Related papers: A Coarse Jacquet-Zagier Trace Formula for GL($n$) …

This is an announcement of results on rank-one Eisenstein cohomology of GL(N), with N > 1 an odd integer, and algebraicity theorems for ratios of successive critical values of certain Rankin-Selberg L-functions for GL(n) x GL(n') when n is…

In this article we study the nonvanishing of cuspidal cohomology for GL(n). Using endoscopic transfer from various classical groups we construct cuspidal representations of GL(n) of cohomological type while working over a totally real field…

In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [1-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph…

We consider the variety of pre-Lie algebra structures on a given n-dimensional vector space. The group GL_n(K) acts on it, and we study the closure of the orbits with respect to the Zariski topology. This leads to the definition of pre-Lie…

We prove both the biquadratic Guo--Jacquet Fundamental Lemma (FL) and the biquadratic linear Arithmetic Fundamental Lemma (AFL) for GL(4) with the unit test function. Our approach relies on a detailed study of pairs of quadratic embeddings,…

Dirac's theorem states that any $n$-vertex graph $G$ with even integer $n$ satisfying $\delta(G) \geq n/2$ contains a perfect matching. We generalize this to $k$-uniform linear hypergraphs by proving the following. Any $n$-vertex…

In this paper we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [$1$-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph $G$ on $n$…

Let $(G_n(x))_{n=0}^\infty$ be a $d$-th order linear recurrence sequence having polynomial characteristic roots, one of which has degree strictly greater than the others. Moreover, let $m\geq 2$ be a given integer. We ask for…

Let $\pi$ be a cuspidal automorphic representation of PGL($2n$) over a number field $F$, and $\eta$ the quadratic idele class character attached to a quadratic extension $E/F$. Guo and Jacquet conjectured a relation between the nonvanishing…

We study certain cases of convoluted Fourier coefficients of $GL_n$-automorphic functions. We establish identities that express them in terms of Fourier coefficients related to unipotent orbits. The most general case that is studied is…

In this paper, we give a purely geometric approach to the local Jacquet-Langlands correspondence for GL(n) over a p-adic field, under the assumption that the invariant of the division algebra is 1/n. We use the l-adic etale cohomology of…

Using the stable twisted trace formula for the triality automorphism, we show the adjoint lifting (to GL(8)) of cuspidal representations of GL(3) with a discrete series local component. We also describe the possible isobaric decompositions…

We formulate Guo--Jacquet type fundamental lemma conjectures and arithmetic transfer conjectures for inner forms of $GL_{2n}$. Our main results confirm these conjectures for division algebras of invariant $1/4$ and $3/4$.

In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [1-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph…

In this note we prove the following three algebraic facts which have applications in the theory of holonomy groups and homogeneous spaces: Any irreducibly acting connected subgroup $G \subset Gl(n,\rr)$ is closed. Moreover, if $G$ admits an…

We prove finite jet determination results for smooth CR embeddings which are of constant degeneracy, using the method of complete systems. As an application, we derive a reflection principle for mappings between a Levi-nondegenerate…

Fix $n \geq 2$ an integer, and let $F$ be a totally real number field. We derive estimates for the finite parts of the $L$-functions of irreducible cuspidal $\operatorname{GL}_n({\bf{A}}_F)$-automorphic representations twisted by class…

We prove the $P=W$ conjecture for $\mathrm{GL}_n$ for all ranks $n$ and curves of arbitrary genus $g\geq 2$. The proof combines a strong perversity result on tautological classes with the curious Hard Lefschetz theorem of Mellit. For the…

The Local Converse Problem is to determine how the family of the local gamma factors $\gamma(s,\pi\times\tau,\psi)$ characterizes the isomorphism class of an irreducible admissible generic representation $\pi$ of $\mathrm{GL}_n(F)$, with…

For a regular $2n$-gon there are $(2n-1)!!$ ways to match and glue the $2n$ sides. The Harer-Zagier bivariate generating function enumerates the gluings by $n$ and the genus $g$ of the attendant surface and leads to a recurrence equation…