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Nicolas Bergeron

Publications and source records attributed to Nicolas Bergeron.

At least 19 recordsLinked to original sources
arXiv2023

Elliptic units for complex cubic fields

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 $\mathrm{SL}_3(\mathbf{Z})$ 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.

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arXiv2023

Cocycles de groupe pour GL$_n$ et arrangements d'hyperplans

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.''

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arXiv2022

On homology torsion growth

We prove new vanishing results on the growth of higher torsion homologies for suitable arithmetic lattices, Artin groups and mapping class groups. The growth is understood along Farber sequences, in particular, along residual chains. For principal congruence subgroups, we also obtain strong asymptotic bounds for the torsion growth. As a central tool, we introduce a quantitative homotopical method called effective rebuilding. This constructs small classifying spaces of finite index subgroups, at the same time controlling the complexity of the homotopy. The method easily applies to free abelian groups and then extends recursively to a wide class of residually finite groups.

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arXiv2022

Eisenstein cohomology classes for $\mathrm{GL}_N$ over imaginary quadratic fields

We study the arithmetic of degree $N-1$ Eisenstein cohomology classes for locally symmetric spaces associated to $\mathrm{GL}_N$ over an imaginary quadratic field $k$. Under natural conditions we evaluate these classes on $(N-1)$-cycles associated to degree $N$ extensions $F/k$ as linear combinations of generalised 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 multiplication by $k$. We recover in particular the algebraicity of these critical values.

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arXiv2022

Eigenfunctions and Random Waves in the Benjamini-Schramm limit

We investigate the asymptotic behavior of eigenfunctions of the Laplacian on Riemannian manifolds. We show that Benjamini-Schramm convergence provides a unified language for the level and eigenvalue aspects of the theory. As a result, we present a mathematically precise formulation of Berry's conjecture for a compact negatively curved manifold and formulate a Berry-type conjecture for sequences of locally symmetric spaces. We prove some weak versions of these conjectures. Using ergodic theory, we also analyze the connections of these conjectures to Quantum Unique Ergodicity.

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arXiv2021

Convergence of normalized Betti numbers in nonpositive curvature

We study the convergence of volume-normalized Betti numbers in Benjamini-Schramm convergent sequences of non-positively curved manifolds with finite volume. In particular, we show that if $X$ is an irreducible symmetric space of noncompact type, $X \neq \mathbb H^3$, and $(M_n)$ is any Benjamini-Schramm convergent sequence of finite volume $X$-manifolds, then the normalized Betti numbers $b_k(M_n)/vol(M_n)$ converge for all $k$. As a corollary, if $X$ has higher rank and $(M_n)$ is any sequence of distinct, finite volume $X$-manifolds, the normalized Betti numbers of $M_n$ converge to the $L^2$ Betti numbers of $X$. This extends our earlier work with Nikolov, Raimbault and Samet, where we proved the same convergence result for uniformly thick sequences of compact $X$-manifolds.

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arXiv2021

Tautological classes of hyper-Kähler manifolds. Erratum

This note is an erratum to the paper "Tautological classes on moduli spaces of hyper-Kähler manifolds." Thorsten Beckman and Mirko Mauri have pointed to us a gap in the proof of \cite[Theorem 8.2.1]{Duke}. We do not know how to correct the proof. We can only recover a partial statement. This gap affects the proof of one of the two main results of \cite{Duke}, we explain how to correct it.

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arXiv2017

Tautological classes on moduli space of hyperkähler manifolds

In this paper, we discuss the cycle theory on moduli spaces $\cF_h$ of $h$-polarized hyperkähler manifolds. Firstly, we construct the tautological ring on $\cF_h$ following the work of Marian, Oprea and Pandharipande on the tautological conjecture on moduli spaces of K3 surfaces. We study the tautological classes in cohomology groups and prove that most of them are linear combinations of Noether-Lefschetz cycle classes. In particular, we prove the cohomological version of the tautological conjecture on moduli space of K3$^{[n]}$-type hyperkähler manifolds with $n\leq 2$. Secondly, we prove the cohomological generalized Franchetta conjecture on universal family of these hyperkähler manifolds.

