cedex

France

Office 417 tour 15-25

Université Paris 6

4 Place Jussieu

75005 Paris

phone number : 33 (1) 44 27 54 41 / Fax +33 (1) 44 27 73 21

- visiting UC Santa Cruz during sept.-oct. 2008. Courtesy of S. Dasgupta.

- visiting UC Santa Cruz during mar.-apr. 2010. Courtesy of S. Dasgupta.

- visiting Tokyo Univ. oct-nov. 2013. Courtesy of T. Oda.

**- co-organizer with
Carine Apparicio of the "prime twins project" (start. June 22nd 2011, Paris) ****"Twins"**, co-starring **Jules** and **Arthur**.

**- co-organizer with Matthew Greenberg and Samit Dasgupta of the BIRS workshop (nov. 2011, Banff) ****"Cycles on modular varieties"**.

**- former co-organizer of the ****"London-Paris Number theory seminar"**.

**- formerly co-organizer with Huayi Chen du ****"séminaire de théorie des nombres de Chevaleret"**.

**- solo organizer of the ****Workshop "Regulators and modularity" **** in Jussieu, Nov 30-Dec 1st 2015**.

**- co-organizer with F. Deglise, G. Freixas, X. Ma and V. Maillot of the ****Conference "Regulators IV" in Paris,**** IMJ-PRG May 23-28 2016**.

**- co-organizer with H. Darmon of the ****Journée "Algebraic cycles, L-values and Euler systems" en l'honneur de Bernadette Perrin-Riou,**** at Sorbonne Univ. , IMJ-PRG Jan. 24th 2023**.

**- co-organizer with L. Garcia of the ****"EisensteinDays", **** at UCL (London) , Dec. 18-19th 2023**.

**- co-organizer with J. Bergdall, G. Chenevier, S. Dasgupta, M. Dimitrov and A. Medvedovsky of the ****conference "Modular forms, L-functions and Eigenvarieties" in memoriam of Joël Bellaïche, **** at ENS Ulm, June 18-21 2024**.

- **2020-2023 :Préparation à l'agrégation externe de mathématiques**. En particulier, préparation de l'épreuve écrite d'algèbre. **Entraînement agreg 2020 **

- 2023-2024 ** Une introduction aux formes modulaires. ** (M2 Maths. Fondamentales)

- 2019-2023 ** Théorie des nombres avancée. ** (M1 Maths. 4M034, Fondamentales)

- 2020-2023 ** M1 4M023 : Algorithmique Algébrique : ** 2nd semestre, 6ECTS. (résultant, factorisation de polynômes, algorithme LLL).

- Licence première année : **1M002** (suites et intégrales, algèbre linéaire)

We introduce and study the main properties of certain Dedekind sums attached to a totally real number field. These are not rational numbers but rather real-analytic functions. They arise in the transformation formula of an ad-hoc analog of the logarithm of the Dedekind eta function for the Hilbert modular group.

We describe a conjectural construction (in the spirit of Hilbert’s 12th problem) of units
in abelian extensions of certain base fields which are neither totally real nor CM. These
base fields are quadratic extensions with exactly one complex place of a totally real number
field F,
and are referred to as Almost Totally Real (ATR) extensions. Our construction involves
certain null-homologous topological cycles on the Hilbert modular variety attached to
F. The special
units are the images of these cycles under a map defined by integration of weight two Eisenstein
series on GL2(F).
This map is formally analogous to the higher Abel–Jacobi maps that arise in the theory
of algebraic cycles.

We show that our conjecture is compatible with Stark’s conjecture
for ATR extensions ; it is, however, a genuine strengthening of Stark’s conjecture in this
context since it gives an analytic formula for the arguments of the Stark units and not
just their absolute values. The last section provides numerical evidences for our conjecture.

We define an integral version of Sczech's Eisenstein cocycle on GLn by smoothing at a prime ell. As a result we obtain a new proof of the integrality of the values at nonpositive integers of the smoothed partial zeta functions associated to ray class extensions of totally real fields. We also obtain a new construction of the p-adic L-functions associated to these extensions. Our cohomological construction allows for a study of the leading term of these p-adic L-functions at s=0.

We apply Spiess's formalism to prove that the order of vanishing at s=0 is at least equal to the expected one, as conjectured by Gross. This result was already known from Wiles' proof of the Iwasawa Main Conjecture.

We define a cocycle on Gln using Shintani's method.
This cocycle is closely related to cocycles defined earlier by Solomon and Hill, but
differs in that the cocycle property is achieved through the introduction of an auxiliary perturbation vector Q.
As a corollary of our result we obtain a new proof of a theorem of Diaz y Diaz and Friedman on signed fundamental
domains, and give a cohomological reformulation of Shintani's proof of the Klingen--Siegel rationality theorem
on partial zeta functions of totally real fields.

Next we prove that the cohomology class represented by our Shintani
cocycle is essentially equal to that represented by the Eisenstein cocycle defined by Sczech. This generalizes
a result of Sczech and Solomon in the case n=2. It implies that the formulae for values of partial zeta functions
arising from Shintani's method and from Sczech's method are identical.

