Related papers: Characterizing Hilbert modular cusp forms by coeff…

We study the question of Eulerianity (factorizability) for Fourier coefficients of automorphic forms, and we prove a general transfer theorem that allows one to deduce the Eulerianity of certain coefficients from that of another…

Let f be a modular form with complex multiplication. If f has critical slope, then Coleman's classicality theorem implies that there is a p-adic overconvergent generalized Hecke eigenform with the same Hecke eigenvalues as f. We give a…

Kaneko's formula expresses the Fourier coefficients of the elliptic modular $j$-function as finite sums of singular moduli. First published as a short article in 1996, it was presented as a consequence of Zagier's work inspired by Borcherds…

It is believed that Dirichlet series with a functional equation and Euler product of a particular form are associated to holomorphic newforms on a Hecke congruence group. We perform computer algebra experiments which find that in certain…

In 1975, Cohen constructed a kind of one-variable modular forms of half-integral weight, says $r+(1/2),$ whose $n$-th Fourier coefficient $H(n)$ only occurs when $(-1)^r n$ is congruent to 0 or 1 modulo 4. The space of modular forms whose…

We give density results for automorphic representations of Hilbert modular groups. In particular, we show that there are infinitely many automorphic representations that have a prescribed discrete series factor at some (but not all) real…

We prove that vector-valued Siegel cusp forms for $\Gamma_0^n(N)$ with certain nebentypus are determined by their fundamental Fourier coefficients with discriminants coprime to the level $N$, assuming $N$ is odd and square-free. In the case…

In this short note, we treat an unbalanced shifted convolution sum of Fourier coefficients of cusp forms by a rather simple argument. Our result improves previous results established by more advanced approaches.

Nesterenko proved, among other results, the algebraic independence over $\QQ$ of the numbers $\pi,e^{\pi},\Gamma(1/4)$. A very important feature of his proof is a multiplicity estimate for quasi-modular forms associated to $\SL_2(\ZZ)$…

We prove that Hecke eigenvalues for any Hilbert and Siegel modular forms are algebraic integers. Our method does not rely on cohomologicality nor Galois representations. We apply the integrality of Hecke eigenvalues for Hilbert modular…

We establish (Theorem 3.6) polynomial-growth estimates for the Fourier coefficients of holomorphic logarithmic vector-valued modular forms.

An algebraic classification is given for spaces of holomorphic vector-valued modular forms of arbitrary real weight and multiplier system, associated to irreducible, T-unitarizable representations of the full modular group, of dimension…

Extending the approach of Iwaniec and Duke, we present strong uniform bounds for Fourier coefficients of half-integral weight cusp forms of level $N$. As an application, we consider a Waring-type problem with sums of mixed powers.

We consider a variant of a question of N. Koblitz. For an elliptic curve $E/\Q$ which is not $\Q$-isogenous to an elliptic curve with torsion, Koblitz has conjectured that there exists infinitely many primes $p$ such that…

Let $A$ be a subset of a finite abelian group such that $A$ has a small difference set $A-A$ and the density of $A$ is small. We prove that, counter--intuitively, the smallness (in terms of $|A-A|$) of the Fourier coefficients of $A$…

In this paper, we introduce new general frameworks for estimating the maximal dimension of Hilbert cubes contained in finite truncations of arbitrary sets. As applications, we investigate Hilbert cubes in a range of arithmetic sets,…

We study the behavior of the shifted convolution sum involving fourth power of the Fourier coefficients of holomorphic cusp forms with a weight function to be the $k$-full kernel function for any fixed integer $k\geq2$.

To study statistical properties of modular forms, including for instance Sato-Tate like problems, it is essential to have a large number of Fourier coefficients. In this article, we exhibit three bases for the space of modular forms of any…

In this paper, we prove some divisibility results for the Fourier coefficients of reduced modular forms of sign vectors. More precisely, we generalize a divisibility result of Siegel on constant terms when the weight is non-positive, which…

For a rational prime $p \geq 3$ we consider $p$-ordinary, Hilbert modular newforms $f$ of weight $k\geq 2$ with associated $p$-adic Galois representations $\rho_f$ and $\mod{p^n}$ reductions $\rho_{f,n}$. Under suitable hypotheses on the…