A conference in honour of Laurent Clozel, LMO, Paris-Saclay
Monday, September 49h45-10h15 Harris
10h15-11h15 Venkatesh
11h30-12h30 Thorne
12h30-14h
Lunch
14h-15hZhao
15h15-16h15 Mézard

Introduction
Some things I learned from Clozel
Level-raising and symmetric power functoriality
The Langlands parametrization for covering groups
Potentially Barsotti-Tate deformations rings

Tuesday, September 5
10h15-11h15 Colmez
11h30-12h30 Burungale
Zeta elements for elliptic curves and applications
12h30-14h
Lunch
14h-15h
Beuzart-Plessis Isolation of the cuspidal spectrum, multipliers
and the real Paley-Wiener theorem
15h15-16h15 Kret
The H 0 of Igusa varieties via automorphic forms
16h30-17h30 Lafforgue
Progress on local shtukas
18h-20h
Cocktail
Wednesday, September 6
9h45-10h45 Calegari The arithmetic of modular forms
11h-12h
Lapid
Some remarks on the geometric side of Arthur’s trace formula
12h15-13h15 Jiang
Godement-Jacquet Kernels and Clozel Theorem
13h15-14h30 Lunch
Thursday, September 7
10h15-11h15 Taylor
The formalism of Shimura varieties
11h30-12h30 Fouquet Celebrating (in) 2023
12h30-14hLunch
14h-15h
Shin
Endoscopic classification for unitary groups
15h15-16h15 Schraen Multiplicities and (non-)classicality for p-adic automorphic forms
Friday, September 8
10h15-11h15 Breuil
Splitting the de Rham complex of the Drinfeld space
11h30-12h30 Bergeron Complex cubic fields and SL3 (Z)-units
12h30-14h
Lunch
14h-15h
Chenevier On level one algebraic cuspforms of GL(n) over Q
15h15-16h15 Emerton

12Abstracts :— Bergeron : Complex cubic fields and SL3 (Z)-units
In the early 2000’s Felder and Varchenko have defined a generalisation of the Euler gamma function that is associated to an elliptic curve. This remarkable elliptic
gamma function is meromorphic in several variables and comes from mathematical physic. It satisfies modular functional equations under the group SL3 (Z) whichmake it an analogue of the Jacobi theta function. In this paper, we unveil the placethat this function and its avatars play in number theory. Our main thesis is thatthese functions play the role of meromorphic modular functions in extending thetheory of complex multiplication to complex cubic fields. In other words we propose a conjectural solution to Hilbert’s 12th problem for complex cubic fields.
We give a lot of numerical evidences that support this conjecture. In addition weprove an analogue of the Kronecker limit formula that expresses the logarithm of
the modulus of these conjectural units as the derivative at s = 0 of a partial zetafunction. This relates our conjectural units to the conjectural Stark units. This isa joint work with Pierre Charollois and Luis Garcia.
— Beuzart-Plessis : Isolation of the cuspidal spectrum, multipliers and the real
Paley-Wiener theorem
In this talk, I will review a construction of convolution operators that isolate
certain cuspidal representations from the rest of the automorphic spectrum. This
allows for drastic simplifications in the analysis of spectral sides of certain (relative)
trace formulas. For this, we combine the action of spherical Hecke algebras at
unramified places with that of an algebra of ”multipliers” at Archimedean places.
In particular, it is crucial that the multiplier algebra we use be sufficiently large.
Time permitting, I will also speculate on a possible extension of Arthur’s PaleyWiener theorem for real reductive groups that fits very naturally in this story.
This is based on joint work with Yifeng Liu, Wei Zhang and Xinwen Zhu.
— Burungale : Zeta elements for elliptic curves and applications
The talk plans to outline the existence of two-variable zeta element over animaginary quadratic field for an elliptic curve defined over Q. Its arithmetic consequences include proof of Kobayashi’s main conjecture for semistable curves. (Jointwith C. Skinner, Y. Tian and X. Wan.)— Calegari : The arithmetic of modular forms
We discuss a number of new applications of transcendental number theory tothe theory of modular forms.
— Fouquet : Celebrating (in) 2023
There are great mathematicians, and there are small mathematicians. Laurent
Clozel is a great mathematician. This talk, as befits the speaker, will be verysmall : it will be concerned mostly about one elliptic curve at one prime - theonly rational elliptic curve E of conductor 2023 and algebraic rank 2, and theprime 17. I will show that the Tate-Shafarevich group of this elliptic curve at thisprime has the order predicted by the Birch and Swinnerton-Dyer Conjecture (notethat E has rank 2 and very bad reduction at 17 so that the result is small, butnonetheless new). Along the way, we will encounter universal deformation rings,
p-adic families of automorphic representations, special values of L-functions andmany other things we could not possibly have studied without Laurent Clozel.
— Jiang : Godement-Jacquet Kernels and Clozel Theorem
In a recent work of L. Clozel, certain kernel functions on GL(1) and its connection with problems in analytic number theory were introduced and studied. In3