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arXiv2017

A note on local rigidity

The aim of this note is to give a geometric proof for classical local rigidity of lattices in semisimple Lie groups. We are reproving well known results in a more geometric (and hopefully clearer) way.

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arXiv2017

On the growth of $L^2$-invariants for sequences of lattices in Lie groups

We study the asymptotic behaviour of Betti numbers, twisted torsion and other spectral invariants of sequences of locally symmetric spaces. Our main results are uniform versions of the DeGeorge--Wallach Theorem, of a theorem of Delorme and various other limit multiplicity theorems. A basic idea is to adapt the notion of Benjamini--Schramm convergence (BS-convergence), originally introduced for sequences of finite graphs of bounded degree, to sequences of Riemannian manifolds, and analyze the possible limits. We show that BS-convergence of locally symmetric spaces implies convergence, in an appropriate sense, of the associated normalized relative Plancherel measures. This then yields convergence of normalized multiplicities of unitary representations, Betti numbers and other spectral invariants. On the other hand, when the corresponding Lie group $G$ is simple and of real rank at least two, we prove that there is only one possible BS-limit, i.e. when the volume tends to infinity, locally symmetric spaces always BS-converge to their universal cover $G/K$. This leads to various general uniform results. When restricting to arbitrary sequences of congruence covers of a fixed arithmetic manifold we prove a strong quantitative version of BS-convergence which in turn implies upper estimates on the rate of convergence of normalized Betti numbers in the spirit of Sarnak--Xue. An important role in our approach is played by the notion of Invariant Random Subgroups. For higher rank simple Lie groups $G$, we exploit rigidity theory, and in particular the Nevo--Stück--Zimmer theorem and Kazhdan's property (T), to obtain a complete understanding of the space of IRSs of $G$.

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arXiv2016

On the growth of L2-invariants of locally symmetric spaces, II: exotic invariant random subgroups in rank one

In the first paper of this series (arxiv.org/abs/1210.2961) we studied the asymptotic behavior of Betti numbers, twisted torsion and other spectral invariants for sequences of lattices in Lie groups G. A key element of our work was the study of invariant random subgroups (IRSs) of G. Any sequence of lattices has a subsequence converging to an IRS, and when G has higher rank, the Nevo-Stuck-Zimmer theorem classifies all IRSs of G. Using the classification, one can deduce asymptotic statments about spectral invariants of lattices. When G has real rank one, the space of IRSs is more complicated. We construct here several uncountable families of IRSs in the groups SO(n,1). We give dimension-specific constructions when n=2,3, and also describe a general gluing construction that works for every n at least 2. Part of the latter construction is inspired by Gromov and Piatetski-Shapiro's construction of non-arithmetic lattices in SO(n,1).

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arXiv2016

Twisted limit formula for torsion and cyclic base change

Let $G$ be the group of complex points of a real semi-simple Lie group whose fundamental rank is equal to 1, e.g. $G= \SL_2 (\C) \times \SL_2 (\C)$ or $\SL_3 (\C)$. Then the fundamental rank of $G$ is $2,$ and according to the conjecture made in \cite{BV}, lattices in $G$ should have 'little' --- in the very weak sense of 'subexponential in the co-volume' --- torsion homology. Using base change, we exhibit sequences of lattices where the torsion homology grows exponentially with the \emph{square root} of the volume. This is deduced from a general theorem that compares twisted and untwisted $L^2$-torsions in the general base-change situation. This also makes uses of a precise equivariant 'Cheeger-Müller Theorem' proved by the second author \cite{Lip1}.

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arXiv2015

The Noether-Lefschetz conjecture and generalizations

We prove the Noether-Lefschetz conjecture on the moduli space of quasi-polarized K3 surfaces. This is deduced as a particular case of a general theorem that states that low degree cohomology classes of arithmetic manifolds of orthogonal type are dual to the classes of special cycles, i.e. sub-arithmetic manifolds of the same type. For compact manifolds this was proved in \cite{BMM11}, here we extend the results of \cite{BMM11} to non-compact manifolds. This allows us to apply our results to the moduli spaces of quasi-polarized K3 surfaces.

