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We show the existence of some non-classical cohomological p-adic automorphic eigenforms for SL(2) using endoscopy and the geometry of eigenvarieties. These forms seem to account for some non-automorphic members of classical global…

For two distinct primes p and l, we investigate the Z_l-cohomology of the Lubin-Tate towers of a p-adic field. We prove that it realizes some version of Langlands and Jacquet-Langlands correspondences for flat families of irreducible…

In this paper, we study the monodromy of Appell hypergeometric partial differential equations, which lead us to find four derivatives which are associated to the group GL(3). Our four derivatives have the remarkable properties. We find that…

Let $K$ be a totally real field and $\pi$ be a regular algebraic polarized cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb A_K)$. Let $\{\rho_{\pi,\lambda}:\mathrm{Gal}_K\to\mathrm{GL}_n(\overline E_\lambda)\}_\lambda$ be the…

For a local field $F$ and an Artinian local coefficient ring $\Lambda$ with the same positive residue characteristic $p$ we define, for any $e\in{\mathbb N}$, a category ${\mathfrak C}^{(e)}(\Lambda)$ of ${\rm GL}_2(F)$-equivariant…

Two outer automorphisms of infinite-dimensional representations of $sl(2)$ algebra are considered. The similar constructions for the loop algebras and yangians are presented. The corresponding linear and quadratic $R$-brackets include the…

We prove an integrality result for the value at s=1 of the adjoint L-function associated to a cohomological cuspidal automorphic representation on GL(n) over any number field. We then show that primes (outside an exceptional set) dividing…

Let V be the space of 2x2x2 complex hypermatrices, endowed with the natural group action of GL=GL(2,C)^3. The category of GL-equivariant coherent D-modules on V is equivalent to the category of representations of a quiver with relations. In…

Let $F/F_0$ be a quadratic extension of non-Archimedean locally compact fields with residual characteristic $p\neq2$, and $\ell$ be a prime number different from $p$. We classify those $\ell$-modular cuspidal irreducible representations of…

We strengthen the compatibility between local and global Langlands correspondences for GL_{n} when n is even and l=p. Let L be a CM field and \Pi\ a cuspidal automorphic representation of GL_{n}(\mathbb{A}_{L}) which is conjugate self-dual…

In this paper, we prove that a $\mathrm{GL}(2n)$-eigenvariety is \'etale over the (pure) weight space at non-critical Shalika points, and construct multi-variable $p$-adic $L$-functions varying over the resulting Shalika components. Our…

Let G be a unitary group over the rationals, associated to a CM-field F with totally real part F^+, with signature (1,1) at all the archimedean places of F^+. Under certain hypotheses on F^+, we show that Jacquet-Langlands correspondences…

In this paper, we study top Fourier coefficients of certain automorphic representations of $\mathrm{GL}_n(\mathbb{A})$. In particular, we prove a conjecture of Jiang on top Fourier coefficients of isobaric automorphic representations of…

In [HJLLZ24], we proposed a new conjecture on the structure of the unitary dual of connected reductive groups over non-Archimedean local fields of characteristic zero based on their Arthur representations and verified it for all the known…

We investigate the representations of the exotic conformal Galilei algebra. This is done by explicitly constructing all singular vectors within the Verma modules, and then deducing irreducibility of the associated highest weight quotient…

Let G be a connected, real, semisimple Lie group contained in its complexification G_C, and let K be a maximal compact subgroup of G. We construct a K_C-G double coset domain in G_C, and we show that the action of G on the K-finite vectors…

In this paper we investigate arithmetic properties of automorphic forms on the group G' = GL_m/D, for a central division-algebra D over an arbitrary number field F. The results of this article are generalizations of results in the split…

Let G be a unitary group over a totally real field, and X a Shimura variety for G. For certain primes p of good reduction for X, we construct cycles on the characteristic p fiber of X. These cycles are defined as the loci on which the…

We give a proof of the existence of Asai, exterior square, and symmetric square local $L$-functions, $\gamma$-factors and root numbers in characteristic $p$, including the case of $p = 2$. Our study is made possible by developing the…

We show that the local exterior square L-functions of GL_n constructed via the theory of integral representations by Jacquet and Shalika coincide with those constructed by the Langlands-Shahidi method for square integrable representations…