this talk, I will explain that the reformulation of the work of Godement-Jacquet
can be used to understand the framework of those kernels and its dual kernels forany irreducible cuspidal automorphic representations π of GL(n). Along the way,
I will also discuss other related problems. This is based on my joint work withZhilin Luo and Zhalin Li.
— Lafforgue : Progress on local shtukas
I will explain recent progress on local shtukas in characteristic p, in a joint workwith Genestier and a work in progress with Eteve, Gaitsgory and Genestier. I willexplain also the compatibility with the cohomology of global shtukas.
— Lapid : Some remarks on the geometric side of Arthur’s trace formula
In this talk I will report on work in progress with Tobias Finis aiming at expressing the unipotent contribution of Arthur’s trace formula in terms of zetafunctions of prehomogeneous vector spaces.
— Mézard : Potentially Barsotti-Tate deformations rings
In collaboration with B. Le Hung and S. Morra, we develop a local model theory
for moduli stacks of étale (ϕ, Γ)-modules corresponding to 2-dimensional nonscalar tame potentially Barsotti–Tate Galois representations of the Galois groupof an unramified extension of Qp . We derive from this explicit presentations ofpotentially Barsotti–Tate deformation rings, in particular the proof of the variousCaruso–David–Mézard conjectures.
— Schraen : Multiplicities and (non-)classicality for p-adic automorphic forms
We prove the existence of non-classical overconvergent p-adic automorphic forms
for the group U(3) with a classical system of eigenvalues. I will explain how thisis related to the existence of multiplicities for locally algebraic vectors in the expected p-adic Langlands correspondence and a relation with a geometric formof this correspondence. This is a joint work with Eugen Hellmann and Valentin
Hernandez.
— Shin : Endoscopic classification for unitary groups
Arthur proved the endoscopic classification theorem for automorphic representations of quasi-split symplectic and orthogonal groups about a decade ago. It
was extended to quasi-split unitary groups by Mok, and partly to non-quasi-split
unitary groups by Kaletha, Minguez, White, and myself. In particular, this strengthens the cohomological base change for unitary groups due to Clozel and Labesse.
All of these theorems, except Clozel-Labesse’s base change, are conditional on certain expected results. I will discuss their current status as well as a related ongoingproject with Atobe, Gan, Ichino, Kaletha, and Minguez.
— Taylor : The formalism of Shimura varieties
This is joint work with Jack Sempliner. In the 1970’s Deligne proposed a formalism for Shimura varieties and Langlands conjectured a formula for the action ofGalois on them. Deligne’s formalism involves ‘Shimura data’ which parametrizes
a Shimura together with a preferred embedding of its field of definition into thecomplex numbers. We propose a different formalism that directly parametrizes a
Shimura variety over any field of characteristic 0. To this end we have to revisit
the theory of conjugation of Shimura varieties conjectured by Langlands and established by Milne and others. We hope that our reformulation of this work also
helps to clarify the theory. A key tool is Kottwitz’s definition of the groups B(G).
In this talk I will sketch the formalism of Deligne and Langlands and pointout what we see as its shortcomings. I will then introduce Kottwitz’s cohomology
theory for extensions of Galois groups, and explain how to make cocycles a canonical object, not just cohomology classes. Finally I shall explain a reformulation of

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the theory of conjugation of Deligne’s Shimura varieties and propose an alternative
formalism for Shimura varieties.
— Thorne : Level-raising and symmetric power functoriality
In 2014 Clozel and I formulated a programme to prove the existence of all symmetric power liftings of Hilbert modular forms of regular weight, conditional on
conjectures concerning the existence of level-raising congruences and cases of Langlands tensor product functoriality. In 2022 I proved the existence of sufficiently
many of these level-raising congruences and, in joint work with James Newton,
applied this to carry out our programme unconditionally. In this talk I will review
the context and discuss the proof of existence of these congruences.
— Venkatesh : Some things I learned from Clozel
In 2002, shortly after I finished my PhD, I attended a series of lectures givenby Clozel at Park City. There he made a beautiful observation to which I havereturned many times over my career, namely, Arthur’s global conjectures havevery surprising ”purity” implications for questions in local harmonic analysis.
This implies, in particular, that many features of the harmonic analysis of G
acting on L2 (G/H) (for G a reductive group over a local field) are controlled bya single nilpotent orbit in the dual group Ĝ. I will sketch Clozel’s argument andwhy I find this phenomenon so interesting. I will then discuss the important roleof this orbit in ”relative Langlands duality”.
— Zhao : The Langlands parametrization for covering groups
In this talk, I will report on some recent progress in the Langlands program forcovering groups, as initiated by Weissman, Gan, and Gao. This extension of theLanglands program puts certain topological covers of reductive groups on equal
footing as linear algebraic groups. I will explain the cohomological origin of thesecovers and how it allows us to transport known constructions of the Langlands
parametrization to covering groups.