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arXiv2015

The Relative Lie Algebra Cohomology of the Weil Representation of SO(n,1)

In Part 1 of this paper we construct a spectral sequence converging to the relative Lie algebra cohomology associated to the action of any subgroup $G$ of the symplectic group on the polynomial Fock model of the Weil representation, see Section 7. These relative Lie algebra cohomology groups are of interest because they map to the cohomology of suitable arithmetic quotients of the symmetric space $G/K$ of $G$. We apply this spectral sequence to the case $G = \mathrm{SO}_0(n,1)$ in Sections 8, 9, and 10 to compute the relative Lie algebra cohomology groups $H^{\bullet} \big(\mathfrak{so}(n,1), \mathrm{SO}(n); \mathcal{P}(V^k) \big)$. Here $V = \mathbb{R}^{n,1}$ is Minkowski space and $\mathcal{P}(V^k)$ is the subspace of $L^2(V^k)$ consisting of all products of polynomials with the Gaussian. In Part 2 of this paper we compute the cohomology groups $H^{\bullet}\big(\mathfrak{so}(n,1), \mathrm{SO}(n); L^2(V^k) \big)$ using spectral theory and representation theory. In Part 3 of this paper we compute the maps between the polynomial Fock and $L^2$ cohomology groups induced by the inclusions $\mathcal{P}(V^k) \subset L^2(V^k)$.

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arXiv2015

Hodge type theorems for arithmetic manifolds associated to orthogonal groups

We show that special cycles generate a large part of the cohomology of locally symmetric spaces associated to orthogonal groups. We prove in particular that classes of totally geodesic submanifolds generate the cohomology groups of degree $n$ of compact congruence $p$-dimensional hyperbolic manifolds "of simple type" as long as $n$ is strictly smaller than $\frac{p}{3}$. We also prove that for connected Shimura varieties associated to $\OO (p,2)$ the Hodge conjecture is true for classes of degree $< \frac{p+1}{3}$. The proof of our general theorem makes use of the recent endoscopic classification of automorphic representations of orthogonal groups by \cite{ArthurBook}. As such our results are conditional on the hypothesis made in this book, whose proofs have only appear on preprint form so far; see the second paragraph of subsection \ref{org2} below.

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arXiv2014

Sur le spectre et la topologie des variétés hyperboliques de congruence : les cas complexe et quaternionien

Building on results of Arthur and Mok, we extend to (finite volume) complex and quaternionic hyperbolic manifolds the results of arXiv:1004.1085. For the spherical spectrum our results are optimal. Finally, as an application we prove a Lefschetz property for the restriction map between arithmetic quotients of complex balls. This generalizes a recent theorem of Arvind Nair and gives an optimal version of it.

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arXiv2014

The Hodge conjecture and arithmetic quotients of complex balls

Let $S$ be a closed Shimura variety uniformized by the complex $n$-ball. The Hodge conjecture predicts that every Hodge class in $H^{2k} (S, \Q)$, $k=0, \ldots, n$, is algebraic. We show that this holds for all degree $k$ away from the neighborhood $]n/3, 2n/3[$ of the middle degree. We also address the Tate conjecture and the generalized form of the Hodge conjecture and extend most of our results to Shimura varieties associated to unitary groups of any signature. The proofs make use of the recent endoscopic classification of automorphic representations of classical groups by \cite{ArthurBook,Mok}. As such our results are conditional on the stabilization of the trace formula for the (disconnected) groups $\GL (N) \rtimes \langle θ\rangle$ associated to base change. Unfortunately, at present the stabilization of the trace formula has been proved only for the case of {\it connected} groups. The extension needed is part of work in progress by the Paris-Marseille team of automorphic form researchers. For more detail, see the second paragraph of subsection \ref{org2} below.

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