Related papers: A discrepancy result for Hilbert modular forms

In this paper we establish a new Fundamental Lemma for Hecke correspondences. Let $F$ be a local field containing the cube roots of unity. We exhibit an algebra isomorphism of the spherical Hecke algebra of $PGL_3(F)$ and the spherical…

The minimal representation $\pi$ of the indefinite orthogonal group $O(m+1,2)$ is realized on the Hilbert space of square integrable functions on $\mathbb R^m$ with respect to the measure $|x|^{-1} dx_1... dx_m$. This article gives an…

Let $F$ be a totally real number field, and $g,f,h$ be Hilbert modular forms over $F$ that are Hecke eigenforms satisfying $g=f\cdot h$. We characterize such product identities among all real quadratic fields of narrow class number one,…

The aim of this paper is twofold. First, we introduce a new method for evaluating the multiplicity of a given discrete series in the space of level $1$ automorphic forms of a split classical group $G$ over $\mathbb{Z}$, and provide…

Consider a Hilbert space obtained as the completion of the polynomials C[z} in m-variables for which the mnonomials are orthogonal. If the commuting weighted shifts defined by the coordinate functions are essentially normal, then the same…

Starting from a Hilbert cusp form of weight $2\kap$, we will construct a Hilbert-Siegel cusp form of weight $\kap+\frac{m}{2}$ and degree $m$ and its transfer to inner forms of symplectic groups.

For a semibounded sesquilinear form ${\mathfrak t}$ in a Hilbert space ${\mathfrak H}$ there exists a representing map $Q$ from ${\mathfrak H}$ to another Hilbert space ${\mathfrak K}$, such that ${\mathfrak t}[\varphi, \psi]-c(\varphi,…

We prove multiplicity one for vector valued holomorphic Siegel modular forms of weights greater or equal to 3 and the full Siegel modular group and give a trace formula for the action of the Hecke operators T(p) in the regular cases.

In this paper we give an example of a noncongruence subgroup whose three-dimensional space of cusp forms of weight 3 has the following properties. For each of the four residue classes of odd primes modulo 8 there is a basis whose Fourier…

We give a general formula of the bias of root numbers for Hilbert modular newforms of cubic level. Explicit calculation is given when the base field is $\mathbb{Q}, \mathbb{Q}(\sqrt{2}), \mathbb{Q}(\sqrt{5})$ and the level is the cube of…

In 1975 prof. Don Zagier derived a preliminary formula for the trace of the Hecke operators acting on the space of cusp forms (\cite{5}, \cite{6}). Actually, it is an expression in terms of an integral over a fundamental domain of…

For each complex number $\nu$, an associative symplectic reflection algebra $\mathcal H:= H_{1,\nu}(I_2(2m+1))$, based on the group generated by root system $I_2(2m+1)$, has an $m$-dimensional space of traces and an $(m+1)$-dimensional…

Let K be a real algebraic number field and let P be the set of Pisot numbers generating K. We show that the elements of P-P are the algebraic integers of K whose images under the action of all embeddings of K into C, other than the identity…

We investigate some key analytic properties of Fourier coefficients and Hecke eigenvalues attached to scalar-valued Siegel cusp forms $F$ of degree 2, weight $k$ and level $N$. First, assuming that $F$ is a Hecke eigenform that is not of…

Let H be a finite-dimensional unimodular pivotal quasi-Hopf algebra over a field k, and let H-mod be the pivotal tensor category of finite-dimensional H-modules. We give a bijection between left (resp. right) modified traces on the tensor…

Let k and n be positive even integers. For a cuspidal Hecke eigenform g in the Kohnen plus subspace of weight k-n/2+1/2 and level 4, let I(g) be the Duke-Imamoglu-Ikeda lift of g in the space of cusp forms of weight k for Sp(n,Z), and f the…

We develop an explicit theory of formal modular forms over arbitrary number fields $K$, as functions of modular points. We define modular points for $\Gamma_0({\mathfrak n})$ and $\Gamma_1({\mathfrak n})$, where the level ${\mathfrak n}$ is…

We generalise the result of Berger and Shaw the trace formula for Hardy Hilbert space to a larger class of rotation invariant Hilbert function spaces on the unit disk. We also demonstrate many meaningful examples of these Hilbert spaces by…

Ahlgren and Samart relate three cusp forms with complex multiplication to certain weakly holomorphic modular forms using $p$-adic bounds related to their Fourier coefficients. In these three examples, their result strengthens a theorem of…

Let G be a reductive algebraic group over Q, and suppose that Gamma is an arithmetic subgroup of G(R) defined by congruence conditions. A basic problem in arithmetic is to determine the multiplicities of discrete series representations in…