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Olivier Taïbi

Publications and source records attributed to Olivier Taïbi.

15 recordsLinked to original sources
arXiv2026

Invariance Galoisienne des zéros centraux de fonctions L

Nous démontrons l'invariance Galoisienne de la propriété d'annulation en $1/2$ des fonctions L standard ou de Rankin-Selberg pour certaines représentations automorphes cuspidales algébriques régulières autoduales ou autoduales conjuguées de groupes linéaires sur un corps de nombres arbitraire. La démonstration repose sur l'utilisation de la cohomologie pondérée de Goresky-Harder-MacPherson et sur la construction de certaines représentations automorphes discrètes pour les groupes classiques comme résidus de séries d'Eisenstein. L'abandon de l'hypothèse ``$F$ totalement réel'' introduit de nouvelles difficultés concernant certains opérateurs d'entrelacement. Celles-ci sont résolues grâce à l'appendice, rédigé par J.-L. Waldspurger et l'un d'entre nous, démontrant l'holomorphie et la non-annulation de certains opérateurs d'entrelacement normalisés. Nous démontrons également l'invariance Galoisienne des facteurs epsilon correspondants, impliquant l'invariance Galoisienne de la parité de l'ordre d'annulation en $1/2$ de ces fonctions $L$. -- We prove the invariance under the Galois group of the vanishing at $1/2$ of standard and Rankin-Selberg L-functions for certain self-dual or conjugate self-dual algebraic cuspidal automorphic representations for general linear groups over an arbitrary number field. The proof uses Goresky-Harder-MacPherson weighted cohomology and the construction of certain discrete automorphic representations for classical groups as residues of Eisenstein series. New difficulties appear concerning certain intertwining operators. These are solved in the appendix by J.-L. Waldspurger and O. Taïbi proving the holomorphy and non-vanishing of these operators. We also prove the Galois invariance of epsilon factors, implying Galois invariance of the parity of the order at $1/2$ of L-functions.

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arXiv2026

Unimodular lattices of rank 29 and related even genera of small determinant

We classify the unimodular Euclidean integral lattices of rank 29 by developing an elementary, yet very efficient, inductive method. As an application, we determine the isometry classes of even lattices of rank at most 28 and prime (half-)determinant at most 7. We also provide new isometry invariants allowing for independent verification of the completeness of our lists, and we give conceptual explanations of some unique orbit phenomena discovered during our computations. Some of the genera classified here are orders of magnitude larger than any genus previously classified. In a forthcoming companion paper, we use these computations to study the cohomology of GL_n(Z).

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arXiv2026

A Nomizu-van Est theorem in Ekedahl's derived $\ell$-adic setting

A theorem of Nomizu and van Est computes the cohomology of a compact nilmanifold, or equivalently the group cohomology of an arithmetic subgroup of a unipotent linear algebraic group over $\mathbb{Q}$. We prove a similar result for the cohomology of a compact open subgroup of a unipotent linear algebraic group over $\mathbb{Q}_{\ell}$ with coefficients in a complex of continuous $\ell$-adic representations. We work with the triangulated categories defined by Ekedahl which play the role of ``derived categories of continuous $\ell$-adic representations''. This is motivated by Pink's formula computing the derived direct image of an $\ell$-adic local system on a Shimura variety in its minimal compactification, and its application to automorphic perverse sheaves on Shimura varieties. The key technical result is the computation of the cohomology with coefficients in a unipotent representation with torsion coefficients by an explicit complex of polynomial cochains which is of finite type.