Finally we introduce an integral version of our Shintani cocycle by smoothing at an auxiliary prime ell.
Applying the formalism of the first paper in this series, we prove that certain specializations of the smoothed class yield the p-adic L-functions
of totally real fields. Combining our cohomological construction with a theorem of Spiess, we
show that the order of vanishing of these p-adic L-functions is at least as large as the
expected one.

Lalin, Rodrigue and Rogers recently introduced the secant zeta function,
as an analog of the Lerch-Berndt-Arakawa cotan zeta function. These two zeta functions are Dirichlet series attached to a real quadratic irrational number α.
They conjectured that, similarly to the cotan zeta function case, the values at a specific range of negative integers of the secant Dirichlet series are algebraic numbers that belong to the field generated by α.

In this short note we provide two different proofs of their conjecture.

This article introduces a set of recently discovered lecture notes from the last course of Leopold Kronecker, delivered a few weeks before his death in December 1891. The notes, written by F. von Dalwigk, elaborate on the late recognition by Kronecker of the importance of the ``Eisenstein summation process'', invented by the ``companion of his youth'', in order to deal with conditionally convergent series known today as Eisenstein series.
We take this opportunity to give a brief update of the well known book by André Weil (1976) that brought these results of
Eisenstein and Kronecker back to light.
We believe that Eisenstein's approach to the theory of elliptic functions was in fact a very important part of Kronecker's planned proof of his visionary ``Jugendtraum''.

In this paper, we explicitly construct harmonic Maass forms that map to the weight one Theta series associated by Hecke to odd ray class group characters of real quadratic fields. From this construction, we give precise arithmetic information contained in the Fourier coefficients of the holomorphic part of the harmonic Maass form, establishing the main part of a conjecture of the second author.

These notes were written to be distributed to the audience of N. Bergeron's
Takagi lectures delivered June 23, 2018. These videos for these two lectures by Nicolas can be found on line : video Takagi 1 and video Takagi 2.
We give a new construction of some Eisenstein classes for GLn(Z) that were first considered by Sczech and Nori. Building on the proof of transgression of the Euler class by Bismut and Cheeger,
we produce a universal form that can be thought of as a kernel for a regularized theta lift for the reductive pair (GLn, GL1). This suggests looking to more general pairs (GLa,GLb), and leads to fascinating lift that relate the geometry/topology of real arithmetic locally symmetric spaces
to the arithmetic world of modular forms.
In these notes we put a lot of emphasis on various examples that are often classical.

We study the arithmetic of degree N-1 Eisenstein cohomology classes for locally symmetric spaces associated to GLN over an imaginary quadratic field k.
Under natural conditions we evaluate these classes on N-1 cycles associated to degree N extensions F/k in terms of generalized Dedekind sums. As a consequence we prove a remarkable conjecture of Sczech and Colmez expressing critical values of L-functions attached to Hecke characters of F as polynomials in Kronecker-Eisenstein series evaluated at torsion points on elliptic curves with CM by l. We recover in particular the algebraicity of these critical values.

Many authors have constructed different, but related, linear group cocycles that are usually referred to as ``Eisenstein cocycles.'' The main goal of this work is to describe a topological construction that is a common source for all these cocycles. One interesting feature of this construction is that, starting from a purely topological class, it leads to the algebraic world of meromorphic forms on hyperplane complements in n-fold products of either the (complex) additive group, the multiplicative group or a (family of) elliptic curve(s). This yields the construction of three types of ``Sczech cocycles.''

Nicolas Bergeron gave a series of 5 lectures as Aisenstadt Professor (2020) at CRM related to this work, together with a colloquium talk. The videos are available here or here.

We propose a conjecture extending the classical construction of elliptic units to complex cubic number fields K. The conjecture concerns special values of the elliptic gamma function, a holomorphic function of three complex variables arising in mathematical physics whose transformation properties under SL3(Z) were studied by Felder and Varchenko in the early 2000s. Using this function we construct complex numbers that we conjecture to be units in narrow ray class fields of K. We also propose a reciprocity law for the action of the Galois group on these units in the style of Shimura.
To support our conjecture we offer numerical evidence and also prove a new type of Kronecker limit formula relating the logarithm of the modulus of these complex numbers to the derivatives at s=0 of partial zeta functions of K. Our constructions unveil the role played by the elliptic gamma function in Hilbert's twelfth problem for complex cubic fields.

A talk by Nicolas on this material was delivered in Bonn, March 2023, in Honour of G. Harder's 85th birthday.

