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Michał Zydor

Publications and source records attributed to Michał Zydor.

9 recordsLinked to original sources
arXiv2020

The global Gan-Gross-Prasad conjecture for unitary groups: the endoscopic case

In this paper, we prove the Gan-Gross-Prasad conjecture and the Ichino-Ikeda conjecture for unitary groups $U_n\times U_{n+1}$ in all the endoscopic cases. Our main technical innovation is the computation of the contributions of certain cuspidal data, called $*$-generic, to the Jacquet-Rallis trace formula for linear groups. We offer two different computations of these contributions: one, based on truncation, is expressed in terms of regularized Rankin-Selberg periods of Eisenstein series and Flicker-Rallis intertwining periods. The other, built upon Zeta integrals, is expressed in terms of functionals on the Whittaker model. A direct proof of the equality between the two expressions is also given. Finally several useful auxiliary results about the spectral expansion of the Jacquet-Rallis trace formula are provided.

math.RT↗
arXiv2019

Periods of automorphic forms over reductive subgroups

We present a regularization procedure of period integrals of automorphic forms on a group $G$ over an arbitrary reductive subgroup $G' \subset G$. As a consequence we obtain an explicit $G'(\mathbb{A})$-invariant functional on the space of automorphic forms on $G$ whose exponents avoid certain prescribed hyperplanes. We also provide a necessary and sufficient condition for convergence of period integrals of automorphic forms in terms of their exponents.

math.NT↗
arXiv2019

On the residue method for period integrals

By applying the residue method for period integrals and Langlands-Shahidi's theory for residues of Eisenstein series, we study the period integrals for six spherical varieties. For each spherical variety, we prove a relation between the period integrals and certain automorphic L-functions. In some cases, we also study the local multiplicity of the spherical varieties.

math.NT↗
arXiv2018

A $\mathrm{G}_2$-period of a Fourier coefficient of an Eisenstein series on $\mathrm{E}_6$

We calculate a $\mathrm{G}_2$-period of a Fourier coefficient of a cuspidal Eisenstein series on the split simply-connected group $\mathrm{E}_6$, and relate this period to the Ginzburg-Rallis period of cusp forms on $\mathrm{GL}_6$. This gives us a relation between the Ginzburg-Rallis period and the central value of the exterior cube L-function of $\mathrm{GL}_6$

math.NT↗
arXiv2016

Le transfert singulier pour la formule des traces de Jacquet-Rallis

The relative trace formula of Jacquet-Rallis (for unitary groups or general linear groups) is an identity between periods of automorphic representations and geometric distributions. In this paper, we prove the transfer between all geometric terms of unitary groups and those of linear groups. We also show that all geometric terms are in the weak closure of local regular semi-simple orbital integrals. We mention an application to the Gan-Gross-Prasad conjecture for unitary groups.

math.RT↗
arXiv2015

La variante infinitésimale de la formule des traces de Jacquet-Rallis pour les groupes unitaires

We establish an infinitesimal version of the Jacquet-Rallis trace formula for unitary groups. Our formula is obtained by integrating a truncated kernel à la Arthur. It has a geometric side which is a sum of distributions $J_{\mathfrak{o}}$ indexed by classes of elements of the Lie algebra of $U(n+1)$ stable by $U(n)$-conjugation as well as the "spectral side" consisting of the Fourier transforms of the aforementioned distributions. We prove that the distributions $J_{\mathfrak{o}}$ are invariant and depend only on the choice of the Haar measure on $U(n)(\mathbb{A})$. For regular semi-simple classes $\mathfrak{o}$, $J_{\mathfrak{o}}$ is a relative orbital integral of Jacquet-Rallis. For classes $\mathfrak{o}$ called relatively regular semi-simple, we express $J_{\mathfrak{o}}$ in terms of relative orbital integrals regularised by means of zêta functions.

math.NT↗
arXiv2015

Les formules des traces relatives de Jacquet-Rallis grossières

We establish the coarse relative trace formulae of Jacquet-Rallis for linear and unitary groups. Both formulae are of the form: a sum of spectral distributions equals a sum of geometric distributions. In order to obtain the spectral decompositions we introduce new truncation operators and we investigate their properties. On the geometric side, by means of the Cayley transform, the decompositions are derived from a procedure of descent to the tangent spaces for which the formulae are known thanks to our previous work.

math.NT↗
arXiv2015

La variante infinitésimale de la formule des traces de Jacquet-Rallis pour les groupes linéaires

We establish an infinitesimal version of the Jacquet-Rallis trace formula for general linear groups. Our formula is obtained by integrating a kernel truncated a la Arthur multiplied by the absolute value of the determinant to the power $s \in \mathbb{C}$. It has a geometric side which is a sum of distributions $I_{\mathfrak{o}}(s, \cdot)$ indexed by the invariants of the adjoint action of $\mathrm{GL}_n(\mathrm{F})$ on $\mathfrak{gl}_{n+1}(\mathrm{F})$ as well as a "spectral side" consisting of the Fourier transforms of the aforementioned distributions. We prove that the distributions $I_{\mathfrak{o}}(s, \cdot)$ are invariant and depend only on the choice of the Haar measure on $\mathrm{GL}_n(\mathbb{A})$. For regular semi-simple classes $\mathfrak{o}$, $I_{\mathfrak{o}}(s, \cdot)$ is a relative orbital integral of Jacquet-Rallis. For classes $\mathfrak{o}$ called relatively regular semi-simple, we express $I_{\mathfrak{o}}(s, \cdot)$ in terms of relative orbital integrals regularised by means of zeta functions.

math.NT↗