Related papers: Algebraic twists of GL(2) automorphic forms

Under a certain condition, we find the explicit formulas for the trace functions of certain intertwining operators among gl(n)-modules, introduced by Etingof in connection with the solutions of the Calogero-Sutherland model. If n=2, the…

We prove a sharp bound for the average value of the triple product of modular functions for the Hecke subgroup \Gamma_0(N). Our result is an extension of the main result in {Bernstein&Reznikov-2004} to a fixed cuspidal representation of the…

In the paper, we introduce and calculate difference Fourier transforms on representations of the double affine Hecke algebras in polynomilas, polynomials multiplied by the Gaussian, and various spaces of delta-functions including…

We show that with respect to the q-Plancherel measure on partitions of size n, the irreducible characters of an Hecke algebra $H_q(S_n)$ are concentrated around the normalized trace of $H_q(S_n)$. More precisely, we prove that the…

Let $f$ be a cuspidal eigenform (holomorphic or Maass) on the full modular group $SL(2, \mathbb{Z})$ . Let $\chi$ be a primitive character of modulus $P$. We shall prove the following results: 1. Suppose $P = p^r$, where $p$ is a prime and…

Representations of braid group obtained from rational conformal field theories can be used to obtain explicit representations of Temperley-Lieb-Jones algebras. The method is described in detail for SU(2)$_k$ Wess - Zumino conformal field…

Let $(\mathbb{T}_f,\mathfrak{m}_f)$ denote the mod $p$ local Hecke algebra attached to a normalised Hecke eigenform $f$, which is a commutative algebra over some finite field $\mathbb{F}_q$ of characteristic $p$ and with residue field…

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…

We establish a relative trace formula on $\mathrm{GL}(n+1)$ weighted by cusp forms on $\mathrm{GL}(n)$ over number fields. The spectral side is a weighted average of Rankin-Selberg $L$-functions for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$…

Let $\pi$ be a Hecke-Maass cusp form for $\rm SL_3(\mathbf{Z})$ and let $g$ be a holomorphic or Maass cusp form for $\rm SL_2(\mathbf{Z})$. Let $\chi$ be a primitive Dirichlet character of modulus $M=M_1M_2$ with $M_i$ prime, $i=1,2$.…

Let $F$ be a number field and $\pi$ an irreducible cuspidal representation of $\mathrm{GL}_{2}(F)\backslash\mathrm{GL}_{2}(\mathbf{A})$ with unitary central character. Then the bound…

We use recently obtained bounds for sums of Kloosterman sums to bound the sum $\sum_{-D\leq d\leq D} \int_{-D}^D |\zeta(1/2+it,\lambda^d)|^4| \sum_{0<|\mu|^2\leq M} A(\mu)\lambda^d((\mu)) |\mu|^{-2it}|^2 {\rm d}t$, where $\lambda^d$ is the…

We solve the sup-norm problem for spherical Hecke-Maass newforms of square-free level for the group GL(2) over a number field, with a power saving over the local geometric bound simultaneously in the eigenvalue and the level aspect. Our…

The trace formula for the density of single-particle levels in the two-dimensional radial power-law potentials, which nicely approximate the radial dependence of the Woods-Saxon potential and quantum spectra in a bound region, was derived…

We define geodesic normal forms for the general series of complex reflection groups G(de,e,n). This requires the elaboration of a combinatorial technique in order to determine minimal word representatives and to compute the length of the…

Consider the affine Hecke algebra $H_l$ corresponding to the group $GL_l$ over a $p$-adic field with the residue field of cardinality $q$. Regard $H_l$ as an associative algebra over the field $C(q)$. Consider the $H_{l+m}$-module $W$…

Given a self-dual cuspidal automorphic representation for GL(2) over a number field, we establish the existence of an infinite number of Hecke eigenvalues that are greater than an explicit positive constant, and an infinite number of Hecke…

Let O^1 be a (cocompact) Fuchsian group, given as the group of units of norm one in a maximal order O in an indefinite quaternion division algebra over Q. Using the (classical) Selberg trace formula, we show that the eigenvalues of the…

Let $\lambda_i (n)$ $i= 1, 2, 3$ denote the normalised Fourier coefficients of holomorphic eigenform or Maass cusp form. In this paper we shall consider the sum: \[ S:= \frac{1}{H}\sum_{h\leq H} V\left( \frac{h}{H}\right)\sum_{n\leq N}…

Using a factorization theorem of Douglas, we prove functional characterizations of trace spaces $H^s(\partial \Omega)$ involving a family of positive self-adjoint operators. Our method is based on the use of a suitable operator by taking…