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Gaëtan Chenevier

Publications and source records attributed to Gaëtan Chenevier.

16 recordsLinked to original sources
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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arXiv2025

Triality and Functoriality

We use the triality automorphism of simple algebraic groups of type $D_4$ to prove some new instances of global Langlands functorial lifting. In particular, we prove the (weak) spin lifting from ${\rm GSp}_6$ to ${\rm GL}_8$ and the tensor product lifting from ${\rm GL}_2 \times {\rm GSp}_4$ to ${\rm GL}_8$. As an arithmetic application, we establish the expected properties of the spinor L-function attached to an arbitrary Siegel modular cusp form for ${\rm Sp}_6(\mathbb{Z})$ generating a holomorphic discrete series.

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arXiv2024

Unimodular Hunting II

Pursuing ideas in a recent work of the second author, we determine the isometry classes of unimodular lattices of rank 28, as well as the isometry classes of unimodular lattices of rank 29 without nonzero vectors of norm <=2.

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arXiv2024

Unimodular Hunting

We develop a method initiated by Bacher and Venkov, and based on a study of the Kneser neighbors of the standard lattice Z^n, which allows to classify the integral unimodular Euclidean lattices of rank n. As an application, of computational flavour, we determine the isometry classes of unimodular lattices of rank 26 and 27.

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arXiv2023

${\rm Spin}(7)$ is unacceptable

We classify the pairs of group morphisms $Γ\rightarrow {\rm Spin}(7)$ which are element conjugate but not globally conjugate. As an application, we study the case where $Γ$ is the Weil group of a $p$-adic local field, which is relevant to the recent approach to the local Langlands correspondence for ${\rm G}_2$ and ${\rm PGSp}_6$ by Gan and Savin. As a second application, we improve some result of Kret and Shin about ${\rm GSpin}_7$-valued Galois representations.

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arXiv2021

Statistics for Kneser p-neighbors

Let L and L' be two integral Euclidean lattices in the same genus. We give an asymptotic formula for the number of Kneser p-neighbors of L which are isometric to L', when the prime p goes to infinity. In the case L is unimodular, and if we fix furthermore a subgroup A of L, we also give an asymptotic formula for the number of p-neighbors of L containing A and which are isomorphic to L'. These statements explain numerical observations in the recent classifications of unimodular lattices of rank 26, 27 and 28, by B. Allombert and the author. In an Appendix, O. Taïbi shows how to deduce from Arthur's results the existence of global parameters associated to automorphic representations of definite orthogonal groups over the rationals.

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arXiv2020

The Characteristic Masses of Niemeier Lattices

Let $L$ be an integral lattice in the Euclidean space $\mathbb{R}^n$ and $W$ an irreducible representation of the orthogonal group of $\mathbb{R}^n$. We give an implemented algorithm computing the dimension of the subspace of invariants in $W$ under the isometry group ${\rm O}(L)$ of $L$. A key step is the determination of the number of elements in ${\rm O}(L)$ having any given characteristic polynomial, a datum that we call the {\it characteristic masses} of $L$. As an application, we determine the characteristic masses of all the Niemeier lattices, and more generally of any even lattice of determinant $\leq 2$ in dimension $n \leq 25$. For Niemeier lattices, as a verification, we provide an alternative (human) computation of the characteristic masses. The main ingredient is the determination, for each Niemeier lattice $L$ with non-empty root system $R$, of the ${\rm G}(R)$-conjugacy classes of the elements of the "umbral" subgroup ${\rm O}(L)/{\rm W}(R)$ of ${\rm G}(R)$, where ${\rm G}(R)$ is the automorphism group of the Dynkin diagram of $R$, and ${\rm W}(R)$ its Weyl group. These results have consequences for the study of the spaces of automorphic forms of the definite orthogonal groups in $n$ variables over $\mathbb{Q}$. As an example, we provide concrete dimension formulas in the level $1$ case, as a function of the weight $W$, up to dimension $n=25$.

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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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arXiv2019

On the minima of positive definite binary hamiltonian forms

Let $A$ be a definite quaternion algebra over $\mathbb Q$, with discriminant $D_A$, and $O$ a maximal order of $A$. We show that the minimum of the positive definite hamiltonian binary forms over $O$ with discrimiminant $-1$ is $\sqrt{D_A}$. When the different of $O$ is principal, we provide an explicit form representing this minimum, and when $O$ is principal, we give the list of the equivalence classes of all such forms. We also give criteria and algorithms to determine when the different of $O$ is principal.

