Related papers: A Converse Theorem for Metaplectic Eisenstein Seri…

We prove that in either the convergent or overconvergent setting, an absolutely irreducible $F$-isocrystal on the absolute product of two or more smooth schemes over perfect fields of characteristic $p$, further equipped with actions of the…

In this paper we obtain a complete description of images and poles of degenerate Eisenstein series attached to maximal parabolic subgroups of $Sp_4(\mathbb A)$, where $\mathbb A$ is the ring of adeles of $\mathbb Q$.

A simple necessary and sufficient condition is given for an absolutely convergent Dirichlet series with imaginary exponents and only real zeros to be a finite product of sines. The proof is based on Meyer's theorem on quasicrystals.

A generalization of the usual ideles group is proposed, namely, we construct certain adelic complexes for sheaves of $K$-groups on schemes. More generally, such complexes are defined for any abelian sheaf on a scheme. We focus on the case…

We give a new presentation of the Drinfeld double of the elliptic Hall algebra introduced in a previous work with I. Burban. This presentation is similar in spirit to Drinfeld's `new realization' of quantum affine algebras. This answers, in…

In this paper, we give a construction of certain induced complementary series of the universal covering of the symplectic groups. There are abundant such representations besides those of linear groups. We achieve this by applying invariant…

Let $S\subset \text{SL}_2(\mathbb Z)\times \text{SL}_2(\mathbb Z)$ or $\text{SL}_2(\mathbb Z)\ltimes \mathbb Z^2$ be finite symmetric and assume $S$ generates a group $G$ which is a Zariski-dense subgroup $\text{SL}_2(\mathbb Z)\times…

Using Cohen's classification of symplectic reflection groups, we prove that the parabolic subgroups, that is, stabilizer subgroups, of a finite symplectic reflection group are themselves symplectic reflection groups. This is the symplectic…

Classical integral representation of the Mellin type kernel in terms of the Laplace integral gives an idea to construct a new class of non-convolution (index) transforms. Particular examples give the Kontorovich-Lebedev-like transformation…

We review the teory of the pseudo-iperbolic functions on the basis of an algebraic point of view which employs the Eisenstein group. We frame the teory within the general context of the number decomposition and discuss the importance of…

The generalized Parseval equality for the Mellin transform is employed to prove the inversion theorem in L_2 with the respective inverse operator related to the Hartley transform on the nonnegative half-axis (the half-Hartley transform).…

Notable results on the special values of $L$-functions of Siegel modular forms were obtained by J. Sturm in the case when the degree $n$ is even and the weight $k$ is an integer. In this paper we extend this method to half-integer weights…

We show that whenever a closed symplectic manifold admits a Hamiltonian diffeomorphism with finitely many simple periodic orbits, the manifold has a spherical homology class of degree two with positive symplectic area and positive integral…

We study the $L^2$ norm of the Eisenstein series $E(z,1/2+iT)$ restricted to a segment of a geodesic connecting infinity and an arbitrary real. We conjecture that on slightly thickened geodesics of this form, the Eisenstein series satifies…

Let ${\bf A}$ be the ring of adeles of a number field $F$. Given a self-dual irreducible, automorphic, cuspidal representation $\tau$ of $\GL_n(\BA)$, with trivial central characters, we construct its full inverse image under the weak…

In this note, we study the infinitesimal forms of Deligne cycle class maps. As an application, we prove that the infinitesimal form of a conjecture by Beilinson is true.

Let $F(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$ be a Dirichlet series in the axiomatically defined class ${\mathfrak A}^{\#}$ . The class ${\mathfrak A}^{\#}$ is known to contain the extended Selberg class ${\mathcal S}^{\#}$, as well as all…

The Shapley-Folkman theorem is a statement about the Minkowski sum of (non-convex) sets, expressing the closeness of the Minkowski sum to convexity in a quantitative manner. This paper establishes similar theorems for integrally convex…

Let $G$ be the $ax+b$ group. There are essentially two irreducible infinite dimensional unitary representations of $G$, $(\mu, L^2(\mathbb R^+))$ and $(\mu^*, L^2(\mathbb R^+))$. In this paper, we give various characterizations about smooth…

We prove the equality of several $\tau$-recurrent sequences, which were first considered by Pellarin, and which have close connections to Drinfeld vectorial modular forms. Our result has several consequences: an $A$-expansion for the…