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arXiv2026

The Euler characteristic of $\ell$-adic local systems on $\mathcal{A}_n$

We study the Euler characteristic of $\ell$-adic local systems on the moduli stack $\mathcal{A}_n$ of principally polarized abelian varieties of dimension $n$ associated to algebraic representations of $\mathbf{GSp}_{2n}$, as virtual representations of the absolute Galois group of $\mathbb{Q}$ and the unramified Hecke algebra of $\mathbf{GSp}_{2n}$. To this end we take the last steps of the Ihara-Langlands-Kottwitz method to compute the intersection cohomology of minimal compactifications of Siegel modular varieties in level one, following work of Kottwitz and Morel, proving an unconditional reformulation of Kottwitz' conjecture in this case. This entails proving the existence of $\mathrm{GSpin}$-valued Galois representations associated to certain level one automorphic representations for $\mathbf{PGSp}_{2n}$ and $\mathbf{SO}_{4n}$. As a consequence we prove the existence of $\mathrm{GSpin}$-valued Galois representations associated to level one Siegel eigenforms, a higher genus analogue of theorems of Deligne (genus one) and Weissauer (genus two). Using Morel's work and Franke's spectral sequence we derive explicit formulas expressing the Euler characteristic of compactly supported cohomology of automorphic $\ell$-adic local systems on Siegel modular varieties in terms of intersection cohomology. Specializing to genus three and level one, we prove an explicit conjectural formula of Bergström, Faber and van der Geer for the compactly supported Euler characteristic in terms of spin Galois representations associated to level one Siegel cusp forms. Specializing to trivial local systems we give explicit formulas for the number of points of $\mathcal{A}_n$ over finite fields for all $n \leq 7$.

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arXiv2026

The low degree cohomology of compactifications of $A_g$

We compute the low degree $\ell$-adic intersection cohomology of symplectic local systems on the Satake compactification of the moduli space $A_g$ of principally polarized abelian varieties. We prove that only a small finite list of irreducible Galois representations can appear in the low degree cohomology of any nonsingular toroidal compactification of $A_g$ or $X_{g,s}$, the $s$-fold fiber product of the universal abelian variety. We give several applications, including to spaces of holomorphic forms on toroidal compactifications and to the cohomology of the interior. In particular, we give a complete characterization of when the cohomology of $X_{g,s}$, or one of its toroidal compactifications, is of Tate type. The result is independent of the choice of toroidal compactification.

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arXiv2025

The local Langlands conjecture

We formulate the local Langlands conjecture for connected reductive groups over local fields, including the internal parametrization of L-packets using endoscopy.

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arXiv2019

Siegel modular forms of weight 13 and the Leech lattice

For $g=8,12,16$ and $24$, there is a nonzero alternating $g$-multilinear form on the ${\rm Leech}$ lattice, unique up to a scalar, which is invariant by the orthogonal group of ${\rm Leech}$. The harmonic Siegel theta series built from these alternating forms are Siegel modular cuspforms of weight $13$ for ${\rm Sp}_{2g}(\mathbb{Z})$. We prove that they are nonzero eigenforms, determine one of their Fourier coefficients, and give informations about their standard ${\rm L}$-functions. These forms are interesting since, by a recent work of the authors, they are the only nonzero Siegel modular forms of weight $13$ for ${\rm Sp}_{2n}(\mathbb{Z})$, for any $n\geq 1$.

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arXiv2019

Discrete series multiplicities for classical groups over Z and level 1 algebraic cusp forms

The aim of this paper is twofold. First, we introduce a new method for evaluating the multiplicity of a given discrete series in the space of level $1$ automorphic forms of a split classical group $G$ over $\mathbb{Z}$, and provide numerical applications in absolute rank $\leq 8$. Second, we prove a classification result for the level one cuspidal algebraic automorphic representations of ${\rm GL}_n$ over $\mathbb{Q}$ ($n$ arbitrary) whose motivic weight is $\leq 24$. In both cases, a key ingredient is a classical method based on the Weil explicit formula, which allows to disprove the existence of certain level one algebraic cusp forms on ${\rm GL}_n$, and that we push further on in this paper. We use these vanishing results to obtain an arguably ``effortless'' computation of the elliptic part of the geometric side of the trace formula of $G$, for an appropriate test function. Thoses results have consequences for the computation of the dimension of the spaces of (possibly vector-valued) Siegel modular cuspforms for ${\rm Sp}_{2g}(\mathbb{Z})$: we recover all the previously known cases without relying on any, and go further, by a unified and ``effortless'' method.