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arXiv2018

An automorphic generalization of the Hermite-Minkowski theorem

We show that for any integer $N$, there are only finitely many cuspidal algebraic automorphic representations of ${\rm GL}_n$ over $\mathbb{Q}$, with $n$ varying, whose conductor is $N$ and whose weights are in the interval $\{0,1,...,23\}$. More generally, we define a simple sequence $(r(w))_{w \geq 0}$ such that for any integer $w$, any number field $E$ whose root-discriminant is less than $r(w)$, and any ideal $N$ in the ring of integers of $E$, there are only finitely many cuspidal algebraic automorphic representations of general linear groups over $E$ whose conductor is $N$ and whose weights are in the interval $\{0,1,...,w\}$. Assuming a version of GRH, we also show that we may replace $r(w)$ with $8 πe^{γ-H_w}$ in this statement, where $γ$ is Euler's constant and $H_w$ the $w$-th harmonic number. The proofs are based on some new positivity properties of certain real quadratic forms which occur in the study of the Weil explicit formula for Rankin-Selberg $L$-functions. Both the effectiveness and the optimality of the methods are discussed.

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arXiv2016

Subgroups of Spin(7) or SO(7) with each element conjugate to some element of G_2, and applications to automorphic forms

As is well-known, the compact groups Spin(7) and SO(7) both have a single conjugacy class of compact subgroups of exceptional type G_2. We first show that if H is a subgroup of Spin(7), and if each element of H is conjugate to some element of G_2, then H itself is conjugate to a subgroup of G_2. The analogous statement for SO(7) turns out be false, and our main result is a classification of all the exceptions. They are the following groups, embedded in each case in SO(7) in a very specific way: GL_2(Z/3Z), SL_2(Z/3Z), Z/4Z x Z/2Z, as well as the nonabelian subgroups of GO_2(C) with compact closure, similitude factors group {-1,1}, and which are not isomorphic to the dihedral group of order 8. More generally, we consider the analogous problems in which the Euclidean space is replaced by a quadratic space of dimension 7 over an arbitrary field. This type of questions naturally arises in some formulation of a converse statement of Langlands' global functoriality conjecture, to which the results above have thus some applications. Moreover, we give necessary and sufficient local conditions on a cuspidal algebraic regular automorphic representation of GL_7 over a totally real number field so that its associated \ell-adic Galois representations can be conjugate into G_2(\bar{Q_\ell}).

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arXiv2015

Formes automorphes et voisins de Kneser des réseaux de Niemeier

In this memoir, we study the even unimodular lattices of rank at most 24, as well as a related collection of automorphic forms of the orthogonal, symplectic and linear groups of small rank. Our guide is the question of determining the number of p-neighborhoods, in the sense of M. Kneser, between two isometry classes of such lattices. We prove a formula for this number, in which occur certain Siegel modular forms of genus 1 and 2. It has several applications, such as the proof of a conjecture of G. Nebe and B. Venkov about the linear span of the higher genus theta series of the Niemeier lattices, the computation of the p-neighborhoods graphs of the Niemeier lattices (the case p = 2 being due to Borcherds), or the proof of a congruence conjectured by G. Harder. Classical arguments reduce the problem to the description of the automorphic representations of a suitable integral form of the Euclidean orthogonal group of R^24 which are unramified at each finite prime and trivial at the archimedean prime. The recent results of J. Arthur suggest several new approaches to this type of questions. This is the other main theme that we develop in this memoir. We give a number of other applications, for instance to the classification of Siegel modular cuspforms of weight at most 12 for the full Siegel modular group.

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arXiv2010

Sur la densité des représentations cristallines du groupe de Galois absolu de Q_p

Let X_d be the p-adic analytic space classifying the d-dimensional (semisimple) p-adic Galois representations of the absolute Galois group of Q_p. We show that the crystalline representations are Zarski-dense in many irreducible components of X_d, including the components made of residually irreducible representations. This extends to any dimension d previous results of Colmez and Kisin for d = 2. For this we construct an analogue of the infinite fern of Gouvêa-Mazur in this context, based on a study of analytic families of trianguline (phi,Gamma)-modules over the Robba ring. We show in particular the existence of a universal family of (framed, regular) trianguline (phi,Gamma)-modules, as well as the density of the crystalline (phi,Gamma)-modules in this family. These results may be viewed as a local analogue of the theory of p-adic families of finite slope automorphic forms, they are new already in dimension 2. The technical heart of the paper is a collection of results about the Fontaine-Herr cohomology of families of trianguline (phi,Gamma)-modules.

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arXiv2009

Représentations potentiellement triangulines de dimension 2

The two main results of this note are on the one hand that if V is a 2-dimensional potentially trianguline representation of G_Qp then V satisfies at least one of the following properties (1) V is split trianguline (2) V is a direct sum of characters or an induced representation (3) V is a twist of a de Rham representation, and on the other hand that there exists some 2-dimensional representations of G_Qp which are not potentially trianguline.

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