Related papers: Automorphic forms for PGL(3) over elliptic functio…

The focus of this article is to develop computationally efficient mathematical morphology operators on hypergraphs. To this aim we consider lattice structures on hypergraphs on which we build morphological operators. We develop a pair of…

This paper surveys the connection between the elliptic curve E_D: x^3 + y^3 = D and a certain metaplectic form on the cubic cover of GL(3) which has the property that its m,n^{th} Whittaker--Fourier coefficient is essentially the L--series…

Let $f $ be a holomorphic Hecke eigenforms or a Hecke-Maass cusp form for the full modular group $ SL(2, \mathbb{Z})$. In this paper we shall use circle method to prove the Weyl exponent for $GL(2)$ $L$-functions. We shall prove that \[ L…

Let $G$ be a finite group, $H \le G$ a subgroup, $R$ a commutative ring, $A$ an $R$-algebra, and $\alpha$ an action of $G$ on $A$ by $R$-algebra automorphisms. We study the associated \emph{skew Hecke algebra}…

A theorem of Kontsevich relates the homology of certain infinite dimensional Lie algebras to graph homology. We formulate this theorem using the language of reversible operads and mated species. All ideas are explained using a pictorial…

We give a derivative version of the relative trace formula on PGL(2) studied in our previous work, and obtain a formula of an average of central values (derivatives) of automorphic $L$-functions for Hilbert cusp forms. As an application, we…

We prove a mean value theorem for the traces of Hecke operators acting on cusp forms of GL(n) over imaginary quadratic number fields together with an upper bound for the error term depending explicitly on the Hecke operator.

The present paper is devoted to the problem about the reduction of hyperelliptic functions of genus 3. Our research was motivated by applications to the theory of equations and dynamical systems integrable in hyperelliptic functions. In…

We develop a Hamiltonian theory for 2D soliton equations. In particular, we identify the spaces of doubly periodic operators on which a full hierarchy of commuting flows can be introduced, and show that these flows are Hamiltonian with…

We investigate a notion of "higher modularity" for elliptic curves over function fields. Given such an elliptic curve $E$ and an integer $r\geq 1$, we say that $E$ is $r$-modular when there is an algebraic correspondence between a stack of…

We provide the explicit formula for orbital integrals associated with elliptic regular semisimple elements in $\mathrm{GL}_n(F) \cap \mathrm{M}_n(\mathfrak{o})$ and associated with arbitrary elements of the spherical Hecke algebra of…

We carry out some algebraic and analytic properties of a new class of orthogonal polyanalytic polynomials, including their operational formulas, recurrence relations, generating functions, integral representations and different…

In this work, we study multiplicity-free induced representations of finite groups. We analyze in great detail the structure of the Hecke algebra corresponding to the commutant of an induced representation and then specialize to the…

We classify the automorphism groups of del Pezzo surfaces of degrees one and two over an algebraically closed field of characteristic two. This finishes the classification of automorphism groups of del Pezzo surfaces in all characteristics.

Let $H$ be a complex Hilbert space. Consider the ortho-Grassmann graph $\Gamma^{\perp}_{k}(H)$ whose vertices are $k$-dimensional subspaces of $H$ (projections of rank $k$) and two subspaces are connected by an edge in this graph if they…

For all odd primes N up to 500000, we compute the action of the Hecke operator T_2 on the space S_2(Gamma_0(N), Q) and determine whether or not the reduction mod 2 (with respect to a suitable basis) has 0 and/or 1 as eigenvalues. We then…

We compute the $L$-functions of a large class of algebraic curves, and verify the expected functional equation numerically. Our computations are based on our previous results on stable reduction to calculate the local $L$-factor and the…

In this paper, we prove uniform bounds for $\rm GL (3)\times GL(2)$ $L$-functions in the $\rm GL(2)$ spectral aspect and the $t$ aspect by a delta method. More precisely, let $\phi$ be a Hecke--Maass cusp form for $\rm SL(3,\mathbb{Z})$ and…

We consider the genus of $20$ classes of unimodular Hermitian lattices of rank $12$ over the Eisenstein integers. This set is the domain for a certain space of algebraic modular forms. We find a basis of Hecke eigenforms, and guess global…

Assuming the Riemann hypothesis for $L$-functions attached to primitive Dirichlet characters, modular cusp forms, and their tensor products and symmetric squares, we write down explicit finite sets of Hecke operators that span the Hecke…