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arXiv2018

Global rigid inner forms vs isocrystals

We compare the cohomology of the global Galois gerbes constructed in [Kot] and [Kal18a], respectively, and give applications to the theory of endoscopy.

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arXiv2017

On rigid inner forms

We give an explicit construction of global Galois gerbes constructed more abstractly by Kaletha to define global rigid inner forms. This notion is crucial to formulate Arthur's multiplicity formula for inner forms of quasi-split reductive groups. As a corollary, we show that any global rigid inner form is almost everywhere unramified, and we give an algorithm to compute the resulting local rigid inner forms at all places in a given finite set. This makes global rigid inner forms as explicit as global pure inner forms, up to computations in local and global class field theory.

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arXiv2016

Galois and Cartan Cohomology of Real Groups

Real forms of a complex reductive group are classified by Galois cohomology H^1(Gamma,G_ad) where G_ad is the adjoint group. Cartan's classification of real forms in terms of maximal compact subgroups can be stated in terms of H^(Z/2Z,G_ad) where the action is by a (holomorphic) Cartan involution. The main result is that for any complex reductive group, possibly disconnected, there is a canonical isomorphism between H^1(Gamma,G) and H^1(Z/2Z,G). As applications we give short proofs of some well known results, including the Sekiguchi correspondence, Matsuki duality, results on Cartan subgroups, the rational Weyl group, and strong real forms. We also compute H^1(Gamma,G) for all simple, simply connected real groups.

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arXiv2014

Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula

We consider the problem of explicitly computing dimensions of spaces of automorphic or modular forms in level one, for a split classical group $\mathbf{G}$ over $\mathbb{Q}$ such that $\mathbf{G}(\R)$ has discrete series. Our main contribution is an algorithm calculating orbital integrals for the characteristic function of $\mathbf{G}(\mathbb{Z}_p)$ at torsion elements of $\mathbf{G}(\mathbb{Q}_p)$. We apply it to compute the geometric side in Arthur's specialisation of his invariant trace formula involving stable discrete series pseudo-coefficients for $\mathbf{G}(\mathbb{R})$. Therefore we explicitly compute the Euler-Poincaré characteristic of the level one discrete automorphic spectrum of $\mathbf{G}$ with respect to a finite-dimensional representation of $\mathbf{G}(\mathbb{R})$. For such a group $\mathbf{G}$, Arthur's endoscopic classification of the discrete spectrum allows to analyse precisely this Euler-Poincaré characteristic. For example one can deduce the number of everywhere unramified automorphic representations $π$ of $\mathbf{G}$ such that $π_{\infty}$ is isomorphic to a given discrete series representation of $\mathbf{G}(\mathbb{R})$. Dimension formulae for the spaces of vector-valued Siegel modular forms are easily derived.

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arXiv2012

Eigenvarieties for classical groups and complex conjugations in Galois representations

The goal of this paper is to remove the irreducibility hypothesis in a theorem of Richard Taylor describing the image of complex conjugations by $p$-adic Galois representations associated with regular, algebraic, essentially self-dual, cuspidal automorphic representations of $\GL_{2n+1}$ over a totally real number field $F$. We also extend it to the case of representations of $\GL_{2n}/F$ whose multiplicative character is "odd". We use a $p$-adic deformation argument, more precisely we prove that on the eigenvarieties for symplectic and even orthogonal groups, there are "many" points corresponding to (quasi-)irreducible Galois representations. The recent work of James Arthur describing the automorphic spectrum for these groups is used to define these Galois representations, and also to transfer self-dual automorphic representations of the general linear group to these classical groups.

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