Related papers: On the Fourier coefficients of 2-dimensional vecto…

H. Aoki showed that any symmetric formal Fourier-Jacobi series for the symplectic group Sp_2(Z) is the Fourier-Jacobi expansion of a holomorphic Siegel modular form. We prove an analogous result for vector valued symmetric formal…

We extend the Euler's totient function (from arithmetic) to any irreducible subfactor planar algebra, using the Mobius function of its biprojection lattice, as Hall did for the finite groups. We prove that if it is nonzero then there is a…

We construct an infinite set of conserved tensor currents of rank $2n$, $n=1,2,\dots$, in the two-dimensional theory of free massive fermions, which are bilinear in the fermionic fields. The one-point functions of these currents on the…

We study Demazure modules which occur in a level $\ell$ irreducible integrable representation of an affine Lie algebra. We also assume that they are stable under the action of the standard maximal parabolic subalgebra of the affine Lie…

Given three irreducible, admissible, infinite dimensional complex representations of GL2(F), with F a local field, the space of trilinear functionals invariant by the group has dimension at most one. When it is one we provide an explicit…

Values of quaternionic modular forms are related to twisted central $L$-values via periods and a theorem of Waldspurger. In particular, certain twisted $L$-values must be non-vanishing for forms with no zeroes. Here we study, theoretically…

In this paper, we explore the modular differential equation $\displaystyle y'' + F(z)y = 0$ on the upper half-plane $\mathbb{H}$, where $F$ is a weight 4 modular form for $\Gamma_0(2)$. Our approach centers on solving the associated…

Extending the method of the paper [FS3] we prove three structure theorems for vector valued modular forms, where two correspond to 4-dimensional cases (two hermitian modular groups, one belonging to the field of Eisenstein numbers, the…

Let g be a complex semisimple Lie algebra, tau a point in the upper half-plane, and h a complex deformation parameter such that the image of h in the elliptic curve E_tau is of infinite order. In this paper, we give an intrinsic definition…

Given a holomorphic newform $f$ of weight $k$ and with rational coefficients, a question of Mazur and van Straten asks if there is an associated Calabi-Yau variety $X$ over ${\mathbb Q}$ of dimension $k-1$ such that the $\ell$-adic Galois…

We prove divisibility results for the Fourier coefficients of canonical basis elements for the spaces of weakly holomorphic modular forms of weight $0$ and levels $6, 10, 12, 18$ with poles only at the cusp at infinity. In addition, we show…

We describe the image of general families of two-dimensional representations over compact semi-local rings. Applying this description to the family carried by the universal Hecke algebra acting on the space of modular forms of level $N$…

We show, for levels of the form $N = p^a q^b N'$ with $N'$ squarefree, that in weights $k \geq 4$ every cusp form $f \in \mathcal{S}_k(N)$ is a linear combination of products of certain Eisenstein series of lower weight. In weight $k=2$ we…

We establish lower bounds for (i) the numbers of positive and negative terms and (ii) the number of sign changes in the sequence of Fourier coefficients at squarefree integers of a half-integral weight modular Hecke eigenform.

A PV number is an algebraic integer $\alpha$ of degree $d \geq 2$ all of whose Galois conjugates other than itself have modulus less than $1$. Erd\"{o}s \cite{erdos} proved that the Fourier transform $\widehat \varphi,$ of a nonzero…

Let $n>1$ be an integer, $\alpha\in{\mathbb C}^n$, $b\in{\mathbb C}$, and $V$ a $\mathfrak{gl}_n$-module. We define a class of weight modules $F^\alpha_{b}(V)$ over $\sl_{n+1}$ using the restriction of modules of tensor fields over the Lie…

Let k be a positive integer divisible by 4, l>k a prime, and f an elliptic cuspidal eigenform of weight k-1, level 4, and non-trivial character. Let \rho_f be the l-adic Galois representation attached to f. In this paper we provide evidence…

Determining Fourier coefficients of modular forms of a finite index noncongruence subgroups of the modular group is still a non-trivial task. In this brief note we describe a new algorithm to reliably calculate an approximation for a…

We classify the quasi-finite irreducible highest weight modules over the infinite rank Lie superalgebras $\hgltwo$, $\hC$ and $\hD$, and determine the necessary and sufficient conditions for quasi-finite irreducible highest weight modules…

Rao and Zhao classified the irreducible integrable modules with finite dimensional weight spaces for the untwisted affine superalgebras which are not $\hat{A}(m,n)$ ($m\ne n$) or $\hat{C}(m)$. Here we treat the latter affine superalgebras…