- Conférence Gauss-Dirichlet, Göttingen (20-24/06/2005)

- Workshop informel sur les conjectures de Stark à Montréal (01-03/11/2005) (co-organisateur)

- Summer school Arithmetic geometry à Göttingen organisée par le Clay Institute (17/7-11/8/2006)

- BIRS workshop on Modular forms: arithmetic and computation. (Banff) (5/6/2007)

- Meeting Canada-France, session théorie des nombres (Montreal) (5/6/2008)

- BIRS workshop "Dedekind sums" (Banff) (10/2009)

- Journée "Dr Honoris Causa" de Martin J. Taylor (Bordeaux) (26/10/2009)

- Conference "Arithmetic cycles and special values of L-functions", (CRM Barcelona) (12/2009)

- Conference "Algorithmic of L-functions", (Univ. Lyon 1) (20/6/2011)

- Workshop "Rational points on curves", (Univ. Oxford) (29/9/2012)

- Workshop "Effective methods for Darmon points", (Benasque) (30/8/2013)

- Workshop "Explicit formulae and Arakelov geometry", (IMJ) (19/9/2013)

- Workshop "Harmonic analysis on spherical homogeneous spaces", (Hakuba, Japon) (8/11/2013)

- Rencontre "Théorie des nombres et applications", (CIRM) (13/3/2014)

- Workshop "Arithmetic of Eisenstein series", (Darmstadt, Allemagne) (21/9/2014)

- Journée de théorie des nombres de l'UEVE , (Evry) (10/11/2016)

- Celebration of the CICMA postdoctoral program, (Montréal) (3-6/7/2018)

- CNTA XV Univ. Laval (Québec) (10-15/7/2018)

- Barcelona mathematical days, number theory session (Barcelone, virtual) (23/10/2020)

- Workshop Arithmetic of locally symmetric spaces (Montréal, virtual) (23/10/2020)

- Séminaire de théorie des nombres de Toulouse 2 Le Mirail (18/11/2004)

- Séminaire d'algèbre et de théorie des nombres de l'EPFL (Lausanne) (25/01/2005)

- Séminaire d'arithmétique de Saint-Etienne (01/02/2005)

- Séminaire de théorie algébrique des nombres de Limoges (07/02/2005)

- Séminaire de théorie des nombres de Québec-Vermont (Montréal) (31/03/2005)

- Séminaire de théorie des nombres de Montpellier (04/04/2005)

- Séminaire de théorie des nombres et combinatoire de Lyon (14/04/2005)

- Séminaire de théorie des nombres de Caen (15/04/2005)

- Galois seminar, University of Pennsylvania (Philadelphie) (17/02/2006)

- Séminaire ''autour de la géométrie d'Arakelov'' de Jussieu (24/4/2006)

- Colloquium Rutgers-Newark university (4/12/2006)

- Séminaire de théorie des nombres de Chevaleret(05/02/2007)

- Séminaire de théorie des nombres de Bordeaux (06/04/2007)

- Number Theory Seminar de Harvard (7/11/2007)

- Algebra seminar, Boston university (4/12/2007)

- Colloquium Rutgers-Newark university (10/12/2007)

- Séminaire d'arithmétique à Lyon (7/4/2008)

- Seminar UC Santa Cruz (oct. 2009)

- Seminar UC Santa Cruz (mar. 2010)

- QVNTS Montreal, mar. 2011

- Univ. Laval à Québec, mar. 2011

- Séminaire Algo INRIA Rocquencourt (mai 2012)

- Séminaire de théorie des nombres de Chevaleret (08/10/2012)

- Séminaire de théorie des nombres de Bordeaux (1/03/2013)

- Séminaire de théorie des nombres de Lyon (6/05/2013)

- Number Theory Seminar, Univ. of Tokyo (30/10/2013)

- Univ. Laval à Québec, (27/2/2015)

- EPFL Lausanne (7/5/2015)

- Univ. Lille 1 (28/6/2016)

- Univ. de Besançon (3/11/2016)

- Institut Fourier, Grenoble (23/5/2017)

- Univ. Bordeaux (16/3/2018)

- Jussieu (2/10/2023)

- Bordeaux (19/1/2024)

- Cologne (22/1/2024)

- IAS/Princeton (22/2/2024)

- Ronan Terpereau (2007)

- Pierre Le Boudec (2008)

- Samuel Le Fourn (2009)

- Nicolas Provost (2009)

- Claire Glanois (2010)

- Laura Rastelli (2011)

- Raphael Achet (2011)

- Macarena Peche Irissarry (2012)

- Giacomo Cherubini (2012)

- Cyril Benezet (2013)

- Victor Cauchois (2013)

- Wouter van de Vijver (2015)

- Pranav Nuti (2016)

- Hao Zhang (2016)

- Orel Cosseron (2018)

- Arthur Brugnon (2018)

- Samuel Laurent (2020)

- Leonard Pelletier (2022)

- Paolo Bordignon (2023)

- Pierre Morain (2023)

- Hao Zhang (2017-2020), co-supervised with Henri Darmon, thesis entitled ** "Elliptic cocycle on GLn(Z) and Hecke operators".** Defense at Sorbonne Univ. on Sept. 10th 2020

- Pierre Morain (2023-), co-supervised with Antonin Guilloux. thesis entitled ** "Arithmétique des fonctions Gamma elliptiques d'ordre supérieur".** In progress

French Keywords :

Soutenue le 13 décembre 2004 à l'Université Bordeaux 1 après avis favorable des rapporteurs :

devant un jury composé de

The complete text of my thesis is available as a pdf file at (thèse.pdf), or as a .ps file at (thèse.ps).

A summary is also avalaible here (résumé.pdf),

To contact me : First Name DOT Full Lastname at imj-prg DOT fr

Page modified on 24/1/2024