Simultaneous behaviour of the Fourier coefficients of two Hilbert modular cusp forms

Archiv der Mathematik - In this article, we study simultaneous sign changes of the Fourier coefficients of two Hilbert cusp forms of different non-parallel weights. We also study simultaneous...     

On the signs of Fourier coefficients of Hilbert cusp forms

The Ramanujan Journal - We prove that given any $$\epsilon > 0$$ and a primitive adelic Hilbert cusp form f of weight $$k=(k_1,k_2,\ldots ,k_n) \in (2 {\mathbb {Z}})^n$$ and full level,...     

On the number of Fourier coefficients that determine a Hilbert modular form

We estimate the number of Fourier coefficients that determine a Hilbert modular cusp form of arbitrary weight and level. The method is spectral (Rayleigh quotient) and avoids the use of the maximum principle.

     

Nonvanishing of Fourier coefficients of modular forms

Let be a cusp form with integer weight that is not a linear combination of forms with complex multiplication. For , let

Improving on work of Balog, Ono, and Serre we show that for almost all , where is any good function (e.g. such as ) monotonically tending to infinity with . Using a result of Fouvry and Iwaniec, if is a weight 2 cusp form for an elliptic curve without complex multiplication, then we show for all that . We also obtain conditional results depending on the Generalized Riemann Hypothesis and the Lang-Trotter Conjecture.

     

Two-divisibility of the coefficients of certain weakly holomorphic modular forms

We study a canonical basis for spaces of weakly holomorphic modular forms of weights 12, 16, 18, 20, 22, and 26 on the full modular group. We prove a relation between the Fourier coefficients of modular forms in this canonical basis and a generalized Ramanujan τ-function, and use this to prove that these Fourier coefficients are often highly divisible by 2.     

Determination of holomorphic modular forms by primitive Fourier coefficients

We prove that Siegel modular forms of degree greater than one, integral weight and level N, with respect to a Dirichlet character of conductor are uniquely determined by their Fourier coefficients indexed by matrices whose contents run over all divisors of . The cases of other major types of holomorphic modular forms are included. The author is supported by the Grant-in-Aid for JSPS fellows.