Determination of holomorphic modular forms by primitive Fourier coefficients

Abstract

We prove that Siegel modular forms of degree greater than one, integral weight and level N , with respect to a Dirichlet character χ of conductor f χ are uniquely determined by their Fourier coefficients indexed by matrices whose contents run over all divisors of N /f χ . The cases of other major types of holomorphic modular forms are included.

Distinguishing Hermitian cusp forms of degree 2 by a certain subset of all Fourier coefficients

“…As a first result in this direction, it was shown by D. Zagier [22] that the Siegel cusp forms of degree 2 are determined by primitive Fourier coefficients. This has been generalized to Siegel and Hermitian cusp forms with levels and of higher degrees by S. Yamana [21]. Similar results along this line, essentially distinguishing Siegel Hecke eigenforms of degree 2 by the so-called 'radial' Fourier coefficients (i.e., by certain subset of matrices of the form mT with T half-integral, m ≥ 1), has been obtained in Breulmann-Kohnen [2], Scharlau-Walling [18], Katsurada [10].…”

On the lifting of elliptic cusp forms to cusp forms on quaternionic unitary groups

“…Consider a function G − I(φ), which belongs to S κ (Γ ϕ ) by Corollary 8.2. Since it has vanishing primitive Fourier coefficients by (ii), Theorem 3 of [36] concludes that…”

On the Fourier Coefficients of Siegel modular Forms

“…The statement below seems to be new only when we start from a vector-valued function. In the scalar-valued case variants have appeared in works of Eichler-Zagier [15], Yamana [41], Ibukiyama [18] and others.…”

“…The results in this part should hold more generally over the classical tube domains I-IV (as in e.g. [41]), but we do not pursue it here mainly because such a treatment may obscure the technical points of the paper. We may return to this point in a future work.…”